LIBRARY 

CWWVERSITY  OF  CALIFORNIA 
DAVIS 


TH  E  O  R  I  A 

PHILOSOPHISE    NATURALIS 

REDACTA   AD  UNICAM   LEGEM   VIRIUM 
IN   NATURA   EXISTENTIUM, 


A    V     C     T    O 

P^ROGERIO  JOSEPHO  BOSCOVICH 

SOCIETATIS       J  £  S  U, 
NUNC    AB    IPSO    PERPOLITA,    ET    AUCTA, 

Ac    a    plurimis    praeccclcntium    edifionum 
mendis  expurgata. 

EDITIO    VENETA     PR1MA 

IPSO    «fUCTORE    PRJESENTE,    ET    CORRIGENTE. 


V  E  N  E  T  I  I  S, 

MDCCLXIII. 

»  ^>     M  «?>     «4  «ft     •*»<?>     «4  V0>     «»  <«k     «»  v«»     *»   «^»    « 

Ex    TTPOCRAPHIA    REMOWDINIANA. 
r^IO^Z/M    PZllMII.yi/,    ac    P  R  IV  1LE  G  1  0, 


A  THEORY  OF 

NATURAL  PHILOSOPHY 


PUT   FORWARD  AND  EXPLAINED  BY 

ROGER  JOSEPH  BOSCOVICH,  S.J. 


LATIN— ENGLISH    EDITION 

FROM  THE  TEXT  OF  THE 

FIRST  VENETIAN  EDITION 

PUBLISHED    UNDER  THE  PERSONAL 

SUPERINTENDENCE  OF  THE   AUTHOR 

IN   1763 


WITH 

A    SHORT    LIFE    OF   BOSCOVICH 


CHICAGO  LONDON 

OPEN    COURT    PUBLISHING    COMPANY 

1922 


LIBRARY 

UNIVERSITY  OF  CALIFORNIA 
DAVIS 


PRINTED   IN    GREAT    BRITAIN 

BY 

BUTLER  &  TANNER,  FROME,  ENGLAND 


Copyright 


PREFACE 

HE  text  presented  in  this  volume  is  that  of  the  Venetian  edition  of  1763. 
This  edition  was  chosen  in  preference  to  the  first  edition  of  1758,  published 
at  Vienna,  because,  as  stated  on  the  title-page,  it  was  the  first  edition  (revised 
and  enlarged)  issued  under  the  personal  superintendence  of  the  author. 

In  the  English  translation,  an  endeavour  has  been  made  to  adhere  as 
closely  as  possible  to  a  literal  rendering  of  the  Latin  ;  except  that  the  some- 
what lengthy  and  complicated  sentences  have  been  broken  up.  This  has 
made  necessary  slight  changes  of  meaning  in  several  of  the  connecting  words.  This  will  be 
noted  especially  with  regard  to  the  word  "  adeoque  ",  which  Boscovich  uses  with  a  variety 
of  shades  of  meaning,  from  "  indeed  ", "  also  "  or  "  further  ",  through  "  thus  ",  to  a  decided 
"  therefore  ",  which  would  have  been  more  correctly  rendered  by  "  ideoque  ".  There  is 
only  one  phrase  in  English  that  can  also  take  these  various  shades  of  meaning,  viz.,  "  and  so  "  ; 
and  this  phrase,  for  the  use  of  which  there  is  some  justification  in  the  word  "  adeo  "  itself, 
has  been  usually  employed. 

The  punctuation  of  the  Latin  is  that  of  the  author.  It  is  often  misleading  to  a  modern 
reader  and  even  irrational ;  but  to  have  recast  it  would  have  been  an  onerous  task  and 
something  characteristic  of  the  author  and  his  century  would  have  been  lost. 

My  translation  has  had  the  advantage  of  a  revision  by  Mr.  A.  O.  Prickard,  M.A.,  Fellow 
of  New  College,  Oxford,  whose  task  has  been  very  onerous,  for  he  has  had  to  watch  not 
only  for  flaws  in  the  translation,  but  also  for  misprints  in  the  Latin.  These  were  necessarily 
many  ;  in  the  first  place,  there  was  only  one  original  copy  available,  kindly  loaned  to  me  by 
the  authorities  of  the  Cambridge  University  Library ;  and,  as  this  copy  could  not  leave 
my  charge,  a  type-script  had  to  be  prepared  from  which  the  compositor  worked,  thus  doub- 
ling the  chance  of  error.  Secondly,  there  were  a  large  number  of  misprints,  and  even 
omissions  of  important  words,  in  the  original  itself  ;  for  this  no  discredit  can  be  assigned  to 
Boscovich  ;  for,  in  the  printer's  preface,  we  read  that  four  presses  were  working  at  the 
same  time  in  order  to  take  advantage  of  the  author's  temporary  presence  in  Venice.  Further, 
owing  to  almost  insurmountable  difficulties,  there  have  been  many  delays  in  the  production 
of  the  present  edition,  causing  breaks  of  continuity  in  the  work  of  the  translator  and  reviser  ; 
which  have  not  conduced  to  success.  We  trust,  however,  that  no  really  serious  faults  remain. 
The  short  life  of  Boscovich,  which  follows  next  after  this  preface,  has  been  written  by 
Dr.  Branislav  Petronievic,  Professor  of  Philosophy  at  the  University  of  Belgrade.  It  is  to 
be  regretted  that,  owing  to  want  of  space  requiring  the  omission  of  several  addenda  to  the 
text  of  the  Theoria  itself,  a  large  amount  of  interesting  material  collected  by  Professor 
Petronievic  has  had  to  be  left  out. 

The  financial  support  necessary  for  the  production  of  such  a  costly  edition  as  the  present 
has  been  met  mainly  by  the  Government  of  the  Kingdom  of  Serbs,  Croats  and  Slovenes ; 
and  the  subsidiary  expenses  by  some  Jugo-Slavs  interested  in  the  publication. 

After  the  "  Life,"  there  follows  an  "  Introduction,"  in  which  I  have  discussed  the  ideas 
of  Boscovich,  as  far  as  they  may  be  gathered  from  the  text  of  the  Tbeoria  alone ;  this 
also  has  been  cut  down,  those  parts  which  are  clearly  presented  to  the  reader  in  Boscovich's 
own  Synopsis  having  been  omitted.  It  is  a  matter  of  profound  regret  to  everyone  that  this 
discussion  comes  from  my  pen  instead  of,  as  was  originally  arranged,  from  that  of  the  late 
Philip  E.  P.  Jourdain,  the  well-known  mathematical  logician  ;  whose  untimely  death  threw 
into  my  far  less  capable  hands  the  responsible  duties  of  editorship. 

I  desire  to  thank  the  authorities  of  the  Cambridge  University  Library,  who  time  after 
time  over  a  period  of  five  years  have  forwarded  to  me  the  original  text  of  this  work  of 
Boscovich.  Great  credit  is  also  due  to  the  staff  of  Messrs.  Butler  &  Tanner,  Frome, 
for  the  care  and  skill  with  which  they  have  carried  out  their  share  of  the  work ;  and 
my  special  thanks  for  the  unfailing  painstaking  courtesy  accorded  to  my  demands,  which  were 
frequently  not  in  agreement  with  trade  custom. 

J.  M.  CHILD. 
MANCHESTER  UNIVERSITY, 

December,  1921. 


LIFE  OF   ROGER  JOSEPH    BOSCOVICH 

By  BRANISLAV    PETRONIEVIC' 

]HE  Slav  world,  being  still  in  its  infancy,  has,  despite  a  considerable  number 
of  scientific  men,  been  unable  to  contribute  as  largely  to  general  science 
as  the  other  great  European  nations.  It  has,  nevertheless,  demonstrated 
its  capacity  of  producing  scientific  works  of  the  highest  value.  Above 
all,  as  I  have  elsewhere  indicated,"  it  possesses  Copernicus,  Lobachevski, 
Mendeljev,  and  Boscovich. 

In  the  following  article,  I  propose  to  describe  briefly  the  life  of  the 
Jugo-Slav,  Boscovich,  whose  principal  work  is  here  published  for  the  sixth  time  ;  the  first 
edition  having  appeared  in  1758,  and  others  in  1759,  1763,  1764,  and  1765.  The  present 
text  is  from  the  edition  of  1763,  the  first  Venetian  edition,  revised  and  enlarged. 

On  his  father's  side,  the  family  of  Boscovich  is  of  purely  Serbian  origin,  his  grandfather, 
Bosko,  having  been  an  orthodox  Serbian  peasant  of  the  village  of  Orakova  in  Herzegovina. 
His  father,  Nikola,  was  first  a  merchant  in  Novi  Pazar  (Old  Serbia),  but  later  settled  in 
Dubrovnik  (Ragusa,  the  famous  republic  in  Southern  Dalmatia),  whither  his  father,  Bosko, 
soon  followed  him,  and  where  Nikola  became  a  Roman  Catholic.  Pavica,  Boscovich's 
mother,  belonged  to  the  Italian  family  of  Betere,  which  for  a  century  had  been  established 
in  Dubrovnik  and  had  become  Slavonicized — Bara  Betere,  Pavica's  father,  having  been  a 
poet  of  some  reputation  in  Ragusa. 

Roger  Joseph  Boscovich  (Rudjer  Josif  Boskovic',  in  Serbo-Croatian)  was  born  at  Ragusa 
on  September  i8th,  1711,  and  was  one  of  the  younger  members  of  a  large  family.  He 
received  his  primary  and  secondary  education  at  the  Jesuit  College  of  his  native  town  ; 
in  1725  he  became  a  member  of  the  Jesuit  order  and  was  sent  to  Rome,  where  from  1728 
to  1733  he  studied  philosophy,  physics  and  mathematics  in  the  Collegium  Romanum. 
From  1733  to  1738  he  taught  rhetoric  and  grammar  in  various  Jesuit  schools ;  he  became 
Professor  of  mathematics  in  the  Collegium  Romanum,  continuing  at  the  same  time  his 
studies  in  theology,  until  in  1744  he  became  a  priest  and  a  member  of  his  order. 

In  1736,  Boscovich  began  his  literary  activity  with  the  first  fragment,  "  De  Maculis 
Solaribus,"  of  a  scientific  poem,  "  De  Solis  ac  Lunse  Defectibus  "  ;  and  almost  every 
succeeding  year  he  published  at  least  one  treatise  upon  some  scientific  or  philosophic  problem. 
His  reputation  as  a  mathematician  was  already  established  when  he  was  commissioned  by 
Pope  Benedict  XIV  to  examine  with  two  other  mathematicians  the  causes  of  the  weakness 
in  the  cupola  of  St.  Peter's  at  Rome.  Shortly  after,  the  same  Pope  commissioned  him  to 
consider  various  other  problems,  such  as  the  drainage  of  the  Pontine  marshes,  the  regulariza- 
tion  of  the  Tiber,  and  so  on.  In  1756,  he  was  sent  by  the  republic  of  Lucca  to  Vienna 
as  arbiter  in  a  dispute  between  Lucca  and  Tuscany.  During  this  stay  in  Vienna,  Boscovich 
was  commanded  by  the  Empress  Maria  Theresa  to  examine  the  building  of  the  Imperial 
Library  at  Vienna  and  the  cupola  of  the  cathedral  at  Milan.  But  this  stay  in  Vienna, 
which  lasted  until  1758,  had  still  more  important  consequences ;  for  Boscovich  found 
time  there  to  finish  his  principal  work,  Theoria  Philosophies  Naturalis  ;  the  publication 
was  entrusted  to  a  Jesuit,  Father  Scherffer,  Boscovich  having  to  leave  Vienna,  and  the 
first  edition  appeared  in  1758,  followed  by  a  second  edition  in  the  following  year.  With 
both  of  these  editions,  Boscovich  was  to  some  extent  dissatisfied  (see  the  remarks  made 
by  the  printer  who  carried  out  the  third  edition  at  Venice,  given  in  this  volume  on  page  3) ; 
so  a  third  edition  was  issued  at  Venice,  revised,  enlarged  and  rearranged  under  the  author's 
personal  superintendence  in  1763.  The  revision  was  so  extensive  that  as  the  printer 
remarks,  "  it  ought  to  be  considered  in  some  measure  as  a  first  and  original  edition  "  ; 
and  as  such  it  has  been  taken  as  the  basis  of  the  translation  now  published.  The  fourth 
and  fifth  editions  followed  in  1764  and  1765. 

One  of  the  most  important  tasks  which  Boscovich  was  commissioned  to  undertake 
was  that  of  measuring  an  arc  of  the  meridian  in  the  Papal  States.  Boscovich  had  designed 
to  take  part  in  a  Portuguese  expedition  to  Brazil  on  a  similar  errand  ;  but  he  was  per- 

"  Slav  Achievements  in  Advanced  Science,  London,  1917. 

vii 


viii  A  THEORY  OF  NATURAL  PHILOSOPHY 

suaded  by  Pope  Benedict  XIV,  in  1750,  to  conduct,  in  collaboration  with  an  English  Jesuit, 
Christopher  Maire,  the  measurements  in  Italy.  The  results  of  their  work  were  published, 
in  1755,  by  Boscovich,  in  a  treatise,  De  Litter  aria  Expedition^  -per  Pontificiam,  &c.  ;  this 
was  translated  into  French  under  the  title  of  Voyage  astronomique  et  geograpbique  dans 
VEtat  de  VEglise,  in  1770. 

By  the  numerous  scientific  treatises  and  dissertations  which  he  had  published  up  to 
1759,  and  by  his  principal  work,  Boscovich  had  acquired  so  high  a  reputation  in  Italy,  nay 
in  Europe  at  large,  that  the  membership  of  numerous  academies  and  learned  societies  had 
already  been  conferred  upon  him.  In  1760,  Boscovich,  who  hitherto  had  been  bound  to 
Italy  by  his  professorship  at  Rome,  decided  to  leave  that  country.  In  this  year  we  find 
him  at  Paris,  where  he  had  gone  as  the  travelling  companion  of  the  Marquis  Romagnosi. 
Although  in  the  previous  year  the  Jesuit  order  had  been  expelled  from  France,  Boscovich 
had  been  received  on  the  strength  of  his  great  scientific  reputation.  Despite  this,  he  did  not 
feel  easy  in  Paris ;  and  the  same  year  we  find  him  in  London,  on  a  mission  to  vindicate 
the  character  of  his  native  place,  the  suspicions  of  the  British  Government,  that  Ragusa  was 
being  used  by  France  to  fit  out  ships  of  war,  having  been  aroused  ;  this  mission  he  carried 
out  successfully.  In  London  he  was  warmly  welcomed,  and  was  made  a  member  of  the 
Royal  Society.  Here  he  published  his  work,  De  Solis  ac  Lunce  defectibus,  dedicating  it  to 
the  Royal  Society.  Later,  he  was  commissioned  by  the  Royal  Society  to  proceed  to  Cali- 
fornia to  observe  the  transit  of  Venus ;  but,  as  he  was  unwilling  to  go,  the  Society  sent 
him  to  Constantinople  for  the  same  purpose.  He  did  not,  however,  arrive  in  time  to 
make  the  observation  ;  and,  when  he  did  arrive,  he  fell  ill  and  was  forced  to  remain  at 
Constantinople  for  seven  months.  He  left  that  city  in  company  with  the  English  ambas- 
sador, Porter,  and,  after  a  journey  through  Thrace,  Bulgaria,  and  Moldavia,  he  arrived 
finally  at  Warsaw,  in  Poland  ;  here  he  remained  for  a  time  as  the  guest  of  the  family  of 
PoniatowsM.  In  1762,  he  returned  from  Warsaw  to  Rome  by  way  of  Silesia  and  Austria. 
The  first  part  of  this  long  journey  has  been  described  by  Boscovich  himself  in  his  Giornale 
di  un  viaggio  da  Constantinopoli  in  Polonia — the  original  of  which  was  not  published  until 
1784,  although  a  French  translation  had  appeared  in  1772,  and  a  German  translation 
in  1779. 

Shortly  after  his  return  to  Rome,  Boscovich  was  appointed  to  a  chair  at  the  University 
of  Pavia  ;  but  his  stay  there  was  not  of  long  duration.  Already,  in  1764,  the  building 
of  the  observatory  of  Brera  had  been  begun  at  Milan  according  to  the  plans  of  Boscovich  ; 
and  in  1770,  Boscovich  was  appointed  its  director.  Unfortunately,  only  two  years  later 
he  was  deprived  of  office  by  the  Austrian  Government  which,  in  a  controversy  between 
Boscovich  and  another  astronomer  of  the  observatory,  the  Jesuit  Lagrange,  took  the  part 
of  his  opponent.  The  position  of  Boscovich  was  still  further  complicated  by  the  disbanding 
of  his  company  ;  for,  by  the  decree  of  Clement  V,  the  Order  of  Jesus  had  been  suppressed  in 
1773.  In  the  same  year  Boscovich,  now  free  for  the  second  time,  again  visited  Paris,  where 
he  was  cordially  received  in  official  circles.  The  French  Government  appointed  him  director 
of  "  Optique  Marine,"  with  an  annual  salary  of  8,000  francs ;  and  Boscovich  became  a 
French  subject.  But,  as  an  ex- Jesuit,  he  was  not  welcomed  in  all  scientific  circles.  The 
celebrated  d'Alembert  was  his  declared  enemy  ;  on  the  other  hand,  the  famous  astronomer, 
Lalande,  was  his  devoted  friend  and  admirer.  Particularly,  in  his  controversy  with  Rochon 
on  the  priority  of  the  discovery  of  the  micrometer,  and  again  in  the  dispute  with  Laplace 
about  priority  in  the  invention  of  a  method  for  determining  the  orbits  of  comets,  did 
the  enmity  felt  in  these  scientific  circles  show  itself.  In  Paris,  in  1779,  Boscovich 
published  a  new  edition  of  his  poem  on  eclipses,  translated  into  French  and  annotated, 
under  the  title,  Les  Eclipses,  dedicating  the  edition  to  the  King,  Louis  XV. 

During  this  second  stay  in  Paris,  Boscovich  had  prepared  a  whole  series  of  new  works, 
which  he  hoped  would  have  been  published  at  the  Royal  Press.  But,  as  the  American 
War  of  Independence  was  imminent,  he  was  forced,  in  1782,  to  take  two  years'  leave  of 
absence,  and  return  to  Italy.  He  went  to  the  house  of  his  publisher  at  Bassano  ;  and  here, 
in  1 785^  were  published  five  volumes  of  his  optical  and  astronomical  works,  Opera  pertinentia 
ad  opticam  et  astronomiam. 

Boscovich  had  planned  to  return  through  Italy  from  Bassano  to  Paris ;  indeed,  he  left 
Bassano  for  Venice,  Rome,  Florence,  and  came  to  Milan.  Here  he  was  detained  by  illness 
and  he  was  obliged  to  ask  the  French  Government  to  extend  his  leave,  a  request  that  was 
willingly  granted.  His  health,  however,  became  worse  ;  and  to  it  was  added  a  melancholia. 
He  died  on  February  I3th,  1787. 

The  great  loss  which  Science  sustained  by  his  death  has  been  fitly  commemorated  in 
the  eulogium  by  his  friend  Lalande  in  the  French  Academy,  of  which  he  was  a  member ; 
and  also  in  that  of  Francesco  Ricca  at  Milan,  and  so  on.  But  it  is  his  native  town,  his 
beloved  Ragusa,  which  has  most  fitly  celebrated  the  death  of  the  greatest  of  her  sons 


A  THEORY  OF  NATURAL  PHILOSOPHY  ix 

in  the  eulogium  of  the  poet,  Bernardo  Zamagna. "  This  magnificent  tribute  from  his  native 
town  was  entirely  deserved  by  Boscovich,  both  for  his  scientific  works,  and  for  his  love  and 
work  for  his  country. 

Boscovich  had  left  his  native  country  when  a  boy,  and  returned  to  it  only  once  after- 
wards, when,  in  1747,  he  passed  the  summer  there,  from  June  20th  to  October  1st ;  but 
he  often  intended  to  return.  In  a  letter,  dated  May  3rd,  1774,  he  seeks  to  secure  a  pension 
as  a  member  of  the  Jesuit  College  of  Ragusa  ;  he  writes  :  "  I  always  hope  at  last  to  find 
my  true  peace  in  my  own  country  and,  if  God  permit  me,  to  pass  my  old  age  there  in 
quietness." 

Although  Boscovich  has  written  nothing  in  his  own  language,  he  understood  it  per- 
fectly ;  as  is  shown  by  the  correspondence  with  his  sister,  by  certain  passages  in  his  Italian 
letters,  and  also  by  his  Giornale  (p.  31  ;  p.  59  of  the  French  edition).  In  a  dispute  with 
d'Alembert,  who  had  called  him  an  Italian,  he  said  :  "  we  will  notice  here  in  the  first  place 
that  our  author  is  a  Dalmatian,  and  from  Ragusa,  not  Italian  ;  and  that  is  the  reason  why 
Marucelli,  in  a  recent  work  on  Italian  authors,  has  made  no  mention  of  him."  *  That  his 
feeling  of  Slav  nationality  was  strong  is  proved  by  the  tributes  he  pays  to  his  native  town 
and  native  land  in  his  dedicatory  epistle  to  Louis  XV. 

Boscovich  was  at  once  philosopher,  astronomer,  physicist,  mathematician,  historian, 
engineer,  architect,  and  poet.  In  addition,  he  was  a  diplomatist  and  a  man  of  the  world  ; 
and  yet  a  good  Catholic  and  a  devoted  member  of  the  Jesuit  order.  His  friend,  Lalande, 
has  thus  sketched  his  appearance  and  his  character  :  "  Father  Boscovich  was  of  great 
stature  ;  he  had  a  noble  expression,  and  his  disposition  was  obliging.  He  accommodated 
himself  with  ease  to  the  foibles  of  the  great,  with  whom  he  came  into  frequent  contact. 
But  his  temper  was  a  trifle  hasty  and  irascible,  even  to  his  friends — at  least  his  manner 
gave  that  impression — but  this  solitary  defect  was  compensated  by  all  those  qualities  which 
make  up  a  great  man.  . .  .  He  possessed  so  strong  a  constitution  that  it  seemed  likely  that 
he  would  have  lived  much  longer  than  he  actually  did  ;  but  his  appetite  was  large,  and  his 
belief  in  the  strength  of  his  constitution  hindered  him  from  paying  sufficient  attention 
to  the  danger  which  always  results  from  this."  From  other  sources  we  learn  that  Boscovich 
had  only  one  meal  daily,  dejeuner. 

Of  his  ability  as  a  poet,  Lalande  says  :  "  He  was  himself  a  poet  like  his  brother,  who  was 
also  a  Jesuit.  .  .  .  Boscovich  wrote  verse  in  Latin  only,  but  he  composed  with  extreme  ease. 
He  hardly  ever  found  himself  in  company  without  dashing  off  some  impromptu  verses  to 
well-known  men  or  charming  women.  To  the  latter  he  paid  no  other  attentions,  for  his 
austerity  was  always  exemplary.  .  .  .  With  such  talents,  it  is  not  to  be  wondered  at  that 
he  was  everywhere  appreciated  and  sought  after.  Ministers,  princes  and  sovereigns  all 
received  him  with  the  greatest  distinction.  M.  de  Lalande  witnessed  this  in  every  part 
of  Italy  where  Boscovich  accompanied  him  in  1765." 

Boscovich  was  acquainted  with  several  languages — Latin,  Italian,  French,  as  well  as 
his  native  Serbo-Croatian,  which,  despite  his  long  absence  from  his  country,  he  did  not 
forget.  Although  he  had  studied  in  Italy  and  passed  the  greater  part  of  his  life  there, 
he  had  never  penetrated  to  the  spirit  of  the  language,  as  his  Italian  biographer,  Ricca,  notices. 
His  command  of  French  was  even  more  defective  ;  but  in  spite  of  this  fact,  French  men 
of  science  urged  him  to  write  in  French.  English  he  did  not  understand,  as  he  confessed 
in  a  letter  to  Priestley ;  although  he  had  picked  up  some  words  of  polite  conversation 
during  his  stay  in  London. 

His  correspondence  was  extensive.  The  greater  part  of  it  has  been  published  in 
the  Memoirs  de  VAcademie  Jougo-Slave  of  Zagrab,  1887  to  1912. 

"  Oratio  in  funere  R.  J.  Boscovichii  ...  a  Bernardo  Zamagna. 

*  Voyage  Astronomique,  p.  750  ;    also  on  pp.  707  seq. 

•  Journal  des  Sfavans,  Fevrier,  1792,  pp.  113-118. 


INTRODUCTION 

ALTHOUGH  the  title  to  this  work  to  a  very  large  extent  correctly  describes 
the  contents,  yet  the  argument  leans  less  towards  the  explanation  of  a 
theory  than  it  does  towards  the  logical  exposition  of  the  results  that  must 
follow  from  the  acceptance  of  certain  fundamental  assumptions,  more  or 
less  generally  admitted  by  natural  philosophers  of  the  time.  The  most 
important  of  these  assumptions  is  the  doctrine  of  Continuity,  as  enunciated 
by  Leibniz.  This  doctrine  may  be  shortly  stated  in  the  words  :  "  Every- 
thing takes  place  by  degrees  "  ;  or,  in  the  phrase  usually  employed  by  Boscovich  :  "  Nothing 
happens  -per  saltum."  The  second  assumption  is  the  axiom  of  Impenetrability ;  that  is  to 
say,  Boscovich  admits  as  axiomatic  that  no  two  material  points  can  occupy  the  same  spatial, 
or  local,  point  simultaneously.  Clerk  Maxwell  has  characterized  this  assumption  as  "  an 
unwarrantable  concession  to  the  vulgar  opinion."  He  considered  that  this  axiom  is  a 
prejudice,  or  prejudgment,  founded  on  experience  of  bodies  of  sensible  size.  This  opinion 
of  Maxwell  cannot  however  be  accepted  without  dissection  into  two  main  heads.  The 
criticism  of  the  axiom  itself  would  appear  to  carry  greater  weight  against  Boscovich  than 
against  other  philosophers ;  but  the  assertion  that  it  is  a  prejudice  is  hardly  warranted. 
For,  Boscovich,  in  accepting  the  truth  of  the  axiom,  has  no  experience  on  which  to  found  his 
acceptance.  His  material  points  have  absolutely  no  magnitude  ;  they  are  Euclidean  points, 
"  having  no  parts."  There  is,  therefore,  no  reason  for  assuming,  by  a  sort  of  induction  (and 
Boscovich  never  makes  an  induction  without  expressing  the  reason  why  such  induction  can 
be  made),  that  two  material  points  cannot  occupy  the  same  local  point  simultaneously ; 
that  is  to  say,  there  cannot  have  been  a  prejudice  in  favour  of  the  acceptance  of  this  axiom, 
derived  from  experience  of  bodies  of  sensible  size ;  for,  since  the  material  points  are  non- 
extended,  they  do  not  occupy  space,  and  cannot  therefore  exclude  another  point  from 
occupying  the  same  space.  Perhaps,  we  should  say  the  reason  is  not  the  same  as  that  which 
makes  it  impossible  for  bodies  of  sensible  size.  The  acceptance  of  the  axiom  by  Boscovich  is 
purely  theoretical ;  in  fact,  it  constitutes  practically  the  whole  of  the  theory  of  Boscovich.  On 
the  other  hand,  for  this  very  reason,  there  are  no  readily  apparent  grounds  for  the  acceptance 
of  the  axiom  ;  and  no  serious  arguments  can  be  adduced  in  its  favour  ;  Boscovich 's  own 
line  of  argument,  founded  on  the  idea  that  infinite  improbability  comes  to  the  same  thing 
as  impossibility,  is  given  in  Art.  361.  Later,  I  will  suggest  the  probable  source  from  which 
Boscovich  derived  his  idea  of  impenetrability  as  applying  to  points  of  matter,  as  distinct 
from  impenetrability  for  bodies  of  sensible  size. 

Boscovich's  own  idea  of  the  merit  of  his  work  seems  to  have  been  chiefly  that  it  met  the 
requirements  which,  in  the  opinion  of  Newton,  would  constitute  "  a  mighty  advance  in 
philosophy."  These  requirements  were  the  "  derivation,  from  the  phenomena  of  Nature, 
of  two  or  three  general  principles  ;  and  the  explanation  of  the  manner  in  which  the  properties 
and  actions  of  all  corporeal  things  follow  from  these  principles,  even  if  the  causes  of  those 
principles  had  not  at  the  time  been  discovered."  Boscovich  claims  in  his  preface  to  the 
first  edition  (Vienna,  1758)  that  he  has  gone  far  beyond  these  requirements ;  in  that  he  has 
reduced  all  the  principles  of  Newton  to  a  single  principle — namely,  that  given  by  his  Law 
of  Forces. 

The  occasion  that  led  to  the  writing  of  this  work  was  a  request,  made  by  Father  Scherffer, 
who  eventually  took  charge  of  the  first  Vienna  edition  during  the  absence  of  Boscovich  ;  he 
suggested  to  Boscovich  the  investigation  of  the  centre  of  oscillation.  Boscovich  applied  to 
this  investigation  the  principles  which,  as  he  himself  states,  "  he  lit  upon  so  far  back  as  the 
year  1745."  Of  these  principles  he  had  already  given  some  indication  in  the  dissertations 
De  Viribus  vivis  (published  in  1745),  De  Lege  Firium  in  Natura  existentium  (1755),  and 
others.  While  engaged  on  the  former  dissertation,  he  investigated  the  production  and 
destruction  of  velocity  in  the  case  of  impulsive  action,  such  as  occurs  in  direct  collision. 
In  this,  where  it  is  to  be  noted  that  bodies  of  sensible  size  are  under  consideration,  Boscovich 
was  led  to  the  study  of  the  distortion  and  recovery  of  shape  which  occurs  on  impact ;  he 
came  to  the  conclusion  that,  owing  to  this  distortion  and  recovery  of  shape,  there  was 
produced  by  the  impact  a  continuous  retardation  of  the  relative  velocity  during  the  whole 
time  of  impact,  which  was  finite  ;  in  other  words,  the  Law  of  Continuity,  as  enunciated  by 


XI 


xii  INTRODUCTION 

Leibniz,  was  observed.  It  would  appear  that  at  this  time  (1745)  Boscovich  was  concerned 
mainly,  if  not  solely,  with  the  facts  of  the  change  of  velocity,  and  not  with  the  causes  for 
this  change.  The  title  of  the  dissertation,  De  Firibus  vivis,  shows  however  that  a  secondary 
consideration,  of  almost  equal  importance  in  the  development  of  the  Theory  of  Boscovich, 
also  held  the  field.  The  natural  philosophy  of  Leibniz  postulated  monads,  without  parts, 
extension  or  figure.  In  these  features  the  monads  of  Leibniz  were  similar  to  the  material 
points  of  Boscovich  ;  but  Leibniz  ascribed  to  his  monads  1  perception  and  appetition  in 
addition  to  an  equivalent  of  inertia.  They  are  centres  of  force,  and  the  force  exerted  is  a 
vis  viva.  Boscovich  opposes  this  idea  of  a  "  living,"  or  "  lively  "  force  ;  and  in  this  first 
dissertation  we  may  trace  the  first  ideas  of  the  formulation  of  his  own  material  points. 
Leibniz  denies  action  at  a  distance  ;  with  Boscovich  it  is  the  fundamental  characteristic  of 
a  material  point. 

The  principles  developed  in  the  work  on  collisions  of  bodies  were  applied  to  the  problem 
of  the  centre  of  oscillation.  During  the  latter  investigation  Boscovich  was  led  to  a  theorem 
on  the  mutual  forces  between  the  bodies  forming  a  system  of  three  ;  and  from  this  theorem 
there  followed  the  natural  explanation  of  a  whole  sequence  of  phenomena,  mostly  connected 
with  the  idea  of  a  statical  moment ;  and  his  initial  intention  was  to  have  published  a 
dissertation  on  this  theorem  and  deductions  from  it,  as  a  specimen  of  the  use  and  advantage 
of  his  principles.  But  all  this  time  these  principles  had  been  developing  in  two  directions, 
mathematically  and  philosophically,  and  by  this  time  included  the  fundamental  notions 
of  the  law  of  forces  for  material  points.  The  essay  on  the  centre  of  oscillation  grew  in  length 
as  it  proceeded  ;  until,  finally,  Boscovich  added  to  it  all  that  he  had  already  published  on 
the  subject  of  his  principles  and  other  matters  which,  as  he  says,  "  obtruded  themselves  on 
his  notice  as  he  was  writing."  The  whole  of  this  material  he  rearranged  into  a  more  logical 
(but  unfortunately  for  a  study  of  development  of  ideas,  non-chronological)  order  before 
publication. 

As  stated  by  Boscovich,  in  Art.  164,  the  whole  of  his  Theory  is  contained  in  his  statement 
that  :  "  Matter  is  composed  of  perfectly  indivisible,  non-extended,  discrete  points."  To  this 
assertion  is  conjoined  the  axiom  that  no  two  material  points  can  be  in  the  same  point  of 
space  at  the  same  time.  As  stated  above,  in  opposition  to  Clerk  Maxwell,  this  is  no  matter 
of  prejudice.  Boscovich,  in  Art.  361,  gives  his  own  reasons  for  taking  this  axiom  as  part 
of  his  theory.  He  lays  it  down  that  the  number  of  material  points  is  finite,  whereas  the 
number  of  local  points  is  an  infinity  of  three  dimensions ;  hence  it  is  infinitely  improbable, 
i.e.,  impossible,  that  two  material  points,  without  the  action  of  a  directive  mind,  should 
ever  encounter  one  another,  and  thus  be  in  the  same  place  at  the  same  time.  He  even  goes 
further  ;  he  asserts  elsewhere  that  no  material  point  ever  returns  to  any  point  of  space  in 
which  it  has  ever  been  before,  or  in  which  any  other  material  point  has  ever  been.  Whether 
his  arguments  are  sound  or  not,  the  matter  does  not  rest  on  a  prejudgment  formed  from 
experience  of  bodies  of  sensible  size  ;  Boscovich  has  convinced  himself  by  such  arguments 
of  the  truth  of  the  principle  of  Impenetrability,  and  lays  it  down  as  axiomatic  ;  and  upon 
this,  as  one  of  his  foundations,  builds  his  complete  theory.  The  consequence  of  this  axiom 
is  immediately  evident ;  there  can  be  no  such  thing  as  contact  between  any  two  material 
points ;  two  points  cannot  be  contiguous  or,  as  Boscovich  states,  no  two  points  of  matter 
can  be  in  mathematical  contact.  For,  since  material  points  have  no 
dimensions,  if,  to  form  an  imagery  of  Boscovich's  argument,  we  take 
two  little  squares  ABDC,  CDFE  to  represent  two  points  in  mathema- 
tical contact  along  the  side  CD,  then  CD  must  also  coincide  with  AB, 
and  EF  with  CD  ;  that  is  the  points  which  we  have  supposed  to  be 
contiguous  must  also  be  coincident.  This  is  contrary  to  the  axiom  of 
Impenetrability  ;  and  hence  material  points  must  be  separated  always  O  U  Ir 
by  a  finite  interval,  no  matter  how  small.  This  finite  interval  however 
has  no  minimum  ;  nor  has  it,  on  the  other  hand,  on  account  of  the  infinity  of  space,  any 
maximum,  except  under  certain  hypothetical  circumstances  which  may  possibly  exist. 
Lastly,  these  points  of  matter  float,  so  to  speak,  in  an  absolute  void. 

Every  material  point  is  exactly  like  every  other  material  point ;  each  is  postulated  to 
have  an  inherent  propensity  (determinatio)  to  remain  in  a  state  of  rest  or  uniform  motion  in 
a  straight  line,  whichever  of  these  is  supposed  to  be  its  initial  state,  so  long  as  the  point  is 
not  subject  to  some  external  influence.  Thus  it  is  endowed  with  an  equivalent  of  inertia 
as  formulated  by  Newton  ;  but  as  we  shall  see,  there  does  not  enter  the  Newtonian  idea 
of  inertia  as  a  characteristic  of  mass.  The  propensity  is  akin  to  the  characteristic  ascribed 
to  the  monad  by  Leibniz  ;  with  this  difference,  that  it  is  not  a  symptom  of  activity,  as  with 
Leibniz,  but  one  of  inactivity. 

1  See  Bertrand  Russell,  Philosophy  of  Leibniz ;   especially  p.  91  for  connection  between  Boscovich  and  Leibniz. 


INTRODUCTION  xiii 

Further,  according  to  Boscovich,  there  is  a  mutual  vis  between  every  pair  of  points, 
the  magnitude  of  which  depends  only  on  the  distance  between  them.  At  first  sight,  there 
would  seem  to  be  an  incongruity  in  this  supposition  ;  for,  since  a  point  has  no  magnitude, 
it  cannot  have  any  mass,  considered  as  "  quantity  of  matter  "  ;  and  therefore,  if  the  slightest 
"  force  "  (according  to  the  ordinary  acceptation  of  the  term)  existed  between  two  points, 
there  would  be  an  infinite  acceleration  or  retardation  of  each  point  relative  to  the  other. 
If,  on  the  other  hand,  we  consider  with  Clerk  Maxwell  that  each  point  of  matter  has  a 
definite  small  mass,  this  mass  must  be  finite,  no  matter  how  small,  and  not  infinitesimal. 
For  the  mass  of  a  point  is  the  whole  mass  of  a  body,  divided  by  the  number  of  points  of 
matter  composing  that  body,  which  are  all  exactly  similar  ;  and  this  number  Boscovich 
asserts  is  finite.  It  follows  immediately  that  the  density  of  a  material  point  must  be  infinite, 
since  the  volume  is  an  infinitesimal  of  the  third  order,  if  not  of  an  infinite  order,  i.e.,  zero. 
Now,  infinite  density,  if  not  to  all  of  us,  to  Boscovich  at  least  is  unimaginable.  Clerk 
Maxwell,  in  ascribing  mass  to  a  Boscovichian  point  of  matter,  seems  to  have  been  obsessed 
by  a  prejudice,  that  very  prejudice  which  obsesses  most  scientists  of  the  present  day,  namely, 
that  there  can  be  no  force  without  mass.  He  understood  that  Boscovich  ascribed  to  each 
pair  of  points  a  mutual  attraction  or  repulsion  ;  and,  in  consequence,  prejudiced  by  Newton's 
Laws  of  Motion,  he  ascribed  mass  to  a  material  point  of  Boscovich. 

This  apparent  incongruity,  however,  disappears  when  it  is  remembered  that  the  word 
vis,  as  used  by  the  mathematicians  of  the  period  of  Boscovich,  had  many  different  meanings ; 
or  rather  that  its  meaning  was  given  by  the  descriptive  adjective  that  was  associated  with  it. 
Thus  we  have  vis  viva  (later  associated  with  energy),  vis  mortua  (the  antithesis  of  vis  viva, 
as  understood  by  Leibniz),  vis  acceleratrix  (acceleration),  vis  matrix  (the  real  equivalent 
of  force,  since  it  varied  with  the  mass  directly),  vis  descensiva  (moment  of  a  weight  hung  at 
one  end  of  a  lever),  and  so  on.  Newton  even,  in  enunciating  his  law  of  universal  gravitation, 
apparently  asserted  nothing  more  than  the  fact  of  gravitation — a  propensity  for  approach — 
according  to  the  inverse  square  of  the  distance  :  and  Boscovich  imitates  him  in  this.  The 
mutual  vires,  ascribed  by  Boscovich  to  his  pairs  of  points,  are  really  accelerations,  i.e. 
tendencies  for  mutual  approach  or  recession  of  the  two  points,  depending  on  the  distance 
between  the  points  at  the  time  under  consideration.  Boscovich's  own  words,  as  given  in 
Art.  9,  are  :  "  Censeo  igitur  bina  quaecunque  materise  puncta  determinari  asque  in  aliis 
distantiis  ad  mutuum  accessum,  in  aliis  ad  recessum  mutuum,  quam  ipsam  determinationem 
apello  vim."  The  cause  of  this  determination,  or  propensity,  for  approach  or  recession, 
which  in  the  case  of  bodies  of  sensible  size  is  more  correctly  called  "  force  "  (vis  matrix), 
Boscovich  does  not  seek  to  explain  ;  he  merely  postulates  the  propensities.  The  measures 
of  these  propensities,  i.e.,  the  accelerations  of  the  relative  velocities,  are  the  ordinates  of 
what  is  usually  called  his  curve  of  forces.  This  is  corroborated  by  the  statement  of  Boscovich 
that  the  areas  under  the  arcs  of  his  curve  are  proportional  to  squares  of  velocities ;  which 
is  in  accordance  with  the  formula  we  should  now  use  for  the  area  under  an  "  acceleration- 
space  "  graph  (Area  =  J  f.ds  =  j-r-ds  =  I  v.dv).  See  Note  (f)  to  Art.  118,  where  it  is 

evident  that  the  word  vires,  translated  "  forces,"  strictly  means "  accelerations ; "  seejalso  Art.64- 
Thus  it  would  appear  that  in  the  Theory  of  Boscovich  we  have  something  totally 
different  from  the  monads  of  Leibniz,  which  are  truly  centres  of  force.  Again,  although 
there  are  some  points  of  similarity  with  the  ideas  of  Newton,  more  especially  in  the 
postulation  of  an  acceleration  of  the  relative  velocity  of  every  pair  of  points  of  matter  due 
to  and  depending  upon  the  relative  distance  between  them,  without  any  endeavour  to 
explain  this  acceleration  or  gravitation  ;  yet  the  Theory  of  Boscovich  differs  from  that  of 
Newton  in  being  purely  kinematical.  His  material  point  is  defined  to  be  without  parts, 
i.e.,  it  has  no  volume  ;  as  such  it  can  have  no  mass,  and  can  exert  no  force,  as  we  understand 
such  terms.  The  sole  characteristic  that  has  a  finite  measure  is  the  relative  acceleration 
produced  by  the  simultaneous  existence  of  two  points  of  matter  ;  and  this  acceleration 
depends  solely  upon  the  distance  between  them.  The  Newtonian  idea  of  mass  is  replaced 
by  something  totally  different ;  it  is  a  mere  number,  without  "  dimension  "  ;  the  "  mass  " 
of  a  body  is  simply  the  number  of  points  that  are  combined  to  "  form  "  the  body. 

Each  of  these  points,  if  sufficiently  close  together,  will  exert  on  another  point  of  matter, 
at  a  relatively  much  greater  distance  from  every  point  of  the  body,  the  same  acceleration 
very  approximately.  Hence,  if  we  have  two  small  bodies  A  and  B,  situated  at  a  distance  s 
from  one  another  (the  wording  of  this  phrase  postulates  that  the  points  of  each  body  are 
very  close  together  as  compared  with  the  distance  between  the  bodies)  :  and  if  the  number 
of  points  in  A  and  B  are  respectively  a  and  b,  and  /  is  the  mutual  acceleration  between  any 
pair  of  material  points  at  a  distance  s  from  one  another  ;  then,  each  point  of  A  will  give  to 
each  point  of  B  an  acceleration  /.  Hence,  the  body  A  will  give  to  each  point  of  B,  and 
therefore  to  the  whole  of  B,  an  acceleration  equal  to  a/.  Similarly  the  body  B  will  give  to 


xiv  INTRODUCTION 

the  body  A  an  acceleration  equal  to  bf.  Similarly,  if  we  placed  a  third  body,  C,  at  a  distance 
j  from  A  and  B,  the  body  A  would  give  the  body  C  an  acceleration  equal  to  af,  and  the  body 
B  would  give  the  body  C  an  acceleration  equal  to  bf.  That  is,  the  accelerations  given  to  a 
standard  body  C  are  proportional  to  the  "  number  of  points  "  in  the  bodies  producing 
these  accelerations ;  thus,  numerically,  the  "  mass  "  of  Boscovich  comes  to  the  same  thing 
as  the  "  mass  "  of  Newton.  Further,  the  acceleration  given  by  C  to  the  bodies  A  and  B 
is  the  same  for  either,  namely,  cf ;  from  which  it  follows  that  all  bodies  have  their  velocities 
of  fall  towards  the  earth  equally  accelerated,  apart  from  the  resistance  of  the  air  ;  and  so  on. 
But  the  term  "  force,"  as  the  cause  of  acceleration  is  not  applied  by  Boscovich  to  material 
points ;  nor  is  it  used  in  the  Newtonian  sense  at  all.  When  Boscovich  investigates  the 
attraction  of  "  bodies,"  he  introduces  the  idea  of  a  cause,  but  then  only  more  or  less  as  a 
convenient  phrase.  Although,  as  a  philosopher,  Boscovich  denies  that  there  is  any  possibility 
of  a  fortuitous  circumstance  (and  here  indeed  we  may  admit  a  prejudice  derived  from 
experience  ;  for  he  states  that  what  we  call  fortuitous  is  merely  something  for  which  we, 
in  our  limited  intelligence,  can  assign  no  cause),  yet  with  him  the  existent  thing  is  motion 
and  not  force.  The  latter  word  is  merely  a  convenient  phrase  to  describe  the  "  product  "  of 
"  mass  "  and  "  acceleration." 

To  sum  up,  it  would  seem  that  the  curve  of  Boscovich  is  an  acceleration-interval  graph ; 
and  it  is  a  mistake  to  refer  to  his  cosmic  system  as  a  system  of  "  force-centres."  His  material 
points  have  zero  volume,  zero  mass,  and  exert  zero  force.  In  fact,  if  one  material  point 
alone  existed  outside  the  mind,  and  there  were  no  material  point  forming  part  of  the  mind, 
then  this  single  external  point  could  in  no  way  be  perceived.  In  other  words,  a  single 
point  would  give  no  sense-datum  apart  from  another  point ;  and  thus  single  points  might 
be  considered  as  not  perceptible  in  themselves,  but  as  becoming  so  in  relation  to  other 
material  points.  This  seems  to  be  the  logical  deduction  from  the  strict  sense  of  the 
definition  given  by  Boscovich  ;  what  Boscovich  himself  thought  is  given  in  the  supplements 
that  follow  the  third  part  of  the  treatise.  Nevertheless,  the  phraseology  of  "  attraction  " 
and  "  repulsion  "  is  so  much  more  convenient  than  that  of  "  acceleration  of  the  velocity  of 
approach  "  and  "  acceleration  of  the  velocity  of  recession,"  that  it  will  be  used  in  what 
follows  :  as  it  has  been  used  throughout  the  translation  of  the  treatise. 

There  is  still  another  point  to  be  considered  before  we  take  up  the  study  of  the  Boscovich 
curve  ;  namely,  whether  we  are  to  consider  Boscovich  as,  consciously  or  unconsciously,  an 
atomist  in  the  strict  sense  of  the  word.  The  practical  test  for  this  question  would  seem 
to  be  simply  whether  the  divisibility  of  matter  was  considered  to  be  limited  or  unlimited. 
Boscovich  himself  appears  to  be  uncertain  of  his  ground,  hardly  knowing  which  point  of 
view  is  the  logical  outcome  of  his  definition  of  a  material  point.  For,  in  Art.  394,  he  denies 
infinite  divisibility  ;  but  he  admits  infinite  componibility.  The  denial  of  infinite  divisibility 
is  necessitated  by  his  denial  of  "  anything  infinite  in  Nature,  or  in  extension,  or  a  self- 
determined  infinitely  small."  The  admission  of  infinite  componibility  is  necessitated  by 
his  definition  of  the  material  point ;  since  it  has  no  parts,  a  fresh  point  can  always  be  placed 
between  any  two  points  without  being  contiguous  to  either.  Now,  since  he  denies  the 
existence  of  the  infinite  and  the  infinitely  small,  the  attraction  or  repulsion  between  two 
points  of  matter  (except  at  what  he  calls  the  limiting  intervals)  must  be  finite  :  hence,  since 
the  attractions  of  masses  are  all  by  observation  finite,  it  follows  that  the  number  of  points 
in  a  mass  must  be  finite.  To  evade  the  difficulty  thus  raised,  he  appeals  to  the  scale  of 
integers,  in  which  there  is  no  infinite  number  :  but,  as  he  says,  the  scale  of  integers  is  a 
sequence  of  numbers  increasing  indefinitely,  and  having  no  last  term.  Thus,  into  any  space, 
however  small,  there  may  be  crowded  an  indefinitely  great  number  of  material  points ;  this 
number  can  be  still  further  increased  to  any  extent ;  and  yet  the  number  of  points  finally 
obtained  is  always  finite.  It  would,  again,  seem  that  the  system  of  Boscovich  was  not  a 
material  system,  but  a  system  of  relations ;  if  it  were  not  for  the  fact  that  he  asserts,  in 
Art.  7,  that  his  view  is  that  "  the  Universe  does  not  consist  of  vacuum  interspersed  amongst 
matter,  but  that  matter  is  interspersed  in  a  vacuum  and  floats  in  it."  The  whole  question 
is  still  further  complicated  by  his  remark,  in  Art.  393,  that  in  the  continual  division  of  a 
body,  "  as  soon  as  we  reach  intervals  less  than  the  distance  between  two  material  points, 
further  sections  will  cut  empty  intervals  and  not  matter  "  ;  and  yet  he  has  postulated  that 
there  is  no  minimum  value  to  the  interval  between  two  material  points.  Leaving,  however, 
this  question  of  the  philosophical  standpoint  of  Boscovich  to  be  decided  by  the  reader,  after 
a  study  of  the  supplements  that  follow  the  third  part  of  the  treatise,  let  us  now  consider  the 
curve  of  Boscovich. 

Boscovich,  from  experimental  data,  gives  to  his  curve,  when  the  interval  is  large,  a 
branch  asymptotic  to  the  axis  of  intervals ;  it  approximates  to  the  "  hyperbola  "  x*y—  c,  in 
which  x  represents  the  interval  between  two  points,  and  y  the  vis  corresponding  to  that 
interval,  which  we  have  agreed  to  call  an  attraction,  meaning  thereby,  not  a  force,  but  an 


INTRODUCTION  xv 

acceleration  of  the  velocity  of  approach.  For  small  intervals  he  has  as  yet  no  knowledge 
of  the  quality  or  quantity  of  his  ordinates.  In  Supplement  IV,  he  gives  some  very  ingenious 
arguments  against  forces  that  are  attractive  at  very  small  distances  and  increase  indefinitely, 
such  as  would  be  the  case  where  the  law  of  forces  was  represented  by  an  inverse  power  of 
the  interval,  or  even  where  the  force  varied  inversely  as  the  interval.  For  the  inverse  fourth 
or  higher  power,  he  shows  that  the  attraction  of  a  sphere  upon  a  point  on  its  surface  would 
be  less  than  the  attraction  of  a  part  of  itself  on  this  point ;  for  the  inverse  third  power,  he  con- 
siders orbital  motion,  which  in  this  case  is  an  equiangular  spiral  motion,  and  deduces  that 
after  a  finite  time  the  particle  must  be  nowhere  at  all.  Euler,  considering  this  case,  asserted 
that  on  approaching  the  centre  of  force  the  particle  must  be  annihilated  ;  Boscovich,  with 
more  justice,  argues  that  this  law  of  force  must  be  impossible.  For  the  inverse  square  law, 
the  limiting  case  of  an  elliptic  orbit,  when  the  transverse  velocity  at  the  end  of  the  major 
axis  is  decreased  indefinitely,  is  taken  ;  this  leads  to  rectilinear  motion  of  the  particle  to  the 
centre  of  force  and  a  return  from  it ;  which  does  not  agree  with  the  otherwise  proved 
oscillation  through  the  centre  of  force  to  an  equal  distance  on  either  side. 

Now  it  is  to  be  observed  that  this  supplement  is  quoted  from  his  dissertation  De  Lege 
Firium  in  Natura  existentium,  which  was  published  in  1755  ;  also  that  in  1743  he  had 
published  a  dissertation  of  which  the  full  title  is  :  De  Motu  Corporis  attracti  in  centrum 
immobile  viribus  decrescentibus  in  ratione  distantiarum  reciproca  duplicata  in  spatiis  non 
resistentibus.  Hence  it  is  not  too  much  to  suppose  that  somewhere  between  1741  and  1755 
he  had  tried  to  find  a  means  of  overcoming  this  discrepancy  ;  and  he  was  thus  led  to  suppose 
that,  in  the  case  of  rectilinear  motion  under  an  inverse  square  law,  there  was  a  departure 
from  the  law  on  near  approach  to  the  centre  of  force  ;  that  the  attraction  was  replaced  by  a 
repulsion  increasing  indefinitely  as  the  distance  decreased  ;  for  this  obviously  would  lead  to 
an  oscillation  to  the  centre  and  back,  and  so  come  into  agreement  with  the  limiting  case  of 
the  elliptic  orbit.  I  therefore  suggest  that  it  was  this  consideration  that  led  Boscovich  to 
the  doctrine  of  Impenetrability.  However,  in  the  treatise  itself,  Boscovich  postulates  the 
axiom  of  Impenetrability  as  applying  in  general,  and  thence  argues  that  the  force  at  infinitely 
small  distances  must  be  repulsive  and  increasing  indefinitely.  Hence  the  ordinate  to  the 
curve  near  the  origin  must  be  drawn  in  the  opposite  direction  to  that  of  the  ordinates  for 
sensible  distances,  and  the  area  under  this  branch  of  the  curve  must  be  indefinitely  great. 
That  is  to  say,  the  branch  must  be  asymptotic  to  the  axis  of  ordinates ;  Boscovich  however 
considers  that  this  does  not  involve  an  infinite  ordinate  at  the  origin,  because  the  interval 
between  two  material  points  is  never  zero  ;  or,  vice  versa,  since  the  repulsion  increases 
indefinitely  for  very  small  intervals,  the  velocity  of  relative  approach,  no  matter  how  great, 
of  two  material  points  is  always  destroyed  before  actual  contact ;  which  necessitates  a  finite 
interval  between  two  material  points,  and  the  impossibility  of  encounter  under  any  circum- 
stances :  the  interval  however,  since  a  velocity  of  mutual  approach  may  be  supposed  to  be 
of  any  magnitude,  can  have  no  minimum.  Two  points  are  said  to  be  in  physical  contact, 
in  opposition  to  mathematical  contact,  when  they  are  so  close  together  that  this  great  mutual 
repulsion  is  sufficiently  increased  to  prevent  nearer  approach. 

Since  Boscovich  has  these  two  asymptotic  branches,  and  he  postulates  Continuity, 
there  must  be  a  continuous  curve,  with  a  one-valued  ordinate  for  any  interval,  to  represent 
the  "  force  "  at  all  other  distances ;  hence  the  curve  must  cut  the  axis  at  some  point  in 
between,  or  the  ordinate  must  become  infinite.  He  does  not  lose  sight  of  this  latter  possi- 
bility, but  apparently  discards  it  for  certain  mechanical  and  physical  reasons.  Now,  it  is 
known  that  as  the  degree  of  a  curve  rises,  the  number  of  curves  of  that  degree  increases  very 
rapidly  ;  there  is  only  one  of  the  first  degree,  the  conic  sections  of  the  second  degree,  while 
Newton  had  found  over  three-score  curves  with  equations  of  the  third  degree,  and  nobody 
had  tried  to  find  all  the  curves  of  the  fourth  degree.  Since  his  curve  is  not  one  of  the  known 
curves,  Boscovich  concludes  that  the  degree  of  its  equation  is  very  high,  even  if  it  is  not 
transcendent.  But  the  higher  the  degree  of  a  curve,  the  greater  the  number  of  possible 
intersections  with  a  given  straight  line  ;  that  is  to  say,  it  is  highly  probable  that  there  are  a 
great  many  intersections  of  the  curve  with  the  axis ;  i.e.,  points  giving  zero  action  for 
material  points  situated  "at  the  corresponding  distance  from  one  another.  Lastly,  since  the 
ordinate  is  one-valued,  the  equation  of  the  curve,  as  stated  in  Supplement  III,  must  be  of 
the  form  P-Qy  =  o,  where  P  and  Q  are  functions  of  x  alone.  Thus  we  have  a  curve  winding 
about  the  axis  for  intervals  that  are  very  small  and  developing  finally  into  the  hyperbola  of 
the  third  degree  for  sensible  intervals.  This  final  branch,  however,  cannot  be  exactly  this 
hyperbola  ;  for,  Boscovich  argues,  if  any  finite  arc  of  the  curve  ever  coincided  exactly  with 
the  hyperbola  of  the  third  degree,  it  would  be  a  breach  of  continuity  if  it  ever  departed  from 
it.  Hence  he  concludes  that  the  inverse  square  law  is  observed  approximately  only,  even 
at  large  distances. 

As  stated  above,  the  possibility  of  other  asymptotes,  parallel  to  the  asymptote  at  the 


INTRODUCTION 

origin,  is  not  lost  sight  of.  The  consequence  of  one  occurring  at  a  very  small  distance  from 
the  origin  is  discussed  in  full.  Boscovich,  however,  takes  great  pains  to  show  that  all  the 
phenomena  discussed  can  be  explained  on  the  assumption  of  a  number  of  points  of  inter- 
section of  his  curve  with  the  axis,  combined  with  different  characteristics  of  the  arcs  that  lie 
between  these  points  of  intersection.  There  is,  however,  one  suggestion  that  is  very 
interesting,  especially  in  relation  to  recent  statements  of  Einstein  and  Weyl.  Suppose  that 
beyond  the  distances  of  the  solar  system,  for  which  the  inverse  square  law  obtains  approxi- 
mately at  least,  the  curve  of  forces,  after  touching  the  axis  (as  it  may  do,  since  it  does  not 
coincide  exactly  with  the  hyperbola  of  the  third  degree),  goes  off  to  infinity  in  the  positive 
direction  ;  or  suppose  that,  after  cutting  the  axis  (as  again  it  may  do,  for  the  reason  given 
above),  it  once  more  begins  to  wind  round  the  axis  and  finally  has  an  asymptotic  attractive 
branch.  Then  it  is  evident  that  the  universe  in  which  we  live  is  a  self-contained  cosmic 
system  ;  for  no  point  within  it  can  ever  get  beyond  the  distance  of  this  further  asymptote. 
If  in  addition,  beyond  this  further  asymptote,  the  curve  had  an  asymptotic  repulsive  branch 
and  went  on  as  a  sort  of  replica  of  the  curve  already  obtained,  then  no  point  outside  our 
universe  could  ever  enter  within  it.  Thus  there  is  a  possibility  of  infinite  space  being 
filled  with  a  succession  of  cosmic  systems,  each  of  which  would  never  interfere  with  any 
other  ;  indeed,  a  mind  existing  in  any  one  of  these  universes  could  never  perceive  the 
existence  of  any  other  universe  except  that  in  which  it  existed.  Thus  space  might  be  in 
reality  infinite,  and  yet  never  could  be  perceived  except  as  finite. 

The  use  Boscovich  makes  of  his  curve,  the  ingenuity  of  his  explanations  and  their  logic, 
the  strength  or  weakness  of  his  attacks  on  the  theories  of  other  philosophers,  are  left  to  the 
consideration  of  the  reader  of  the  text.  It  may,  however,  be  useful  to  point  out  certain 
matters  which  seem  more  than  usually  interesting.  Boscovich  points  out  that  no  philosopher 
has  attempted  to  prove  the  existence  of  a  centre  of  gravity.  It  would  appear  especially  that 
he  is,  somehow  or  other,  aware  of  the  mistake  made  by  Leibniz  in  his  early  days  (a  mistake 
corrected  by  Huygens  according  to  the  statement  of  Leibniz),  and  of  the  use  Leibniz  later 
made  of  the  principle  of  moments ;  Boscovich  has  apparently  considered  the  work  of  Pascal 
and  others,  especially  Guldinus ;,  it  looks  almost  as  if  (again,  somehow  or  other)  he  had  seen 
some  description  of  "  The  Method  "  of  Archimedes.  For  he  proceeds  to  define  the  centre 
of  gravity  geometrically,  and  to  prove  that  there  is  always  a  centre  of  gravity,  or  rather  a 
geometrical  centroid  ;  whereas,  even  for  a  triangle,  there  is  no  centre  of  magnitude,  with 
which  Leibniz  seems  to  have  confused  a  centroid  before  his  conversation  with  Huygens. 
This  existence  proof,  and  the  deductions  from  it,  are  necessary  foundations  for  the  centro- 
baryc  analysis  of  Leibniz.  The  argument  is  shortly  as  follows  :  Take  a  plane  outside,  say 
to  the  right  of,  all  the  points  of  all  the  bodies  under  consideration  ;  find  the  sum  of  all  the 
distances  of  all  the  points  from  this  plane  ;  divide  this  sum  by  the  number  of  points ;  draw 
a  plane  to  the  left  of  and  parallel  to  the  chosen  plane,  at  a  distance  from  it  equal  to  the 
quotient  just  found.  Then,  observing  algebraic  sign,  this  is  a  plane  such  that  the  sum  of 
the  distances  of  all  the  points  from  it  is  zero  ;  i.e.,  the  sum  of  the  distances  of  all  the  points 
on  one  side  of  this  plane  is  equal  arithmetically  to  the  sum  of  the  distances  of  all  the  points  on 
the  other  side.  Find  a  similar  plane  of  equal  distances  in  another  direction  ;  this  intersects 
the  first  plane  in  a  straight  line.  A  third  similar  plane  cuts  this  straight  line  in  a  point ; 
this  point  is  the  centroid  ;  it  has  the  unique  property  that  all  planes  through  it  are  planes 
of  equal  distances.  If  some  of  the  points  are  conglomerated  to  form  a  particle,  the  sum 
of  the  distances  for  each  of  the  points  is  equal  to  the  distance  of  the  particle  multiplied  by 
the  number  of  points  in  the  particle,  i.e.,  by  the  mass  of  the  particle.  Hence  follows  the 
theorem  for  the  statical  moment  for  lines  and  planes  or  other  surfaces,  as  well  as  for  solids 
that  have  weight. 

Another  interesting  point,  in  relation  to  recent  work,  is  the  subject-matter  of  Art.  230- 
236 ;  where  it  is  shown  that,  due  solely  to  the  mutual  forces  exerted  on  a  third  point  by 
two  points  separated  by  a  proper  interval,  there  is  a  series  of  orbits,  approximately  confocal 
ellipses,  in  which  the  third  point  is  in  a  state  of  steady  motion  ;  these  orbits  are  alternately 
stable  and  stable.  If  the  steady  motion  in  a  stable  orbit  is  disturbed,  by  a  sufficiently  great 
difference  of  the  velocity  being  induced  by  the  action  of  a  fourth  point  passing  sufficiently 
near  the  third  point,  this  third  point  will  leave  its  orbit  and  immediately  take  up  another 
stable  orbit,  after  some  initial  oscillation  about  it.  This  elegant  little  theorem  does  not 
depend  in  any  way  on  the  exact  form  of  the  curve  of  forces,  so  long  as  there  are  •portions  of  the 
curve  winding  about  the  axis  for  very  small  intervals  between  the  points. 

It  is  sufficient,  for  the  next  point,  to  draw  the  reader's  attention  to  Art.  266-278,  on 
collision,  and  to  the  articles  which  follow  on  the  agreement  between  .resolution  and  com- 
position of  forces  as  a  working  hypothesis.  From  what  Boscovich  says,  it  would  appear  that 
philosophers  of  his  time  were  much  perturbed  over  the  idea  that,  when  a  force  was  resolved 
into  two  forces  at  a  sufficiently  obtuse  angle,  the  force  itself  might  be  less  than  either  of 


INTRODUCTION  xvii 

the  resolutes.  Boscovich  points  out  that,  in  his  Theory,  there  is  no  resolution,  only  com- 
position ;  and  therefore  the  difficulty  does  not  arise.  In  this  connection  he  adds  that  there 
are  no  signs  in  Nature  of  anything  approaching  the  vires  viva  of  Leibniz. 

In  Art.  294  we  have  Boscovich's  contribution  to  the  controversy  over  the  correct 
measure  of  the  "  quantity  of  motion  "  ;  but,  as  there  is  no  attempt  made  to  follow  out  the 
change  in  either  the  velocity  or  the  square  of  the  velocity,  it  cannot  be  said  to  lead  to  any- 
thing conclusive.  As  a  matter  of  fact,  Boscovich  uses  the  result  to  prove  the  non-existence 
of  vires  vivce. 

In  Art.  298-306  we  have  a  mechanical  exposition  of  reflection  and  refraction  of  light. 
This  comes  under  the  section  on  Mechanics,  because  with  Boscovich  light  is  matter  moving 
with  a  very  high  velocity,  and  therefore  reflection  is  a  case  of  impact,  in  that  it  depends 
upon  the  destruction  of  the  whole  of  the  perpendicular  velocity  upon  entering  the  "  surface  " 
of  a  denser  medium,  the  surface  being  that  part  of  space  in  front  of  the  physical  surface  of 
the  medium  in  which  the  particles  of  light  are  near  enough  to  the  denser  medium  to  feel  the 
influence  of  the  last  repulsive  asymptotic  branch  of  the  curve  of  forces.  If  this  perpendicular 
velocity  is  not  all  destroyed,  the  particle  enters  the  medium,  and  is  refracted  ;  in  which 
case,  the  existence  of  a  sine  law  is  demonstrated.  It  is  to  be  noted  that  the  "  fits  "  of 
alternate  attraction  and  repulsion,  postulated  by  Newton,  follow  as  a  natural  consequence 
of  the  winding  portion  of  the  curve  of  Boscovich. 

In  Art.  328-346  we  have  a  discussion  of  the  centre  of  oscillation,  and  the  centre  of 
percussion  is  investigated  as  well  for  masses  in  a  plane  perpendicular  to  the  axis  of  rotation, 
and  masses  lying  in  a  straight  line,  where  each  mass  is  connected  with  the  different  centres. 
Boscovich  deduces  from  his  theory  the  theorems,  amongst  others,  that  the  centres  of  suspen- 
sion and  oscillation  are  interchangeable,  and  that  the  distance  between  them  is  equal  to  the 
distance  of  the  centre  of  percussion  from  the  axis  of  rotation  ;  he  also  gives  a  rule  for  finding 
the  simple  equivalent  pendulum.  The  work  is  completed  in  a  letter  to  Fr.  Scherffer,  which 
is  appended  at  the  end  of  this  volume. 

In  the  third  section,  which  deals  with  the  application  of  the  Theory  to  Physics,  we 
naturally  do  not  look  for  much  that  is  of  value.  But,  in  Art.  505,  Boscovich  evidently  has 
the  correct  notion  that  sound  is  a  longitudinal  vibration  of  the  air  or  some  other  medium  ; 
and  he  is  able  to  give  an  explanation  of  the  propagation  of  the  disturbance  purely  by  means 
of  the  mutual  forces  between  the  particles  of  the  medium.  In  Art.  507  he  certainly  states 
that  the  cause  of  heat  is  a  "  vigorous  internal  motion  "  ;  but  this  motion  is  that  of  the 
"  particles  of  fire,"  if  it  is  a  motion  ;  an  alternative  reason  is  however  given,  namely,  that  it 
may  be  a  "  fermentation  of  a  sulphurous  substance  with  particles  of  light."  "  Cold  is 
a  lack  of  this  substance,  or  of  a  motion  of  it."  No  attention  will  be  called  to  this  part 
of  the  work,  beyond  an  expression  of  admiration  for  the  great  ingenuity  of  a  large  part 
of  it. 

There  is  a  metaphysical  appendix  on  the  seat  of  the  mind,  and  its  nature,  and  on  the 
existence  and  attributes  of  GOD.  This  is  followed  by  two  short  discussions  of  a  philosophical 
nature  on  Space  and  Time.  Boscovich  does  not  look  on  either  of  these  as  being  in  themselves 
existent ;  his  entities  are  modes  of  existence,  temporal  and  local.  These  three  sections  are 
full  of  interest  for  the  modern  philosophical  reader. 

Supplement  V  is  a  theoretical  proof,  purely  derived  from  the  theory  of  mutual  actions 
between  points  of  matter,  of  the  law  of  the  lever  ;  this  is  well  worth  study. 

There  are  two  points  of  historical  interest  beyond  the  study  of  the  work  of  Boscovich 
that  can  be  gathered  from  this  volume.  The  first  is  that  at  this  time  it  would  appear  that 
the  nature  of  negative  numbers  and  quantities  was  not  yet  fully  understood.  Boscovich,  to 
make  his  curve  more  symmetrical,  continues  it  to  the  left  of  the  origin  as  a  reflection  in  the 
axis  of  ordinates.  It  is  obvious,  however,  that,  if  distances  to  the  left  of  the  origin  stand  for 
intervals  measured  in  the  opposite  direction  to  the  ordinary  (remembering  that  of  the  two 
points  under  consideration  one  is  supposed  to  be  at  the  origin),  then  the  force  just  the  other 
side  of  the  axis  of  ordinates  must  be  repulsive  ;  but  the  repulsion  is  in  the  opposite  direction 
to  the  ordinary  way  of  measuring  it,  and  therefore  should  appear  on  the  curve  represented 
by  an  ordinate  of  attraction.  Thus,  the  curve  of  Boscovich,  if  completed,  should  have  point 
symmetry  about  the  origin,  and  not  line  symmetry  about  the  axis  of  ordinates.  Boscovich, 
however,  avoids  this  difficulty,  intentionally  or  unintentionally,  when  showing  how  the 
equation  to  the  curve  may  be  obtained,  by  taking  z  =  x*  as  his  variable,  and  P  and  Q  as 
functions  of  z,  in  the  equation  P-Qy  =  o,  referred  to  above.  Note. — In  this  connection 
(p.  410,  Art.  25,  1.  5),  Boscovich  has  apparently  made  a  slip  over  the  negative  sign  :  as  the 
intention  is  clear,  no  attempt  has  been  made  to  amend  the  Latin. 

The  second  point  is  that  Boscovich  does  not  seem  to  have  any  idea  of  integrating  between 
limits.  He  has  to  find  the  area,  in  Fig.  I  on  p.  134,  bounded  by  the  axes,  the  curve  and  the 
ordinate  ag  ;  this  he  does  by  the  use  of  the  calculus  in  Note  (1)  on  p.  141.  He  assumes  that 


xviii  INTRODUCTION 

gt 

the  equation  of  the  curve  is  xmyn  =  I,  and  obtains  the  integral  -   -  xy  +  A,  where  A  is  the 

n—m 

constant  of  integration.  He  states  that,  if  n  is  greater  than  m,  A  =  o,  being  the  initial  area 
at  the  origin.  He  is  then  faced  with  the  necessity  of  making  the  area  infinite  when  n  =  m, 
and  still  more  infinite  when  n<jn.  He  says  :  "  The  area  is  infinite,  when  n  =  m,  because 
this  makes  the  divisor  zero  ;  and  thus  the  area  becomes  still  more  infinite  if  n<^m."  Put 

into  symbols,  the  argument  is  :  Since  «-OT<O,  >-  >  oo  .     The  historically  interesting 

n— m      o 

point  about  this  is  that  it  represents  the  persistance  of  an  error  originally  made  by  Wallis 
in  his  Ariihmetica  Infinitorum  (it  was  Wallis  who  invented  the  sign  oc  to  stand  for  "  simple 
infinity,"  the  value  of  i/o,  and  hence  of  «/o).  Wallis  had  justification  for  his  error,  if 
indeed  it  was  an  error  in  his  case  ;  for  his  exponents  were  characteristics  of  certain  infinite 
series,  and  he  could  make  his  own  laws  about  these  so  that  they  suited  the  geometrical 
problems  to  which  they  were  applied  ;  it  was  not  necessary  that  they  should  obey  the  laws 
of  inequality  that  were  true  for  ordinary  numbers.  Boscovich's  mistake  is,  of  course,  that 
of  assuming  that  the  constant  is  zero  in  every  case  ;  and  in  this  he  is  probably  deceived  by 

using  the  formula xy  -f-  A,  instead  of ^B/("-*l)  -}-  A,  for  the  area.     From  the  latter 

n—m  n — m 

it  is  easily  seen  that  since  the  initial  area  is  zero,  we  must  have  A  = ow/("~m).      If  n  is 

m— n 

equal  to  or  greater  than  m,  the  constant  A  is  indeed  zero  ;  but  if  n  is  less  than  m,  the  constant 
is  infinite.  The  persistence  of  this  error  for  so  long  a  time,  from  1655  to  175%>  during  which 
we  have  the  writings  of  Newton,  Leibniz,  the  Bernoullis  and  others  on  the  calculus,  seems 
to  lend  corroboration  to  a  doubt  as  to  whether  the  integral  sign  was  properly  understood  as 
a  summation  between  limits,  and  that  this  sum  could  be  expressed  as  the  difference  of  two 
values  of  the  same  function  of  those  limits.  It  appears  to  me  that  this  point  is  one  of 
very  great  importance  in  the  history  of  the  development  of  mathematical  thought. 

Some  idea  of  how  prolific  Boscovich  was  as  an  author  may  be  gathered  from  the  catalogue 
of  his  writings  appended  at  the  end  of  this  volume.  This  catalogue  has  been  taken  from  the 
end  of  the  original  first  Venetian  edition,  and  brings  the  list  up  to  the  date  of  its  publication, 
1763.  It  was  felt  to  be  an  impossible  task  to  make  this  list  complete  up  to  the  time  of  the 
death  of  Boscovich  ;  and  an  incomplete  continuation  did  not  seem  desirable.  Mention 
must  however  be  made  of  one  other  work  of  Boscovich  at  least ;  namely,  a  work  in  five 
quarto  volumes,  published  in  1785,  under  the  title  of  Opera  pertinentia  ad  Opticam  et 
Astronomiam. 

Finally,  in  order  to  bring  out  the  versatility  of  the  genius  of  Boscovich,  we  may  mention 
just  a  few  of  his  discoveries  in  science,  which  seem  to  call  for  special  attention.  In  astro- 
nomical science,  he  speaks  of  the  use  of  a  telescope  filled  with  liquid  for  the  purpose  of 
measuring  the  aberration  of  light ;  he  invented  a  prismatic  micrometer  contemporaneously 
with  Rochon  and  Maskelyne.  He  gave  methods  for  determining  the  orbit  of  a  comet  from 
three  observations,  and  for  the  equator  of  the  sun  from  three  observations  of  a  "  spot  "  ; 
he  carried  out  some  investigations  on  the  orbit  of  Uranus,  and  considered  the  rings  of  Saturn. 
In  what  was  then  the  subsidiary  science  of  optics,  he  invented  a  prism  with  a  variable  angle 
for  measuring  the  refraction  and  dispersion  of  different  kinds  of  glass ;  and  put  forward  a 
theory  of  achromatism  for  the  objectives  and  oculars  of  the  telescope.  In  mechanics  and 
geodesy,  he  was  apparently  the  first  to  solve  the  problem  of  the  "  body  of  greatest  attraction  " ; 
he  successfully  attacked  the  question  of  the  earth's  density  ;  and  perfected  the  apparatus 
and  advanced  the  theory  of  the  measurement  of  the  meridian.  In  mathematical  theory, 
he  seems  to  have  recognized,  before  Lobachevski  and  Bolyai,  the  impossibility  of  a  proof  of 
Euclid's  "  parallel  postulate  "  ;  and  considered  the  theory  of  the  logarithms  of  negative 
numbers. 

J.  M.  C. 

N.B. — The  page  numbers  on  the  left-hand  pages  of  the  index  are  the  pages  of  the 
original  Latin  Edition  of  1763  ;  they  correspond  with  the  clarendon  numbers  inserted 
throughout  the  Latin  text  of  this  edition. 


CORRIGENDA 

Attention  is  called  to  the  following  important  corrections,  omissions,  and  alternative  renderings ;  misprints 
involving  a  single  letter  or  syllable  only  are  given  at  the  end  of  the  volume. 

p.    27, 1.    8,  for  in  one  plane  read  in  the  same  direction 

p.    47, 1.  62,  literally  on  which  ...  is  exerted 

p.    49,  1.  33,  for  just  as  ...  is  read  so  that  .  .  .  may  be 

P-    S3>  1-    9>  after  a  line  add  but  not  parts  of  the  line  itself 

p.    61,  Art.  47,  Alternative  rendering:  These  instances  make  good  the  same  point  as  water  making  its  way  through 

the  pores  of  a  sponge  did  for  impenetrability ; 

p.    67,  1.    5,  for  it  is  allowable  for  me  read  I  am  disposed  ;  unless  in  the  original  libet  is  taken  to  be  a  misprint  for  licet 
p.    73, 1.  26,  after  nothing  add  in  the  strict  meaning  of  the  term 
p.    85,  1.  27,  after  conjunction  add  of  the  same  point  of  space 

p.  91,    1.  25,  Alternative  rendering  :  and  these  properties  might  distinguish  the  points  even  in  the  view  of  the  followers 
of  Leibniz 

1.    5  from  bottom,  Alternative  rendering :  Not  to  speak  of  the  actual  form  of  the  leaves  present  in  the  seed 
p.  115,  1.  25,  after  the  left  add  but  that  the  two  outer  elements  do  not  touch  each  other 

1.  28,  for  two  little  spheres  read  one  little  sphere 
p.  117,  1.  41,  for  precisely  read  abstractly 
p.  125,  1.  29,  for  ignored  read  urged  in  reply 
p.  126,  1.  6  from  bottom,  it  is  -possible  that  acquirere  is  intended  for  acquiescere,  with  a  corresponding  change  in  the 

translation 

p.  129,  Art.  162,  marg.  note,  for  on  what  they  may  be  founded  read  in  what  it  consists, 
p.  167,  Art.  214,  1.  2  of  marg.  note,  transpose  by  and  on 

footnote,  1.  I,  for  be  at  read  bisect  it  at 
p.  199,  1.  24,  for  so  that  read  just  as 
p.  233,  1.    4  from  bottom,  for  base  to  the  angle  read  base  to  the  sine  of  the  angle 

last  line,  after  vary  insert  inversely 

p.  307,  1.    5   from  end,  for  motion,  as  (with  fluids)  takes  place  read  motion  from  taking  place 
p.  323,  1.  39,  for  the  agitation  will  read  the  fluidity  will 
P-  345»  1-  32>  for  described  read  destroyed 
p.  357, 1.  44,  for  others  read  some,  others  of  others 

1.    5  from  end,  for  fire  read  a  fiery  and  insert  a  comma  before  substance 


XIX 


THEORIA 
PHILOSOPHIC    NATURALIS 


TYPOGRAPHUS 

VENETUS 

LECTORI 


PUS,  quod  tibi  offero,  jam  ab  annis  quinque  Viennse  editum,  quo  plausu 
exceptum  sit  per  Europam,  noveris  sane,  si  Diaria  publica  perlegeris,  inter 
quse  si,  ut  omittam  caetera,  consulas  ea,  quae  in  Bernensi  pertinent  ad 
initium  anni  1761  ;  videbis  sane  quo  id  loco  haberi  debeat.  Systema 
continet  Naturalis  Philosophise  omnino  novum,  quod  jam  ab  ipso  Auctore 
suo  vulgo  Boscovichianum  appellant.  Id  quidem  in  pluribus  Academiis 
jam  passim  publice  traditur,  nee  tantum  in  annuis  thesibus,  vel  disserta- 
tionibus  impressis,  ac  propugnatis  exponitur,  sed  &  in  pluribus  elementaribus  libris  pro 
juventute  instituenda  editis  adhibetur,  exponitur,  &  a  pluribus  habetur  pro  archetype. 
Verum  qui  omnem  systematis  compagem,  arctissimum  partium  nexum  mutuum,  fcecun- 
ditatem  summam,  ac  usum  amplissimum  ac  omnem,  quam  late  patet,  Naturam  ex  unica 
simplici  lege  virium  derivandam  intimius  velit  conspicere,  ac  contemplari,  hoc  Opus 
consulat,  necesse  est. 


Haec  omnia  me  permoverant  jam  ab  initio,  ut  novam  Operis  editionem  curarem  : 
accedebat  illud,  quod  Viennensia  exemplaria  non  ita  facile  extra  Germaniam  itura  videbam, 
&  quidem  nunc  etiam  in  reliquis  omnibus  Europse  partibus,  utut  expetita,  aut  nuspiam 
venalia  prostant,  aut  vix  uspiam  :  systema  vero  in  Italia  natum,  ac  ab  Auctore  suo  pluribus 
hie  apud  nos  jam  dissertationibus  adumbratum,  &  casu  quodam  Viennae,  quo  se  ad  breve 
tempus  contulerat,  digestum,  ac  editum,  Italicis  potissimum  typis,  censebam,  per  univer- 
sam  Europam  disseminandum.  Et  quidem  editionem  ipsam  e  Viennensi  exemplari  jam 
turn  inchoaveram  ;  cum  illud  mihi  constitit,  Viennensem  editionem  ipsi  Auctori,  post  cujus 
discessum  suscepta  ibi  fuerat,  summopere  displicere  :  innumera  obrepsisse  typorum  menda  : 
esse  autem  multa,  inprimis  ea,  quas  Algebraicas  formulas  continent,  admodum  inordinata, 
&  corrupta  :  ipsum  eorum  omnium  correctionem  meditari,  cum  nonnullis  mutationibus, 
quibus  Opus  perpolitum  redderetur  magis,  &  vero  etiam  additamentis. 


Illud  ergo  summopere  desideravi,  ut  exemplar  acquirerem  ab  ipso  correctum,  &  auctum 
ac  ipsum  edition!  praesentem  haberem,  &  curantem  omnia  per  sese.  At  id  quidem  per 
hosce  annos  obtinere  non  licuit,  eo  universam  fere  Europam  peragrante ;  donee  demum 
ex  tarn  longa  peregrinatione  redux  hue  nuper  se  contulit,  &  toto  adstitit  editionis  tempore, 
ac  praeter  correctores  nostros  omnem  ipse  etiam  in  corrigendo  diligentiam  adhibuit ; 
quanquam  is  ipse  haud  quidem  sibi  ita  fidit,  ut  nihil  omnino  effugisse  censeat,  cum  ea  sit 
humanas  mentis  conditio,  ut  in  eadem  re  diu  satis  intente  defigi  non  possit. 


Haec  idcirco  ut  prima  quaedam,  atque  originaria  editio  haberi  debet,  quam  qui  cum 
Viennensi  contulerit,  videbit  sane  discrimen.  E  minoribus  mutatiunculis  multae  pertinent 
ad  expolienda,  &  declaranda  plura  loca ;  sunt  tamen  etiam  nonnulla  potissimum  in  pagin- 
arum  fine  exigua  additamenta,  vel  mutatiunculas  exiguae  factae  post  typographicam 
constructionem  idcirco  tantummodo,  ut  lacunulae  implerentur  quae  aliquando  idcirco 
supererant,  quod  plures  ph'ylirae  a  diversis  compositoribus  simul  adornabantur,  &  quatuor 
simul  praela  sudabant;  quod  quidem  ipso  praesente  fieri  facile  potuit,  sine  ulla  pertur- 
batione  sententiarum,  &  ordinis. 


THE   PRINTER  AT  VENICE 

TO 

THE    READER 

!  OU  will  be  well  aware,  if  you  have  read  the  public  journals,  with  what  applause 
the  work  which  I  now  offer  to  you  has  been  received  throughout  Europe 
since  its  publication  at  Vienna  five  years  ago.  Not  to  mention  others,  if 
you  refer  to  the  numbers  of  the  Berne  Journal  for  the  early  part  of  the 
year  1761,  you  will  not  fail  to  see  how  highly  it  has  been  esteemed.  It 
contains  an  entirely  new  system  of  Natural  Philosophy,  which  is  already 
commonly  known  as  the  Boscovichian  theory,  from  the  name  of  its  author, 
As  a  matter  of  fact,  it  is  even  now  a  subject  of  public  instruction  in  several  Universities  in 
different  parts  ;  it  is  expounded  not  only  in  yearly  theses  or  dissertations,  both  printed  & 
debated  ;  but  also  in  several  elementary  books  issued  for  the  instruction  of  the  young  it  is 
introduced,  explained,  &  by  many  considered  as  their  original.  Any  one,  however,  who 
wishes  to  obtain  more  detailed  insight  into  the  whole  structure  of  the  theory,  the  close 
relation  that  its  several  parts  bear  to  one  another,  or  its  great  fertility  &  wide  scope  for 
the  purpose  of  deriving  the  whole  of  Nature,  in  her  widest  range,  from  a  single  simple  law 
of  forces ;  any  one  who  wishes  to  make  a  deeper  study  of  it  must  perforce  study  the  work 
here  offered. 

All  these  considerations  had  from  the  first  moved  me  to  undertake  a  new  edition  of 
the  work  ;  in  addition,  there  was  the  fact  that  I  perceived  that  it  would  be  a  matter  of  some 
difficulty  for  copies  of  the  Vienna  edition  to  pass  beyond  the  confines  of  Germany — indeed, 
at  the  present  time,  no  matter  how  diligently  they  are  inquired  for,  they  are  to  be  found 
on  sale  nowhere,  or  scarcely  anywhere,  in  the  rest  of  Europe.  The  system  had  its  birth  in 
Italy,  &  its  outlines  had  already  been  sketched  by  the  author  in  several  dissertations  pub- 
lished here  in  our  own  land  ;  though,  as  luck  would  have  it,  the  system  itself  was  finally 
put  into  shape  and  published  at  Vienna,  whither  he  had  gone  for  a  short  time.  I  therefore 
thought  it  right  that  it  should  be  disseminated  throughout  the  whole  of  Europe,  &  that 
preferably  as  the  product  of  an  Italian  press.  I  had  in  fact  already  commenced  an  edition 
founded  on  a  copy  of  the  Vienna  edition,  when  it  came  to  my  knowledge  that  the  author 
was  greatly  dissatisfied  with  the  Vienna  edition,  taken  in  hand  there  after  his  departure ; 
that  innumerable  printer's  errors  had  crept  in ;  that  many  passages,  especially  those  that 
contain  Algebraical  formulae,  were  ill-arranged  and  erroneous ;  lastly,  that  the  author 
himself  had  in  mind  a  complete  revision,  including  certain  alterations,  to  give  a  better 
finish  to  the  work,  together  with  certain  additional  matter. 

That  being  the  case,  I  was  greatly  desirous  of  obtaining  a  copy,  revised  &  enlarged 
by  himself ;  I  also  wanted  to  have  him  at  hand  whilst  the  edition  was  in  progress,  &  that 
he  should  superintend  the  whole  thing  for  himself.  This,  however,  I  was  unable  to  procure 
during  the  last  few  years,  in  which  he  has  been  travelling  through  nearly  the  whole  of 
Europe  ;  until  at  last  he  came  here,  a  little  while  ago,  as  he  returned  home  from  his  lengthy 
wanderings,  &  stayed  here  to  assist  me  during  the  whole  time  that  the  edition  was  in 
hand.  He,  in  addition  to  our  regular  proof-readers,  himself  also  used  every  care  in  cor- 
recting the  proof ;  even  then,  however,  he  has  not  sufficient  confidence  in  himself  as  to 
imagine  that  not  the  slightest  thing  has  escaped  him.  For  it  is  a  characteristic  of  the  human 
mind  that  it  cannot  concentrate  long  on  the  same  subject  with  sufficient  attention. 

It  follows  that  this  ought  to  be  considered  in  some  measure  as  a  first  &  original 
edition  ;  any  one  who  compares  it  with  that  issued  at  Vienna  will  soon  see  the  difference 
between  them.  Many  of  the  minor  alterations  are  made  for  the  purpose  of  rendering 
certain  passages  more  elegant  &  clear  ;  there  are,  however,  especially  at  the  foot  of  a 
page,  slight  additions  also,  or  slight  changes  made  after  the  type  was  set  up,  merely  for 
the  purpose  of  filling  up  gaps  that  were  left  here  &  there — these  gaps  being  due  to  the 
fact  that  several  sheets  were  being  set  at  the  same  time  by  different  compositors,  and  four 
presses  were  kept  hard  at  work  together.  As  he  was  at  hand,  this  could  easily  be  done 
without  causing  any  disturbance  of  the  sentences  or  the  pagination. 


4  TYPOGRAPHUS  VENETUS  LECTORI 

Inter  mutationes  occurret  ordo  numerorum  mutatus  in  paragraphis  :  nam  numerus  82 
de  novo  accessit  totus  :  deinde  is,  qui  fuerat  261  discerptus  est  in  5  ;  demum  in  Appendice 
post  num.  534  factse  sunt  &  mutatiunculse  nonnullae,  &  additamenta  plura  in  iis,  quse 
pertinent  ad  sedem  animse. 

Supplementorum  ordo  mutatus  est  itidem  ;  quse  enim  fuerant  3,  &  4,  jam  sunt  I,  & 
2  :  nam  eorum  usus  in  ipso  Opere  ante  alia  occurrit.  Illi  autem,  quod  prius  fuerat  primum, 
nunc  autem  est  tertium,  accessit  in  fine  Scholium  tertium,  quod  pluribus  numeris  complec- 
titur  dissertatiunculam  integram  de  argumento,  quod  ante  aliquot  annos  in  Parisiensi 
Academia  controversise  occasionem  exhibuit  in  Encyclopedico  etiam  dictionario  attactum, 
in  qua  dissertatiuncula  demonstrat  Auctor  non  esse,  cur  ad  vim  exprimendam  potentia 
quaepiam  distantiae  adhibeatur  potius,  quam  functio. 

Accesserunt  per  totum  Opus  notulae  marginales,  in  quibus  eorum,  quae  pertractantur 
argumenta  exponuntur  brevissima,  quorum  ope  unico  obtutu  videri  possint  omnia,  &  in 
memoriam  facile  revocari. 

Postremo  loco  ad  calcem  Operis  additus  est  fusior  catalogus  eorum  omnium,  quse  hue 
usque  ab  ipso  Auctore  sunt  edita,  quorum  collectionem  omnem  expolitam,  &  correctam, 
ac  eorum,  quse  nondum  absoluta  sunt,  continuationem  meditatur,  aggressurus  illico  post 
suum  regressum  in  Urbem  Romam,  quo  properat.  Hie  catalogus  impressus  fuit  Venetisis 
ante  hosce  duos  annos  in  reimpressione  ejus  poematis  de  Solis  ac  Lunae  defectibus. 
Porro  earn,  omnium  suorum  Operum  Collectionem,  ubi  ipse  adornaverit,  typis  ego  meis 
excudendam  suscipiam,  quam  magnificentissime  potero. 

Haec  erant,  quae  te  monendum  censui ;    tu  laboribus  nostris  fruere,  &  vive  felix. 


THE  PRINTER  AT  VENICE  TO  THE  READER  5 

Among  the  more  Important  alterations  will  be  found  a  change  in  the  order  of  numbering 
the  paragraphs.  Thus,  Art.  82  is  additional  matter  that  is  entirely  new  ;  that  which  was 
formerly  Art.  261  is  now  broken  up  into  five  parts  ;  &,  in  the  Appendix,  following  Art. 
534,  both  some  slight  changes  and  also  several  additions  have  been  made  in  the  passages 
that  relate  to  the  Seat  of  the  Soul. 

The  order  of  the  Supplements  has  been  altered  also  :  those  that  were  formerly  num- 
bered III  and  IV  are  now  I  and  II  respectively.  This  was  done  because  they  are  required 
for  use  in  this  work  before  the  others.  To  that  which  was  formerly  numbered  I,  but  is 
now  III,  there  has  been  added  a  third  scholium,  consisting  of  several  articles  that  between 
them  give  a  short  but  complete  dissertation  on  that  point  which,  several  years  ago  caused 
a  controversy  in  the  University  of  Paris,  the  same  point  being  also  discussed  in  the 
Dictionnaire  Encydopedique.  In  this  dissertation  the  author  shows  that  there  is  no  reason 
why  any  one  power  of  the  distance  should  be  employed  to  express  the  force,  in  preference 
to  a  function. 

Short  marginal  summaries  have  been  inserted  throughout  the  work,  in  which  the 
arguments  dealt  with  are  given  in  brief ;  by  the  help  of  these,  the  whole  matter  may  be 
taken  in  at  a  glance  and  recalled  to  mind  with  ease. 

Lastly,  at  the  end  of  the  work,  a  somewhat  full  catalogue  of  the  whole  of  the  author's 
publications  up  to  the  present  time  has  been  added.  Of  these  publications  the  author 
intends  to  make  a  full  collection,  revised  and  corrected,  together  with  a  continuation  of 
those  that  are  not  yet  finished  ;  this  he  proposes  to  do  after  his  return  to  Rome,  for  which 
city  he  is  preparing  to  set  out.  This  catalogue  was  printed  in  Venice  a  couple  of  years  ago 
in  connection  with  a  reprint  of  his  essay  in  verse  on  the  eclipses  of  the  Sun  and  Moon. 
Later,  when  his  revision  of  them  is  complete,  I  propose  to  undertake  the  printing  of  this 
complete  collection  of  his  works  from  my  own  type,  with  all  the  sumptuousness  at  my 
command. 

Such  were  the  matters  that  I  thought  ought  to  be  brought  to  your  notice.  May  you 
enjoy  the  fruit  of  our  labours,  &  live  in  happiness. 


EPISTOLA  AUCTORIS    DEDICATORIA 

EDITIONIS   VIENNENSIS 


AD  CELSISSIMUM  TUNC  PRINCIPEM  ARCHIEPISCOPUM 

VIENNENSEM,  NUNC  PR^TEREA  ET  CARDINALEM 

EMINENTISSIMUM,    ET    EPISCOPUM    VACCIENSEM 

CHRISTOPHORUM  E  COMITATIBUS  DE  MIGAZZI 

IA.BIS  veniam,  Princeps  Celsissime,  si  forte  inter  assiduas  sacri  regirninis  curas 
importunus  interpellator  advenio,  &  libellum  Tibi  offero  mole  tenuem,  nee 
arcana  Religionis  mysteria,  quam  in  isto  tanto  constitutus  fastigio  adminis- 
tras,  sed  Naturalis  Philosophise  principia  continentem.  Novi  ego  quidem, 
quam  totus  in  eo  sis,  ut,  quam  geris,  personam  sustineas,  ac  vigilantissimi 
sacrorum  Antistitis  partes  agas.  Videt  utique  Imperialis  haec  Aula,  videt 
universa  Regalis  Urbs,  &  ingenti  admiratione  defixa  obstupescit,  qua  dili- 
gentia,  quo  labore  tanti  Sacerdotii  munus  obire  pergas.  Vetus  nimirum  illud  celeberrimum 
age,  quod  agis,  quod  ab  ipsa  Tibi  juventute,  cum  primum,  ut  Te  Romas  dantem  operam 
studiis  cognoscerem,  mihi  fors  obtigit,  altissime  jam  insederat  animo,  id  in  omni 
reliquo  amplissimorum  munerum  Tibi  commissorum  cursu  haesit  firmissime,  atque  idipsum 
inprimis  adjectum  tarn  multis  &  dotibus,  quas  a  Natura  uberrime  congestas  habes,  & 
virtutibus,  quas  tute  diuturna  Tibi  exercitatione,  atque  assiduo  labore  comparasti,  sanc- 
tissime  observatum  inter  tarn  varias  forenses,  Aulicas,  Sacerdotales  occupationes,  istos  Tibi 
tarn  celeres  dignitatum  gradus  quodammodo  veluti  coacervavit,  &  omnium  una  tarn 
populorum,  quam  Principum  admirationem  excitavit  ubique,  conciliavit  amorem  ;  unde 
illud  est  factum,  ut  ab  aliis  alia  Te,  sublimiora  semper,  atque  honorificentiora  munera 
quodammodo  velut  avulsum,  atque  abstractum  rapuerint.  Dum  Romse  in  celeberrimo  illo, 
quod  Auditorum  Rotae  appellant,  collegio  toti  Christiano  orbi  jus  diceres,  accesserat 
Hetrusca  Imperialis  Legatio  apud  Romanum  Pontificem  exercenda  ;  cum  repente  Mech- 
liniensi  Archiepiscopo  in  amplissima  ilia  administranda  Ecclesia  Adjutor  datus,  &  destinatus 
Successor,  possessione  prsestantissimi  muneris  vixdum  capta,  ad  Hispanicum  Regem  ab 
Augustissima  Romanorum  Imperatrice  ad  gravissima  tractanda  negotia  Legatus  es  missus, 
in  quibus  cum  summa  utriusque  Aulae  approbatione  versatum  per  annos  quinque  ditissima 
Vacciensis  Ecclesia  adepta  est  ;  atque  ibi  dum  post  tantos  Aularum  strepitus  ea,  qua 
Christianum  Antistitem  decet,  &  animi  moderatione,  &  demissione  quadam,  atque  in  omne 
hominum  genus  charitate,  &  singular!  cura,  ac  diligentia  Religionem  administras,  &  sacrorum 
exceres  curam  ;  non  ea  tantum  urbs,  atque  ditio,  sed  universum  Hungariae  Regnum, 
quanquam  exterum  hominem,  non  ut  civem  suum  tantummodo,  sed  ut  Parentem  aman- 
tissimum  habuit,  quern  adhuc  ereptum  sibi  dolet,  &  angitur  ;  dum  scilicet  minore,  quam 
unius  anni  intervallo  ab  Ipsa  Augustissima  Imperatrice  ad  Regalem  hanc  Urbem,  tot 
Imperatorum  sedem,  ac  Austriacae  Dominationis  caput,  dignum  tantis  dotibus  explicandis 
theatrum,  eocatum  videt,  atque  in  hac  Celsissima  Archiepiscopali  Sede,  accedente  Romani 
Pontificis  Auctoritate  collocatum  ;  in  qua  Tu  quidem  personam  itidem,  quam  agis,  diligen- 
tissime  sustinens,  totus  es  in  gravissimis  Sacerdotii  Tui  expediendis  negotiis,  in  iis  omnibus, 
quae  ad  sacra  pertinent,  curandis  vel  per  Te  ipsum  usque  adeo,  ut  saepe,  raro  admodum  per 


AUTHOR'S   EPISTLE  DEDICATING 

THE   FIRST   VIENNA    EDITION 

TO 

CHRISTOPHER,  COUNT  DE  MIGAZZI,  THEN  HIS  HIGHNESS 
THE  PRINCE  ARCHBISHOP  OF  VIENNA,  AND   NOW  ALSO 
IN  ADDITION  HIS  EMINENCE  THE  CARDINAL, 

BISHOP  OF  VACZ 


OU  will  pardon  me,  Most  Noble  Prince,  if  perchance  I  come  to  disturb  at  an 
inopportune  moment  the  unremitting  cares  of  your  Holy  Office,  &  offer 
you  a  volume  so  inconsiderable  in  size ;   one  too  that  contains  none  of  the 
inner  mysteries  of  Religion,  such  as  you  administer  from  the  highly  exalted 
position  to  which  you  are  ordained ;   one  that  merely  deals  with  the  prin- 
ciples of  Natural  Philosophy.     I  know  full  well  how  entirely  your  time  is 
taken  up  with  sustaining  the  reputation  that  you  bear,  &  in  performing 
the  duties  of  a  highly  conscientious  Prelate.     This  Imperial  Court  sees,  nay,  the  whole  of 
this  Royal  City  sees,  with  what  care,  what  toil,  you  exert  yourself  to  carry  out  the  duties  of 
so  great  a  sacred  office,  &  stands  wrapt  with  an  overwhelming  admiration.     Of  a  truth, 
that  well-known  old  saying,  "  What  you  do,  DO,"  which  from  your  earliest  youth,  when 
chance  first  allowed  me  to  make  your  acquaintance  while  you  were  studying  in  Rome,  had 
already  fixed  itself  deeply  in  your  mind,  has  remained  firmly  implanted  there  during  the 
whole  of  the  remainder  of  a  career  in  which  duties  of  the  highest  importance  have  been 
committed  to  your  care.     Your  strict  observance  of  this  maxim  in  particular,  joined  with 
those  numerous  talents  so  lavishly  showered  upon  you  by  Nature,  &  those  virtues  which 
you   have    acquired   for  yourself  by  daily  practice  &  unremitting  toil,  throughout  your 
whole  career,  forensic,  courtly,  &  sacerdotal,  has  so  to  speak  heaped  upon  your  shoulders 
those  unusually  rapid  advances  in  dignity  that  have  been  your  lot.     It  has  aroused  the 
admiration  of  all,  both  peoples  &  princes  alike,  in  every  land  ;  &  at  the  same  time  it  has 
earned  for  you  their  deep  affection.     The  consequence  was  that  one  office  after  another, 
each  ever  more  exalted  &  honourable  than  the  preceding,  has  in  a  sense  seized  upon  you 
&  borne  you  away  a  captive.     Whilst  you  were  in  Rome,  giving  judicial  decisions  to  the 
whole  Christian  world  in  that  famous  College,  the  Rota  of  Auditors,  there  was  added  the 
duty  of  acting  on  the  Tuscan  Imperial  Legation  at  the  Court  of  the  Roman  Pontiff.     Sud- 
denly you  were  appointed  coadjutor  to  the  Archbishop  of  Malines  in  the  administration  of 
that  great  church,  &  his  future  successor.     Hardly  had  you  entered  upon  the  duties  of 
that  most  distinguished  appointment,  than  you  were  despatched  by  the  August  Empress  of 
the  Romans  as  Legate  on  a  mission  of  the  greatest  importance.     You  occupied  yourself  on 
this  mission  for  the  space   of  five  years,  to  the  entire  approbation  of  both  Courts,  &  then 
the  wealthy  church  of  Vacz  obtained  your  services.     Whilst  there,  the  great  distractions  of 
a  life  at  Court  being  left  behind,  you  administer  the  offices  of  religion  &  discharge  the 
sacred  rights  with  that  moderation  of  spirit  &  humility  that  befits  a  Christian  prelate,  in 
charity  towards  the  whole  race  of  mankind,  with  a  singularly  attentive  care.     So  that  not 
only  that  city  &  the  district  in  its  see,  but  the  whole  realm  of  Hungary  as  well,  has  looked 
upon  you,  though  of  foreign  race,  as  one  of  her  own  citizens ;  nay,  rather  as  a  well  beloved 
father,  whom  she  still  mourns  &  sorrows  for,  now  that  you  have  been  taken  from  her. 
For,  after  less  than  a  year  had  passed,  she  sees  you  recalled  by  the  August  Empress  herself  to 
this  Imperial  City,  the  seat  of  a  long  line  of  Emperors,  &  the  capital  of  the  Dominions  of 
Austria,  a  worthy  stage  for  the  display  of  your  great  talents ;  she  sees  you  appointed,  under 
the  auspices  of  the  authority  of  the  Roman  Pontiff,  to  this  exalted  Archiepiscopal  see. 
Here  too,  sustaining  with  the  utmost  diligence  the  part  you  play  so  well,  you  throw  your- 
self heart  and  soul  into  the  business  of  discharging  the  weighty  duties  of  your  priesthood, 
or  in  attending  to  all  those  things  that  deal  with  the  sacred  rites  with  your  own  hands  :   so 
much  so  that  we  often  see  you  officiating,  &  even  administering  the  Sacraments,  in  our 


8    EPISTOLA  AUCTORIS  DEDICATORIA  PRI1VLE  EDITIONIS  VIENNENSIS 

haec  nostra  tempora  exemplo,  &  publico  operatum,  ac  ipsa  etiam  Sacramenta  administrantem 
videamus  in  templis,  &  Tua  ipsius  voce  populos,  e  superiore  loco  docentum  audiamus,  atque 
ad  omne  virtutum  genus  inflammantem. 

Novi  ego  quidem  haec  omnia  ;  novi  hanc  indolem,  hanc  animi  constitutionem  ;  nee 
sum  tamen  inde  absterritus,  ne,  inter  gravissimas  istas  Tuas  Sacerdotales  curas,  Philosophicas 
hasce  meditationes  meas,  Tibi  sisterem,  ac  tantulae  libellum  molis  homini  ad  tantum  culmen 
evecto  porrigerem,  ac  Tuo  vellem  Nomine  insignitum.  Quod  enim  ad  primum  pertinet 
caput,  non  Theologicas  tantum,  sed  Philosophicas  etiam  perquisitiones  Christiano  Antistite 
ego  quidem  dignissimas  esse  censeo,  &  universam  Naturae  contemplationem  omnino 
arbitror  cum  Sacerdotii  sanctitate  penitus  consentire.  Mirum  enim,  quam  belle  ab  ipsa 
consideratione  Naturae  ad  caslestium  rerum  contemplationem  disponitur  animus,  &  ad 
ipsum  Divinum  tantae  molis  Conditorem  assurgit,  infinitam  ejus  Potentiam  Sapientiam, 
Providentiam  admiratus,  quae  erumpunt  undique,  &  utique  se  produnt. 


Est  autem  &  illud,  quod  ad  supremi  sacrorum  Moderatoris  curam  pertinet  providere, 
ne  in  prima  ingenuae  juventutis  institutione,  quae  semper  a  naturalibus  studiis  exordium 
ducit,  prava  teneris  mentibus  irrepant,  ac  perniciosa  principia,  quae  sensim  Religionem 
corrumpant,  &  vero  etiam  evertant  penitus,  ac  eruant  a  fundamentis ;  quod  quidem  jam 
dudum  tristi  quodam  Europae  fato  passim  evenire  cernimus,  gliscente  in  dies  malo,  ut  fucatis 
quibusdam,  profecto  perniciosissimis,  imbuti  principiis  juvenes,  turn  demum  sibi  sapere 
videantur,  cum  &  omnem  animo  religionem,  &  Deum  ipsum  sapientissimum  Mundi 
Fabricatorem,  atque  Moderatorem  sibi  mente  excusserint.  Quamobrem  qui  veluti  ad 
tribunal  tanti  Sacerdotum  Principis  Universae  Physicae  Theoriam,  &  novam  potissimum 
Theoriam  sistat,  rem  is  quidem  praestet  sequissimam,  nee  alienum  quidpiam  ab  ejus  munere 
Sacerdotali  offerat,  sed  cum  eodem  apprime  consentiens. 


Nee  vero  exigua  libelli  moles  deterrere  me  debuit,  ne  cum  eo  ad  tantum  Principem 
accederem.  Est  ille  quidem  satis  tenuis  libellus,  at  non  &  tenuem  quoque  rem  continet. 
Argumentum  pertractat  sublime  admodum,  &  nobile,  in  quo  illustrando  omnem  ego  quidem 
industriam  coUocavi,  ubi  si  quid  praestitero,  si  minus  infiliclter  me  gessero,  nemo  sane  me 
impudentiae  arguat,  quasi  vilem  aliquam,  &  tanto  indignam  fastigio  rem  offeram.  Habetur 
in  eo  novum  quoddam  Universae  Naturalis  Philosophiae  genus  a  receptis  hue  usque,  usi- 
tatisque  plurimam  discrepans,  quanquam  etiam  ex  iis,  quae  maxime  omnium  per  haec  tempora 
celebrantur,  casu  quodam  praecipua  quasque  mirum  sane  in  modum  compacta,  atque  inter 
se  veluti  coagmentata  conjunguntur  ibidem,  uti  sunt  simplicia  atque  inextensa  Leibnitian- 
orum  elementa,  cum  Newtoni  viribus  inducentibus  in  aliis  distantiis  accessum  mutuum,  in 
aliis  mutuum  recessum,  quas  vulgo  attractiones,  &  repulsiones  appellant  :  casu,  inquam  : 
neque  enim  ego  conciliandi  studio  hinc,  &  inde  decerpsi  quaedam  ad  arbitrium  selecta,  quae 
utcumque  inter  se  componerem,  atque  compaginarem  :  sed  omni  praejudicio  seposito,  a 
principiis  exorsus  inconcussis,  &  vero  etiam  receptis  communiter,  legitima  ratiocinatione 
usus,  &  continue  conclusionum  nexu  deveni  ad  legem  virium  in  Natura  existentium  unicam, 
simplicem ,  continuam,  quae  mihi  &  constitutionem  elementorum  materiae,  &  Mechanicae 
leges,  &  generales  materiae  ipsius  proprietates,  &  praecipua  corporum  discrimina,  sua 
applicatione  ita  exhibuit,  ut  eadem  in  iis  omnibus  ubique  se  prodat  uniformis  agendi  ratio, 
non  ex  arbitrariis  hypothesibus,  &  fictitiis  commentationibus,  sed  ex  sola  continua  ratio- 
cinatione deducta.  Ejusmodi  autem  est  omnis,  ut  eas  ubique  vel  definiat,  vel  adumbret 
combinationes  elementorum,  quae  ad  diversa  prasstanda  phaenomena  sunt  adhibendas,  ad 
quas  combinationes  Conditoris  Supremi  consilium,  &  immensa  Mentis  Divinae  vis  ubique 
requiritur,  quae  infinites  casus  perspiciat,  &  ad  rem  aptissimos  seligat,  ac  in  Naturam 
inducat. 


Id  mihi  quidem  argumentum  est  operis,  in  quo  Theoriam  meam  expono,  comprobo, 
vindico  :  turn  ad  Mechanicam  primum,  deinde  ad  Physicam  applico,  &  uberrimos  usus 
expono,  ubi  brevi  quidem  libello,  sed  admodum  diuturnas  annorum  jam  tredecim  medita- 
tiones complector  meas,  eo  plerumque  tantummodo  rem  deducens,  ubi  demum  cum 


AUTHOR'S    EPISTLE  DEDICATING  THE   FIRST  VIENNA  EDITION       9 

churches  (a  somewhat  unusual  thing  at  the  present  time),  and  also  hear  you  with  your  own 
voice  exhorting  the  people  from  your  episcopal  throne,  &  inciting  them  to  virtue  of 
every  kind. 

I  am  well  aware  of  all  this ;  I  know  full  well  the  extent  of  your  genius,  &  your  con- 
stitution of  mind  ;  &  yet  I  am  not  afraid  on  that  account  of  putting  into  your  hands, 
amongst  all  those  weighty  duties  of  your  priestly  office,  these  philosophical  meditations  of 
mine  ;  nor  of  offering  a  volume  so  inconsiderable  in  bulk  to  one  who  has  attained  to  such 
heights  of  eminence  ;  nor  of  desiring  that  it  should  bear  the  hall-mark  of  your  name.  With 
regard  to  the  first  of  these  heads,  I  think  that  not  only  theological  but  also  philosophical 
investigations  are  quite  suitable  matters  for  consideration  by  a  Christian  prelate  ;  &  in 
my  opinion,  a  contemplation  of  all  the  works  of  Nature  is  in  complete  accord  with  the 
sanctity  of  the  priesthood.  For  it  is  marvellous  how  exceedingly  prone  the  mind  becomes 
to  pass  from  a  contemplation  of  Nature  herself  to  the  contemplation  of  celestial,  things,  & 
to  give  honour  to  the  Divine  Founder  of  such  a  mighty  structure,  lost  in  astonishment  at 
His  infinite  Power  &  Wisdom  &  Providence,  which  break  forth  &  disclose  themselves 
in  all  directions  &  in  all  things. 

There  is  also  this  further  point,  that  it  is  part  of  the  duty  of  a  religious  superior  to  take 
care  that,  in  the  earliest  training  of  ingenuous  youth,  which  always  takes  its  start  from  the 
study  of  the  wonders  of  Nature,  improper  ideas  do  not  insinuate  themselves  into  tender 
minds ;  or  such  pernicious  principles  as  may  gradually  corrupt  the  belief  in  things  Divine, 
nay,  even  destroy  it  altogether,  &  uproot  it  from  its  very  foundations.  This  is  what  we 
have  seen  for  a  long  time  taking  place,  by  some  unhappy  decree  of  adverse  fate,  all  over 
Europe ;  and,  as  the  canker  spreads  at  an  ever  increasing  rate,  young  men,  who  have  been 
made  to  imbibe  principles  that  counterfeit  the  truth  but  are  actually  most  pernicious  doc- 
trines, do  not  think  that  they  have  attained  to  wisdom  until  they  have  banished  from  their 
minds  all  thoughts  of  religion  and  of  God,  the  All- wise  Founder  and  Supreme  Head  of  the 
Universe.  Hence,  one  who  so  to  speak  sets  before  the  judgment-seat  of  such  a  prince  of 
the  priesthood  as  yourself  a  theory  of  general  Physical  Science,  &  more  especially  one  that 
is  new,  is  doing  nothing  but  what  is  absolutely  correct.  Nor  would  he  be  offering  him 
anything  inconsistent  with  his  priestly  office,  but  on  the  contrary  one  that  is  in  complete 
harmony  with  it. 

Nor,  secondly,  should  the  inconsiderable  size  of  my  little  book  deter  me  from  approach- 
ing with  it  so  great  a  prince.  It  is  true  that  the  volume  of  the  book  is  not  very  great,  but 
the  matter  that  it  contains  is  not  unimportant  as  well.  The  theory  it  develops  is  a  strik- 
ingly sublime  and  noble  idea  ;  &  I  have  done  my  very  best  to  explain  it  properly.  If  in 
this  I  have  somewhat  succeeded,  if  I  have  not  failed  altogether,  let  no  one  accuse  me  of 
presumption,  as  if  I  were  offering  some  worthless  thing,  something  unworthy  of  such  dis- 
tinguished honour.  In  it  is  contained  a  new  kind  of  Universal  Natural  Philosophy,  one  that 
differs  widely  from  any  that  are  generally  accepted  &  practised  at  the  present  time  ; 
although  it  so  happens  that  the  principal  points  of  all  the  most  distinguished  theories  of  the 
present  day,  interlocking  and  as  it  were  cemented  together  in  a  truly  marvellous  way,  are 
combined  in  it ;  so  too  are  the  simple  unextended  elements  of  the  followers  of  Leibniz, 
as  well  as  the  Newtonian  forces  producing  mutual  approach  at 'some  distances  &  mutual 
separation  at  others,  usually  called  attractions  and  repulsions.  I  use  the  words  "  it  so 
happens  "  because  I  have  not,  in  eagerness  to  make  the  whole  consistent,  selected  one  thing 
here  and  another  there,  just  as  it  suited  me  for  the  purpose  of  making  them  agree  &  form 
a  connected  whole.  On  the  contrary,  I  put  on  one  side  all  prejudice,  &  started  from 
fundamental  principles  that  are  incontestable,  &  indeed  are  those  commonly  accepted  ;  I 
used  perfectly  sound  arguments,  &  by  a  continuous  chain  of  deduction  I  arrived  at  a 
single,  simple,  continuous  law  for  the  forces  that  exist  in  Nature.  The  application  of  this 
law  explained  to  me  the  constitution  of  the  elements  of  matter,  the  laws  of  Mechanics,  the 
general  properties  of  matter  itself,  &  the  chief  characteristics  of  bodies,  in  such  a  manner 
that  the  same  uniform  method  of  action  in  all  things  disclosed  itself  at  all  points ;  being 
deduced,  not  from  arbitrary  hypotheses,  and  fictitibus  explanations,  but  from  a  single  con- 
tinuous chain  of  reasoning.  Moreover  it  is  in  all  its  parts  of  such  a  kind  as  defines,  or 
suggests,  in  every  case,  the  combinations  of  the  elements  that  must  be  employed  to  produce 
different  phenomena.  For  these  combinations  the  wisdom  of  the  Supreme  Founder  of  the 
Universe,  &  the  mighty  power  of  a  Divine  Mind  are  absolutely  necessary ;  naught  but 
one  that  could  survey  the  countless  cases,  select  those  most  suitable  for  the  purpose,  and 
introduce  them  into  the  scheme  of  Nature. 

This  then  is  the  argument  of  my  work,  in  which  I  explain,  prove  &  defend  my  theory  ; 
then  I  apply  it,  in  the  first  instance  to  Mechanics,  &  afterwards  to  Physics,  &  set  forth 
the  many  advantages  to  be  derived  from  it.  Here,  although  the  book  is  but  small,  I  yet 
include  the  well-nigh  daily  meditations  of  the  last  thirteen  years,  carrying  on  my  conclu- 


io    EPISTOLA  AUCTORIS  DEDICATORIA  PRIM.£  EDITIONIS  VIENNENSIS 

communibus  Philosophorum  consentio  placitis,  &  ubi  ea,  quae  habemus  jam  pro  compertis, 
ex  meis  etiam  deductionibus  sponte  fluunt,  quod  usque  adeo  voluminis  molem  contraxit. 
Dederam  ego  quidem  dispersa  dissertatiunculis  variis  Theorise  meae  qusedam  velut  specimina, 
quae  inde  &  in  Italia  Professores  publicos  nonnullos  adstipulatores  est  nacta,  &  jam  ad 
exteras  quoque  gentes  pervasit ;  sed  ea  nunc  primum  tota  in  unum  compacta,  &  vero  etiam 
plusquam  duplo  aucta,  prodit  in  publicum,  quern  laborem  postremo  hoc  mense,  molestiori- 
bus  negotiis,  quae  me  Viennam  adduxerant,  &  curis  omnibus  exsolutus  suscepi,  dum  in 
Italiam  rediturus  opportunam  itineri  tempus  inter  assiduas  nives  opperior,  sed  omnem  in 
eodem  adornando,  &  ad  communem  mediocrum  etiam  Philosophorum  captum  accommo- 
dando  diligentiam  adhibui. 


Inde  vero  jam  facile  intelliges,  cur  ipsum  laborem  meum  ad  Te  deferre,  &  Tuo 
nuncupare  Nomini  non  dubitaverim.  Ratio  ex  iis,  quae  proposui,  est  duplex  :  primo  quidem 
ipsum  argumenti  genus,  quod  Christianum  Antistitem  non  modo  non  dedecet,  sed  etiam 
apprime  decet  :  turn  ipsius  argumenti  vis,  atque  dignitas,  quae  nimirum  confirmat,  &  erigit 
nimium  fortasse  impares,  sed  quantum  fieri  per  me  potuit,  intentos  conatus  meos ;  nam 
quidquid  eo  in  genere  meditando  assequi  possum,  totum  ibidem  adhibui,  ut  idcirco  nihil 
arbitrer  a  mea  tenuitate  proferri  posse  te  minus  indignum,  cui  ut  aliquem  offerrem  laborum 
meorum  fructum  quantumcunque,  exposcebat  sane,  ac  ingenti  clamore  quodam  efnagitabat 
tanta  erga  me  humanitas  Tua,  qua  jam  olim  immerentem  complexus  Romae,  hie  etiam 
fovere  pergis,  nee  in  tanto  dedignatus  fastigio,  omni  benevolentiae  significatione  prosequeris. 
Accedit  autem  &  illud,  quod  in  hisce  terris  vix  adhuc  nota,  vel  etiam  ignota  penitus  Theoria 
mea  Patrocinio  indiget,  quod,  si  Tuo  Nomine  insignata  prodeat  in  publicum,  obtinebit  sane 
validissimum,  &  secura  vagabitur  :  Tu  enim  illam,  parente  velut  hie  orbatam  suo,  in  dies 
nimirum  discessuro,  &  quodammodo  veluti  posthumam  post  ipsum  ejus  discessum  typis 
impressam,  &  in  publicum  prodeuntem  tueberis,  fovebisque. 


Haec  sunt,  quae  meum  Tibi  consilium  probent,  Princeps  Celsissime  :  Tu,  qua  soles 
humanitate  auctorem  excipere,  opus  excipe,  &  si  forte  adhuc  consilium  ipsum  Tibi  visum 
fuerit  improbandum ;  animum  saltern  aequus  respice  obsequentissimum  Tibi,  ac  devinct- 
issimum.  Vale. 

Dabam  Viennce  in  Collegia  Academico  Soc.  JESU 
Idibus  Febr.  MDCCLFIIL 


AUTHOR'S  EPISTLE  DEDICATING  THE  FIRST  VIENNA  EDITION      11 

sions  for  the  most  part  only  up  to  the  point  where  I  finally  agreed  with  the  opinions  com- 
monly held  amongst  philosophers,  or  where  theories,  now  accepted  as  established,  are  the 
natural  results  of  my  deductions  also  ;  &  this  has  in  some  measure  helped  to  diminish  the 
size  of  the  volume.  I  had  already  published  some  instances,  so  to  speak,  of  my  general 
theory  in  several  short  dissertations  issued  at  odd  times ;  &  on  that  account  the  theory 
has  found  some  supporters  amongst  the  university  professors  in  Italy,  &  has  already  made 
its  way  into  foreign  countries.  But  now  for  the  first  time  is  it  published  as  a  whole  in  a 
single  volume,  the  matter  being  indeed  more  than  doubled  in  amount.  This  work  I  have 
carried  out  during  the  last  month,  being  quit  of  the  troublesome  business  that  brought  me 
to  Vienna,  and  of  all  other  cares ;  whilst  I  wait  for  seasonable  time  for  my  return  journey 
through  the  everlasting  snow  to  Italy.  I  have  however  used  my  utmost  endeavours  in 
preparing  it,  and  adapting  it  to  the  ordinary  intelligence  of  philosophers  of  only  moderate 
attainments. 

From  this  you  will  readily  understand  why  I  have  not  hesitated  to  bestow  this  book 
of  mine  upon  you,  &  to  dedicate  it  to  you.  My  reason,  as  can  be  seen  from  what  I  have 
said,  was  twofold  ;  in  the  first  place,  the  nature  of  my  theme  is  one  that  is  not  only  not 
unsuitable,  but  is  suitable  in  a  high  degree,  for  the  consideration  of  a  Christian  priest ; 
secondly,  the  power  &  dignity  of  the  theme  itself,  which  doubtless  gives  strength  & 
vigour  to  my  efforts — perchance  rather  feeble,  but,  as  far  as  in  me  lay,  earnest.  What- 
ever in  that  respect  I  could  gain  by  the  exercise  of  thought,  I  have  applied  the  whole  of  it 
to  this  matter  ;  &  consequently  I  think  that  nothing  less  unworthy  of  you  can  be  pro- 
duced by  my  poor  ability ;  &  that  I  should  offer  to  you  some  such  fruit  of  my  labours 
was  surely  required  of  me,  &  as  it  were  clamorously  demanded  by  your  great  kindness 
to  me ;  long  ago  in  Rome  you  had  enfolded  my  unworthy  self  in  it,  &  here  now  you 
continue  to  be  my  patron,  &  do  not  disdain,  from  your  exalted  position,  to  honour  me 
with  every  mark  of  your  goodwill.  There  is  still  a  further  consideration,  namely,  that  my 
Theory  is  as  yet  almost,  if  not  quite,  unknown  in  these  parts,  &  therefore  needs  a  patron's 
support ;  &  this  it  will  obtain  most  effectually,  &  will  go  on  its  way  in  security  if  it 
comes  before  the  public  franked  with  your  name.  For  you  will  protect  &  cherish  it, 
on  its  publication  here,  bereaved  as  it  were  of  that  parent  whose  departure  in  truth  draws 
nearer  every  day ;  nay  rather  posthumous,  since  it  will  be  seen  in  print  only  after  he  has 
gone. 

Such  are  my  grounds  for  hoping  that  you  will  approve  my  idea,  most  High  Prince. 
I  beg  you  to  receive  the  work  with  the  same  kindness  as  you  used  to  show  to  its  author ; 
&,  if  perchance  the  idea  itself  should  fail  to  meet  with  your  approval,  at  least  regard 
favourably  the  intentions  of  your  most  humble  &  devoted  servant.  Farewell. 

University  College  of  the  Society  of  Jesus, 
VIENNA, 

February  i$th,  1758. 


AD    LECTOREM 

EX   EDITIONS   VIENNENSI 

amice  Lector,  Philosophic  Naturalis  Theoriam  ex  unica  lege  virium 
deductam,  quam  &  ubi  jam  olim  adumbraverim,  vel  etiam  ex  parte  explica- 
verim,  y  qua  occasione  nunc  uberius  pertractandum,  atque  augendam  etiam, 
susceperim,  invenies  in  ipso  -primes  •partis  exordia.  Libuit  autem  hoc  opus 
dividere  in  partes  tres,  quarum  prima  continet  explicationem  Theories  ipsius, 
ac  ejus  analyticam  deductionem,  &  vindicationem  :  secunda  applicationem- 
satis  uberem  ad  Mechanicam  ;  tertia  applicationem  ad  Physicam. 

Porro  illud  inprimis  curandum  duxi,  ut  omnia,  quam  liceret,  dilucide  exponerentur,  nee 
sublimiore  Geometria,  aut  Calculo  indigerent.  Et  quidem  in  prima,  ac  tertia  parte  non  tantum 
nullcs  analyticee,  sed  nee  geometries  demonstrations  occurrunt,  paucissimis  qiiibusdam,  quibus 
indigeo,  rejectis  in  adnotatiunculas,  quas  in  fine  paginarum  quarundam  invenies.  Queedam 
autem  admodum  pauca,  quce  majorem  Algebra,  &  Geometries  cognitionem  requirebant,  vel  erant 
complicatiora  aliquando,  &  alibi  a  me  jam  edita,  in  fine  operis  apposui,  quce  Supplementorum 
appellavi  nomine,  ubi  W  ea  addidi,  quce  sentio  de  spatio,  ac  tempore,  Theories  mece  consentanea, 
ac  edita  itidem  jam  alibi.  In  secunda  parte,  ubi  ad  Mechanicam  applicatur  Theoria,a  geome- 
tricis,  W  aliquando  etiam  ab  algebraicis  demonstrationibus  abstinere  omnino  non  potui  ;  sed 
ece  ejusmodi  sunt,  ut  vix  unquam  requirant  aliud,  quam  Euclideam  Geometriam,  &  primas 
Trigonometries  notiones  maxime  simplices,  ac  simplicem  algorithmum. 


In  prima  quidem  parte  occurrunt  Figures  geometricce  complures,  quce  prima  fronte  vide- 
buntur  etiam  complicate?  rem  ipsam  intimius  non  perspectanti  ;  verum  ece  nihil  aliud  exhibent, 
nisi  imaginem  quandam  rerum,  quce  ipsis  oculis  per  ejusmodi  figuras  sistuntur  contemplandce. 
Ejusmodi  est  ipsa  ilia  curva,  quce  legem  virium  exhibet.  Invenio  ego  quidem  inter  omnia 
materice  puncta  vim  quandam  mutuam,  quce  a  distantiis  pendet,  £5"  mutatis  distantiis  mutatur 
ita,  ut  in  aliis  attractiva  sit,  in  aliis  repulsiva,  sed  certa  quadam,  y  continua  lege.  Leges 
ejusmodi  variationis  binarum  quantitatum  a  se  invicem  pendentium,  uti  Jiic  sunt  distantia, 
y  vis,  exprimi  possunt  vel  per  analyticam  formulam,  vel  per  geometricam  curvam  ;  sed  ilia 
prior  expressio  &  multo  plures  cognitiones  requirit  ad  Algebram  pertinentes,  &  imaginationem 
non  ita  adjuvat,  ut  heec  posterior,  qua  idcirco  sum  usus  in  ipsa  prima  operis  parte,  rejecta  in 
Supplementa  formula  analytica,  quce  y  curvam,  &  legem  virium  ab  ilia  expressam  exhibeat. 


Porro  hue  res  omnis  reducitur.  Habetur  in  recta  indefinita,  quce  axis  dicitur,  punctum 
quoddam,  a  quo  abscissa  ipsius  rectce  segmenta  referunt  distantias.  Curva  linea  protenditur 
secundum  rectam  ipsam,  circa  quam  etiam  serpit,  y  eandem  in  pluribus  secat  punctis :  rectce 
a  fine  segmentorum  erectce  perpendiculariter  usque  ad  curvam,  exprimunt  vires,  quce  majores 
sunt,  vel  minores,  prout  ejusmodi  rectce  sunt  itidem  majores,  vel  minores  ;  ac  eesdem  ex  attrac- 
tivis  migrant  in  repulsivis,  vel  vice  versa,  ubi  illce  ipsce  perpendiculares  rectce  directionem 
mutant,  curva  ab  alter  a  axis  indefiniti  plaga  migrante  ad  alter  am.  Id  quidem  nullas  requirit 
geometricas  demonstrations,  sed  meram  cognitionem  vocum  quarundam,  quce  vel  ad  prima  per- 
tinent Geometries  elementa,  y  notissimce  sunt,  vel  ibi  explicantur,  ubi  adhibentur.  Notissima 
autem  etiam  est  significatio  vocis  Asymptotus,  unde  &  crus  asymptoticum  curvce  appellatur  ; 
dicitur  nimirum  recta  asymptotus  cruris  cujuspiam  curvce,  cum  ipsa  recta  in  infinitum  producta, 
ita  ad  curvilineum  arcum  productum  itidem  in  infinitum  semper  accedit  magis,  ut  distantia 
minuatur  in  infinitum,  sed  nusquam  penitus  evanescat,  illis  idcirco  nunquam  invicem  con- 
venientibus. 


Consider atio  porro  attenta  curvce  propositce  in  Fig.  I,  &rationis,  qua  per  illam  exprimitur 

12 


THE   PREFACE   TO   THE   READER 

THAT  APPEARED   IN  THE  VIENNA  EDITION 

EAR  Reader,  you  have  before  you  a  Theory  of  Natural  Philosophy  deduced 
from  a  single  law  of  Forces.  You  will  find  in  the  opening  paragraphs  of 
the  first  section  a  statement  as  to  where  the  Theory  has  been  already 
published  in  outline,  &  to  a  certain  extent  explained  ;  &  also  the  occasion 
that  led  me  to  undertake  a  more  detailed  treatment  &  enlargement  of  it. 
For  I  have  thought  fit  to  divide  the  work  into  three  parts ;  the  first  of 
these  contains  the  exposition  of  the  Theory  itself,  its  analytical  deduction 
&  its  demonstration  ;  the  second  a  fairly  full  application  to  Mechanics ;  &  the  third  an 
application  to  Physics. 

The  most  important  point,  I  decided,  was  for  me  to  take  the  greatest  care  that  every- 
thing, as  far  as  was  possible,  should  be  clearly  explained,  &  that  there  should  be  no  need  for 
higher  geometry  or  for  the  calculus.  Thus,  in  the  first  part,  as  well  as  in  the  third,  there 
are  no  proofs  by  analysis ;  nor  are  there  any  by  geometry,  with  the  exception  of  a  very  few 
that  are  absolutely  necessary,  &  even  these  you  will  find  relegated  to  brief  notes  set  at  the 
foot  of  a  page.  I  have  also  added  some  very  few  proofs,  that  required  a  knowledge  of 
higher  algebra  &  geometry,  or  were  of  a  rather  more  complicated  nature,  all  of  which  have 
been  already  published  elsewhere,  at  the  end  of  the  work ;  I  have  collected  these  under 
the  heading  Supplements  ;  &  in  them  I  have  included  my  views  on  Space  &  Time,  which 
are  in  accord  with  my  main  Theory,  &  also  have  been  already  published  elsewhere.  In 
the  second  part,  where  the  Theory  is  applied  to  Mechanics,  I  have  not  been  able  to  do 
without  geometrical  proofs  altogether  ;  &  even  in  some  cases  I  have  had  to  give  algebraical 
proofs.  But  these  are  of  such  a  simple  kind  that  they  scarcely  ever  require  anything  more 
than  Euclidean  geometry,  the  first  and  most  elementary  ideas  of  trigonometry,  and  easy 
analytical  calculations. 

It  is  true  that  in  the  first  part  there  are  to  be  found  a  good  many  geometrical  diagrams, 
which  at  first  sight,  before  the  text  is  considered  more  closely,  will  appear  to  be  rather 
complicated.  But  these  present  nothing  else  but  a  kind  of  image  of  the  subjects  treated, 
which  by  means  of  these  diagrams  are  set  before  the  eyes  for  contemplation.  The  very 
curve  that  represents  the  law  of  forces  is  an  instance  of  this.  I  find  that  between  all  points 
of  matter  there  is  a  mutual  force  depending  on  the  distance  between  them,  &  changing  as 
this  distance  changes ;  so  that  it  is  sometimes  attractive,  &  sometimes  repulsive,  but  always 
follows  a  definite  continuous  law.  Laws  of  variation  of  this  kind  between  two  quantities 
depending  upon  one  another,  as  distance  &  force  do  in  this  instance,  may  be  represented 
either  by  an  analytical  formula  or  by  a  geometrical  curve ;  but  the  former  method  of 
representation  requires  far  more  knowledge  of  algebraical  processes,  &  does  not  assist  the 
imagination  in  the  way  that  the  latter  does.  Hence  I  have  employed  the  latter  method  in 
the  first  part  of  the  work,  &  relegated  to  the  Supplements  the  analytical  formula  which 
represents  the  curve,  &  the  law  of  forces  which  the  curve  exhibits. 

The  whole  matter  reduces  to  this.  In  a  straight  line  of  indefinite  length,  which  is 
called  the  axis,  a  fixed  point  is  taken ;  &  segments  of  the  straight  line  cut  off  from  this 
point  represent  the  distances.  A  curve  is  drawn  following  the  general  direction  of  this 
straight  line,  &  winding  about  it,  so  as  to  cut  it  in  several  places.  Then  perpendiculars  that 
are  drawn  from  the  ends  of  the  segments  to  meet  the  curve  represent  the  forces ;  these 
forces  are  greater  or  less,  according  as  such  perpendiculars  are  greater  or  less  ;  &  they  pass 
from  attractive  forces  to  repulsive,  and  vice  versa,  whenever  these  perpendiculars  change 
their  direction,  as  the  curve  passes  from  one  side  of  the  axis  of  indefinite  length  to  the  other 
side  of  it.  Now  this  requires  no  geometrical  proof,  but  only  a  knowledge  of  certain  terms, 
which  either  belong  to  the  first  elementary  principles  of  "geometry,  &  are  thoroughly  well 
known,  or  are  such  as  can  be  defined  when  they  are  used.  The  term  Asymptote  is  well 
known,  and  from  the  same  idea  we  speak  of  the  branch  of  a  curve  as  being  asymptotic  ; 
thus  a  straight  line  is  said  to  be  the  asymptote  to  any  branch  of  a  curve  when,  if  the  straight 
line  is  indefinitely  produced,  it  approaches  nearer  and  nearer  to  the  curvilinear  arc  which 
is  also  prolonged  indefinitely  in  such  manner  that  the  distance  between  them  becomes 
indefinitely  diminished,  but  never  altogether  vanishes,  so  that  the  straight  line  &  the  curve 
never  really  meet. 

A  careful  consideration  of  the  curve  given  in  Fig.  I,  &  of  the  way  in  which  the  relation 


14  AD  LECTOREM  EX  EDITIONE  VIENNENSI 

nexus  inter  vires,  y  distantias,  est  utique  admodum  necessaria  ad  intelligendam  Theoriam  ipsam, 
cujus  ea  est  prcecipua  qucedam  veluti  clavis,  sine  qua  omnino  incassum  tentarentur  cetera  ;  sed 
y  ejusmodi  est,  ut  tironum,  &  sane  etiam  mediocrium,  immo  etiam  longe  infra  mediocritatem 
collocatorum,  captum  non  excedat,  potissimum  si  viva  accedat  Professoris  vox  mediocriter  etiam 
versati  in  Mechanica,  cujus  ope,  pro  certo  habeo,  rem  ita  patentem  omnibus  reddi  posse,  ut 
ii  etiam,  qui  Geometric  penitus  ignari  sunt,  paucorum  admodum  explicatione  vocabulorum 
accidente,  earn  ipsis  oculis  intueantur  omnino  perspicuam. 

In  tertia  parte  supponuntur  utique  nonnulla,  quce  demonstrantur  in  secunda  ;  sed  ea  ipsa 
sunt  admodum  pauca,  &  Us,  qui  geometricas  demonstrationes  fastidiunt,  facile  admodum  exponi 
possunt  res  ipsce  ita,  ut  penitus  etiam  sine  ullo  Geometries  adjumento  percipiantur,  quanquam 
sine  Us  ipsa  demonstratio  baberi  non  poterit ;  ut  idcirco  in  eo  differre  debeat  is,  qui  secundam 
partem  attente  legerit,  &  Geometriam  calleat,  ab  eo,  qui  earn  omittat,  quod  ille  primus  veritates 
in  tertia  parte  adhibitis,  ac  ex  secunda  erutas,  ad,  explicationem  Physicce,  intuebitur  per  evi- 
dentiam  ex  ipsis  demonstrationibus  haustam,  hie  secundus  easdem  quodammodo  per  fidem  Geo- 
metris  adhibitam  credet.  Hujusmodi  inprimis  est  illud,  particulam  compositam  ex  punctis 
etiam  homogeneis,  prceditis  lege  virium  proposita,  posse  per  solam  diversam  ipsorum  punctorum 
dispositionem  aliam  particulam  per  certum  intervallum  vel  perpetuo  attrahere,  vel  perpetuo 
repellere,  vel  nihil  in  earn  agere,  atque  id  ipsum  viribus  admodum  diversis,  y  quce  respectu  diver- 
sarum  particularum  diver  see  sint,  &  diver  see  respectu  partium  diver sarum  ejusdem  particulce, 
ac  aliam  particulam  alicubi  etiam  urgeant  in  latus,  unde  plurium  phcenomenorum  explicatio  in 
Physica  sponte  fluit. 


Verum  qui  omnem  Theories,  y  deductionum  compagem  aliquanto  altius  inspexerit,  ac 
diligentius  perpenderit,  videbit,  ut  spero,  me  in  hoc  perquisitionis  genere  multo  ulterius 
progressum  esse,  quam  olim  Newtonus  ipse  desideravit.  Is  enim  in  postremo  Opticce  questione 
prolatis  Us,  quce  per  vim  attractivam,  &  vim  repulsivam,  mutata  distantia  ipsi  attractive  suc- 
cedentem,  explicari  poterant,  hcec  addidit :  "  Atque  hcec  quidem  omnia  si  ita  sint,  jam  Natura 
universa  valde  erit  simplex,  y  consimilis  sui,  perficiens  nimirum  magnos  omnes  corporum 
ccelestium  motus  attractione  gravitatis,  quce  est  mutua  inter  corpora  ilia  omnia,  &  minores  fere 
omnes  particularum  suarum  motus  alia  aliqua  vi  attrahente,  &  repellente,  qua  est  inter  particulas 
illas  mutua"  Aliquanto  autem  inferius  de  primigeniis  particulis  agens  sic  habet :  "  Porro 
videntur  mihi  hce  particulce  primigenice  non  modo  in  se  vim  inertice  habere,  motusque  leges  passivas 
illas,  quce  ex  vi  ista  necessario  oriuntur  ;  verum  etiam  motum  perpetuo  accipere  a  certis  principiis 
actuosis,  qualia  nimirum  sunt  gravitas,  £ff  causa  fermentationis,  &  cohcerentia  corporum.  Atque 
hcec  quidem  principia  considero  non  ut  occultas  qualitates,  quce  ex  specificis  rerum  formis  oriri 
fingantur,  sed  ut  universales  Naturce  leges,  quibus  res  ipsce  sunt  formatce.  Nam  principia 
quidem  talia  revera  existere  ostendunt  phenomena  Naturce,  licet  ipsorum  causce  quce  sint, 
nondum  fuerit  explicatum.  Affirmare,  singulas  rerum  species  specificis  prceditas  esse  qualita- 
tibus  occultis,  per  quas  eae  vim  certam  in  agenda  habent,  hoc  utique  est  nihil  dicere :  at  ex 
phcenomenis  Naturce  duo,  vel  tria  derivare  generalia  motus  principia,  &  deinde  explicare, 
quemadmodum  proprietates,  &  actiones  rerum  corporearum  omnium  ex  istis  principiis  conse- 
quantur,  id  vero  magnus  esset  factus  in  Philosophia  progressus,  etiamsi  principiorum  istorum 
causce  nondum  essent  cognitce.  Quare  motus  principia  supradicta  proponere  non  dubito,  cum 
per  Naturam  universam  latissime  pateant" 


Hcec  ibi  Newtonus,  ubi  is  quidem  magnos  in  Philosophia  progressus  facturum  arbitratus 
est  eum,  qui  ad  duo,  vel  tria  generalia  motus  principia  ex  Naturce  phcenomenis  derivata  pheeno- 
menorum  explicationem  reduxerit,  &  sua  principia  protulit,  ex  quibus  inter  se  diversis  eorum 
aliqua  tantummodo  explicari  posse  censuit.  Quid  igitur,  ubi  tf?  ea  ipsa  tria,  &  alia  prcecipua 
quceque,  ut  ipsa  etiam  impenetrabilitas,  y  impulsio  reducantur  ad  principium  unicum  legitima 
ratiocinatione  deductum  ?  At  id -per  meam  unicam,  &  simplicem  virium  legemprcestari,  patebit 
sane  consideranti  operis  totius  Synopsim  quandam,  quam  hie  subjicio  ;  sed  multo  magis  opus 
ipsum  diligentius  pervolventi. 


THE   PRINTER   AT   VENICE 

TO 

THE    READER 

\  OU  will  be  well  aware,  if  you  have  read  the  public  journals,  with  what  applause 
the  work  which  I  now  offer  to  you  has  been  received  throughout  Europe 
since  its  publication  at  Vienna  five  years  ago.  Not  to  mention  others,  if 
you  refer  to  the  numbers  of  the  Berne  Journal  for  the  early  part  of  the 
year  1761,  you  will  not  fail  to  see  how  highly  it  has  been  esteemed.  It 
contains  an  entirely  new  system  of  Natural  Philosophy,  which  is  already 
commonly  known  as  the  Boscovichian  theory,  from  the  name  of  its  author, 
As  a  matter  of  fact,  it  is  even  now  a  subject  of  public  instruction  in  several  Universities  in 
different  parts  ;  it  is  expounded  not  only  in  yearly  theses  or  dissertations,  both  printed  & 
debated  ;  but  also  in  several  elementary  books  issued  for  the  instruction  of  the  young  it  is 
introduced,  explained,  &  by  many  considered  as  their  original.  Any  one,  however,  who 
wishes  to  obtain  more  detailed  insight  into  the  whole  structure  of  the  theory,  the  close 
relation  that  its  several  parts  bear  to  one  another,  or  its  great  fertility  &  wide  scope  for 
the  purpose  of  deriving  the  whole  of  Nature,  in  her  widest  range,  from  a  single  simple  law 
of  forces ;  any  one  who  wishes  to  make  a  deeper  study  of  it  must  perforce  study  the  work 
here  offered. 

All  these  considerations  had  from  the  first  moved  me  to  undertake  a  new  edition  of 
the  work  ;  in  addition,  there  was  the  fact  that  I  perceived  that  it  would  be  a  matter  of  some 
difficulty  for  copies  of  the  Vienna  edition  to  pass  beyond  the  confines  of  Germany — indeed, 
at  the  present  time,  no  matter  how  diligently  they  are  inquired  for,  they  are  to  be  found 
on  sale  nowhere,  or  scarcely  anywhere,  in  the  rest  of  Europe.  The  system  had  its  birth  in 
Italy,  &  its  outlines  had  already  been  sketched  by  the  author  in  several  dissertations  pub- 
lished here  in  our  own  land  ;  though,  as  luck  would  have  it,  the  system  itself  was  finally 
put  into  shape  and  published  at  Vienna,  whither  he  had  gone  for  a  short  time.  I  therefore 
thought  it  right  that  it  should  be  disseminated  throughout  the  whole  of  Europe,  &  that 
preferably  as  the  product  of  an  Italian  press.  I  had  in  fact  already  commenced  an  edition 
founded  on  a  copy  of  the  Vienna  edition,  when  it  came  to  my  knowledge  that  the  author 
was  greatly  dissatisfied  with  the  Vienna  edition,  taken  in  hand  there  after  his  departure ; 
that  innumerable  printer's  errors  had  crept  in  ;  that  many  passages,  especially  those  that 
contain  Algebraical  formulae,  were  ill-arranged  and  erroneous ;  lastly,  that  the  author 
himself  had  in  mind  a  complete  revision,  including  certain  alterations,  to  give  a  better 
finish  to  the  work,  together  with  certain  additional  matter. 

That  being  the  case,  I  was  greatly  desirous  of  obtaining  a  copy,  revised  &  enlarged 
by  himself ;  I  also  wanted  to  have  him  at  hand  whilst  the  edition  was  in  progress,  &  that 
he  should  superintend  the  whole  thing  for  himself.  This,  however,  I  was  unable  to  procure 
during  the  last  few  years,  in  which  he  has  been  travelling  through  nearly  the  whole  of 
Europe  ;  until  at  last  he  came  here,  a  little  while  ago,  as  he  returned  home  from  his  lengthy 
wanderings,  &  stayed  here  to  assist  me  during  the  whole  time  that  the  edition  was  in 
hand.  He,  in  addition  to  our  regular  proof-readers,  himself  also  used  every  care  in  cor- 
recting the  proof ;  even  then,  however,  he  has  not  sufficient  confidence  in  himself  as  to 
imagine  that  not  the  slightest  thing  has  escaped  him.  For  it  is  a  characteristic  of  the  human 
mind  that  it  cannot  concentrate  long  on  the  same  subject  with  sufficient  attention. 

It  follows  that  this  ought  to  be  considered  in  some  measure  as  a  first  &  original 
edition  ;  any  one  who  compares  it  with  that  issued  at  Vienna  will  soon  see  the  difference 
between  them.  Many  of  the  minor  alterations  are  made  for  the  purpose  of  rendering 
certain  passages  more  elegant  &  clear  ;  there  are,  however,  especially  at  the  foot  of  a 
page,  slight  additions  also,  or  slight  changes  made  after  the  type  was  set  up,  merely  for 
the  purpose  of  filling  up  gaps  that  were  left  here  &  there — these  gaps  being  due  to  the 
fact  that  several  sheets  were  being  set  at  the  same  time  by  different  compositors,  and  four 
presses  were  kept  hard  at  work  together.  As  he  was  at  hand,  this  could  easily  be  done 
without  causing  any  disturbance  of  the  sentences  or  the  pagination. 


14  AD  LECTOREM  EX  EDITIONE  VIENNENSI 

nexus  inter  vires,  &  distantias,  est  utique  admodum  necessaria  ad  intelligendam  Theoriam  ipsam, 
cujus  ea  est  prcecipua  queedam  veluti  clavis,  sine  qua  omnino  incassum  tentarentur  cetera  ;  sed 
y  ejusmodi  est,  ut  tironum,  &  sane  etiam  mediocrium,  immo  etiam  longe  infra  mediocritatem 
collocatorum,  captum  non  excedat,  potissimum  si  viva  accedat  Professoris  vox  mediocriter  etiam 
versati  in  Mechanics,  cujus  ope,  pro  certo  habeo,  rem  ita  patentem  omnibus  reddi  posse,  ut 
ii  etiam,  qui  Geometric?  penitus  ignari  sunt,  paucorum  admodum  explicatione  vocabulorum 
accidente,  earn  ipsis  oculis  intueantur  omnino  perspicuam, 

In  tertia  parte  supponuntur  utique  nonnulla,  que?  demonstrantur  in  secunda  ;  sed  ea  ipsa 
sunt  admodum  pauca,  &  Us,  qui  geometricas  demonstrationes  fastidiunt,  facile  admodum  exponi 
possunt  res  ipsee  ita,  ut  penitus  etiam  sine  ullo  Geometric  adjumento  percipiantur,  quanquam 
sine  Us  ipsa  demonstratio  haberi  non  poterit  ;  ut  idcirco  in  eo  differre  debeat  is,  qui  secundam 
partem  attente  legerit,  y  Geometriam  calleat,  ab  eo,  qui  earn  omittat,  quod  ille  primus  veritates 
in  tertia  parte  adhibitis,  ac  ex  secunda  erutas,  ad  explicationem  Physics,  intuebitur  per  evi- 
dentiam  ex  ipsis  demonstrationibus  baustam,  hie  secundus  easdem  quodammodo  per  fidem  Geo- 
metris  adhibitam  credet.  Hujusmodi  inprimis  est  illud,  particulam  compositam  ex  punctis 
etiam  bomogeneis,  preeditis  lege  virium  proposita,  posse  per  solam  diversam  ipsorum  punctorum 
dispositionem  aliam  particulam  per  cerium  intervallum  vel  perpetuo  attrahere,  vel  perpetuo 
repellere,  vel  nihil  in  earn  agere,  atque  id  ipsum  viribus  admodum  diversis,  y  que?  respectu  diver- 
sarum  particularum  diver  see  sint,  y  diverse?  respectu  partium  diver sarum  ejusdem  particulce, 
ac  aliam  particulam  alicubi  etiam  urgeant  in  latus,  unde  plurium  pheenomenorum  explicatio  in 
Physica  sponte  ftuit. 


Ferum  qui  omnem  Theorie?,  y  deductionum  compagem  aliquanto  altius  inspexerit,  ac 
diligentius  perpenderit,  videbit,  ut  spero,  me  in  hoc  perquisitionis  genere  multo  ulterius 
progressum  esse,  quam  olim  Newtonus  ipse  desideravit.  Is  enim  in  postremo  Opticce  questione 
prolatis  Us,  qua  per  vim  attractivam,  y  vim  repulsivam,  mutata  distantia  ipsi  attractive?  suc- 
cedentem,  explicari  poterant,  he?c  addidit :  "  Atque  he?c  quidem  omnia  si  ita  sint,  jam  Natura 
universa  valde  erit  simplex,  y  consimilis  sui,  perficiens  nimirum  magnos  omnes  corporum 
ccelestium  motus  attractione  gravitatis,  quee  est  mutua  inter  corpora  ilia  omnia,  y  minores  fere 
omnes  particularum  suarum  motus  alia  aliqua  vi  attrabente,  y  repellente,  quiz  est  inter  particulas 
illas  mutua."  Aliquanto  autem  inferius  de  primigeniis  particulis  agens  sic  habet :  "  Porro 
videntur  mihi  he?  particule?  primigeniee  non  modo  in  se  vim  inertice  habere,  motusque  leges  passivas 
illas,  que?  ex  vi  ista  necessario  oriuntur  ;  verum  etiam  motum  perpetuo  accipere  a  certis  principiis 
actuosis,  qualia  nimirum  sunt  gravitas,  y  causa  fermentationis,  y  cohcerentia  corporum.  Atque 
heec  quidem  principia  considero  non  ut  occultas  qualitates,  que?  ex  specificis  rerum  formis  oriri 
fingantur,  sed  ut  universales  Nature?  leges,  quibus  res  ipse?  sunt  formates.  Nam  principia 
quidem  talia  revera  existere  ostendunt  phenomena  Nature?,  licet  ipsorum  cause?  que?  sint, 
nondum  fuerit  explicatum.  Affirmare,  singulas  rerum  species  specificis  preeditas  esse  qualita- 
tibus  occultis,  per  quas  eae  vim  certam  in  agenda  habent,  hoc  utique  est  nihil  dicere  :  at  ex 
phcenomenis  Nature?  duo,  vel  tria  derivare  generalia  motus  principia,  y  deinde  explicare, 
quemadmodum  proprietates,  y  actiones  rerum  corporearum  omnium  ex  istis  principiis  conse- 
quantur,  id  vero  magnus  esset  factus  in  Philosophia  progressus,  etiamsi  principiorum  istorum 
cause?  nondum  essent  cognite?.  Quare  motus  principia  supradicta  proponere  non  dubito,  cum 
per  Naturam  universam  latissime  pateant" 


Hc?c  ibi  Newtonus,  ubi  is  quidem  magnos  in  Philosophia  progressus  facturum  arbitratus 
est  eum,  qui  ad  duo,  vel  tria  generalia  motus  principia  ex  Nature?  pheenomenis  derivata  phe?no- 
menorum  explicationem  reduxerit,  y  sua  principia  protulit,  ex  quibus  inter  se  diversis  eorum 
aliqua  tantummodo  explicari  posse  censuit.  Quid  igitur,  ubi  y  ea  ipsa  tria,  y  alia  preecipua 
quczque,  ut  ipsa  etiam  impenetrabilitas,  y  impulsio  reducantur  ad  principium  unicum  legitima 
ratiocinatione  deductum  ?  At  id  per  meam  unicam,  y  simplicem  virium  legempr<zstari,patebit 
sane  consideranti  operis  totius  Synopsim  quandam,  quam  hie  subjicio  ;  sed  multo  magis  opus 
ipsum  diligentius  pervolventi. 


THE   PRINTER   AT   VENICE 

TO 

THE    READER 

|JOU  will  be  well  aware,  if  you  have  read  the  public  journals,  with  what  applause 
the  work  which  I  now  offer  to  you  has  been  received  throughout  Europe 
since  its  publication  at  Vienna  five  years  ago.  Not  to  mention  others,  if 
you  refer  to  the  numbers  of  the  Berne  Journal  for  the  early  part  of  the 
year  1761,  you  will  not  fail  to  see  how  highly  it  has  been  esteemed.  It 
contains  an  entirely  new  system  of  Natural  Philosophy,  which  is  already 
commonly  known  as  the  Boscovicbian  theory,  from  the  name  of  its  author, 
As  a  matter  of  fact,  it  is  even  now  a  subject  of  public  instruction  in  several  Universities  in 
different  parts  ;  it  is  expounded  not  only  in  yearly  theses  or  dissertations,  both  printed  & 
debated  ;  but  also  in  several  elementary  books  issued  for  the  instruction  of  the  young  it  is 
introduced,  explained,  &  by  many  considered  as  their  original.  Any  one,  however,  who 
wishes  to  obtain  more  detailed  insight  into  the  whole  structure  of  the  theory,  the  close 
relation  that  its  several  parts  bear  to  one  another,  or  its  great  fertility  &  wide  scope  for 
the  purpose  of  deriving  the  whole  of  Nature,  in  her  widest  range,  from  a  single  simple  law 
of  forces ;  any  one  who  wishes  to  make  a  deeper  study  of  it  must  perforce  study  the  work 
here  offered. 

All  these  considerations  had  from  the  first  moved  me  to  undertake  a  new  edition  of 
the  work  ;  in  addition,  there  was  the  fact  that  I  perceived  that  it  would  be  a  matter  of  some 
difficulty  for  copies  of  the  Vienna  edition  to  pass  beyond  the  confines  of  Germany — indeed, 
at  the  present  time,  no  matter  how  diligently  they  are  inquired  for,  they  are  to  be  found 
on  sale  nowhere,  or  scarcely  anywhere,  in  the  rest  of  Europe.  The  system  had  its  birth  in 
Italy,  &  its  outlines  had  already  been  sketched  by  the  author  in  several  dissertations  pub- 
lished here  in  our  own  land  ;  though,  as  luck  would  have  it,  the  system  itself  was  finally 
put  into  shape  and  published  at  Vienna,  whither  he  had  gone  for  a  short  time.  I  therefore 
thought  it  right  that  it  should  be  disseminated  throughout  the  whole  of  Europe,  &  that 
preferably  as  the  product  of  an  Italian  press.  I  had  in  fact  already  commenced  an  edition 
founded  on  a  copy  of  the  Vienna  edition,  when  it  came  to  my  knowledge  that  the  author 
was  greatly  dissatisfied  with  the  Vienna  edition,  taken  in  hand  there  after  his  departure ; 
that  innumerable  printer's  errors  had  crept  in ;  that  many  passages,  especially  those  that 
contain  Algebraical  formulae,  were  ill-arranged  and  erroneous ;  lastly,  that  the  author 
himself  had  in  mind  a  complete  revision,  including  certain  alterations,  to  give  a  better 
finish  to  the  work,  together  with  certain  additional  matter. 

That  being  the  case,  I  was  greatly  desirous  of  obtaining  a  copy,  revised  &  enlarged 
by  himself ;  I  also  wanted  to  have  him  at  hand  whilst  the  edition  was  in  progress,  &  that 
he  should  superintend  the  whole  thing  for  himself.  This,  however,  I  was  unable  to  procure 
during  the  last  few  years,  in  which  he  has  been  travelling  through  nearly  the  whole  of 
Europe  ;  until  at  last  he  came  here,  a  little  while  ago,  as  he  returned  home  from  his  lengthy 
wanderings,  &  stayed  here  to  assist  me  during  the  whole  time  that  the  edition  was  in 
hand.  He,  in  addition  to  our  regular  proof-readers,  himself  also  used  every  care  in  cor- 
recting the  proof ;  even  then,  however,  he  has  not  sufficient  confidence  in  himself  as  to 
imagine  that  not  the  slightest  thing  has  escaped  him.  For  it  is  a  characteristic  of  the  human 
mind  that  it  cannot  concentrate  long  on  the  same  subject  with  sufficient  attention. 

It  follows  that  this  ought  to  be  considered  in  some  measure  as  a  first  &  original 
edition  ;  any  one  who  compares  it  with  that  issued  at  Vienna  will  soon  see  the  difference 
between  them.  Many  of  the  minor  alterations  are  made  for  the  purpose  of  rendering 
certain  passages  more  elegant  &  clear  ;  there  are,  however,  especially  at  the  foot  of  a 
page,  slight  additions  also,  or  slight  changes  made  after  the  type  was  set  up,  merely  for 
the  purpose  of  filling  up  gaps  that  were  left  here  &  there — these  gaps  being  due  to  the 
fact  that  several  sheets  were  being  set  at  the  same  time  by  different  compositors,  and  four 
presses  were  kept  hard  at  work  together.  As  he  was  at  hand,  this  could  easily  be  done 
without  causing  any  disturbance  of  the  sentences  or  the  pagination. 


4  TYPOGRAPHUS  VENETUS  LECTORI 

Inter  mutationes  occurret  ordo  numerorum  mutatus  in  paragraphis  :  nam  numerus  82 
de  novo  accessit  totus  :  deinde  is,  qui  fuerat  261  discerptus  est  in  5  ;  demum  in  Appendice 
post  num.  534  factae  sunt  &  mutatiunculae  nonnullae,  &  additamenta  plura  in  iis,  quae 
pertinent  ad  sedem  animse. 

Supplementorum  ordo  mutatus  est  itidem  ;  quae  enim  fuerant  3,  &  4,  jam  sunt  i,  & 
2  :  nam  eorum  usus  in  ipso  Opere  ante  alia  occurrit.  UK  autem,  quod  prius  fuerat  primum, 
nunc  autem  est  tertium,  accessit  in  fine  Scholium  tertium,  quod  pluribus  numeris  complec- 
titur  dissertatiunculam  integrant  de  argumento,  quod  ante  aliquot  annos  in  Parisiensi 
Academia  controversiae  occasionem  exhibuit  in  Encyclopedico  etiam  dictionario  attactum, 
in  qua  dissertatiuncula  demonstrat  Auctor  non  esse,  cur  ad  vim  exprimendam  potentia 
quaepiam  distantice  adhibeatur  potius,  quam  functio. 

Accesserunt  per  totum  Opus  notulae  marginales,  in  quibus  eorum,  quae  pertractantur 
argumenta  exponuntur  brevissima,  quorum  ope  unico  obtutu  videri  possint  omnia,  &  in 
memoriam  facile  revocari. 

Postremo  loco  ad  calcem  Operis  additus  est  fusior  catalogus  eorum  omnium,  quae  hue 
usque  ab  ipso  Auctore  sunt  edita,  quorum  collectionem  omnem  expolitam,  &  correctam, 
ac  eorum,  quse  nondum  absoluta  sunt,  continuationem  meditatur,  aggressurus  illico  post 
suum  regressum  in  Urbem  Romam,  quo  properat.  Hie  catalogus  impressus  fuit  Venetisis 
ante  hosce  duos  annos  in  reimpressione  ejus  poematis  de  Solis  ac  Lunae  defectibus. 
Porro  earn  omnium  suorum  Operum  Collectionem,  ubi  ipse  adornaverit,  typis  ego  meis 
excudendam  suscipiam,  quam  magnificentissime  potero. 

Haec  erant,  quae  te  monendum  censui ;    tu  laboribus  nostris  fruere,  &  vive  felix. 


THE   PREFACE  TO  THE  READER 

THAT   APPEARED   IN  THE  VIENNA  EDITION 


Reader,  you  have  before  you  a  Theory  of  Natural  Philosophy  deduced 
from  a  single  law  of  Forces.  You  will  find  in  the  opening  paragraphs  of 
the  first  section  a  statement  as  to  where  the  Theory  has  been  already 
published  in  outline,  &  to  a  certain  extent  explained  ;  &  also  the  occasion 
that  led  me  to  undertake  a  more  detailed  treatment  &  enlargement  of  it. 
For  I  have  thought  fit  to  divide  the  work  into  three  parts  ;  the  first  of 
these  contains  the  exposition  of  the  Theory  itself,  its  analytical  deduction 
&  its  demonstration  ;  the  second  a  fairly  full  application  to  Mechanics  ;  &  the  third  an 
application  to  Physics. 

The  most  important  point,  I  decided,  was  for  me  to  take  the  greatest  care  that  every- 
thing, as  far  as  was  possible,  should  be  clearly  explained,  &  that  there  should  be  no  need  for 
higher  geometry  or  for  the  calculus.  Thus,  in  the  first  part,  as  well  as  in  the  third,  there 
are  no  proofs  by  analysis  ;  nor  are  there  any  by  geometry,  with  the  exception  of  a  very  few 
that  are  absolutely  necessary,  &  even  these  you  will  find  relegated  to  brief  notes  set  at  the 
foot  of  a  page.  I  have  also  added  some  very  few  proofs,  that  required  a  knowledge  of 
higher  algebra  &  geometry,  or  were  of  a  rather  more  complicated  nature,  all  of  which  have 
been  already  published  elsewhere,  at  the  end  of  the  work  ;  I  have  collected  these  under 
the  heading  Supplements  ;  &  in  them  I  have  included  my  views  on  Space  &  Time,  which 
are  in  accord  with  my  main  Theory,  &  also  have  been  already  published  elsewhere.  In 
the  second  part,  where  the  Theory  is  applied  to  Mechanics,  I  have  not  been  able  to  do 
without  geometrical  proofs  altogether  ;  &  even  in  some  cases  I  have  had  to  give  algebraical 
proofs.  But  these  are  of  such  a  simple  kind  that  they  scarcely  ever  require  anything  more 
than  Euclidean  geometry,  the  first  and  most  elementary  ideas  of  trigonometry,  and  easy 
analytical  calculations. 

It  is  true  that  in  the  first  part  there  are  to  be  found  a  good  many  geometrical  diagrams, 
which  at  first  sight,  before  the  text  is  considered  more  closely,  will  appear  to  be  rather 
complicated.  But  these  present  nothing  else  but  a  kind  of  image  of  the  subjects  treated, 
which  by  means  of  these  diagrams  are  set  before  the  eyes  for  contemplation.  The  very 
curve  that  represents  the  law  of  forces  is  an  instance  of  this.  I  find  that  between  all  points 
of  matter  there  is  a  mutual  force  depending  on  the  distance  between  them,  &  changing  as 
this  distance  changes  ;  so  that  it  is  sometimes  attractive,  &  sometimes  repulsive,  but  always 
follows  a  definite  continuous  law.  Laws  of  variation  of  this  kind  between  two  quantities 
depending  upon  one  another,  as  distance  &  force  do  in  this  instance,  may  be  represented 
either  by  an  analytical  formula  or  by  a  geometrical  curve  ;  but  the  former  method  of 
representation  requires  far  more  knowledge  of  algebraical  processes,  &  does  not  assist  the 
imagination  in  the  way  that  the  latter  does.  Hence  I  have  employed  the  latter  method  in 
the  first  part  of  the  work,  &  relegated  to  the  Supplements  the  analytical  formula  which 
represents  the  curve,  &  the  law  of  forces  which  the  curve  exhibits. 

The  whole  matter  reduces  to  this.  In  a  straight  line  of  indefinite  length,  which  is 
called  the  axis,  a  fixed  point  is  taken  ;  &  segments  of  the  straight  line  cut  off  from  this 
point  represent  the  distances.  A  curve  is  drawn  following  the  general  direction  of  this 
straight  line,  &  winding  about  it,  so  as  to  cut  it  in  several  places.  Then  perpendiculars  that 
are  drawn  from  the  ends  of  the  segments  to  meet  the  curve  represent  the  forces  ;  these 
forces  are  greater  or  less,  according  as  such  perpendiculars  are  greater  or  less  ;  &  they  pass 
from  attractive  forces  to  repulsive,  and  vice  versa,  whenever  these  perpendiculars  change 
their  direction,  as  the  curve  passes  from  one  side  of  the  axis  of  indefinite  length  to  the  other 
side  of  it.  Now  this  requires  no  geometrical  proof,  but  only  a  knowledge  of  certain  terms, 
which  either  belong  to  the  first  elementary  principles  of  geometry,  &  are  thoroughly  well 
known,  or  are  such  as  can  be  defined  when  they  are  used.  The  term  Asymptote  is  well 
known,  and  from  the  same  idea  we  speak  of  the  branch  of  a  curve  as  being  asymptotic  ; 
thus  a  straight  line  is  said  to  be  the  asymptote  to  any  branch  of  a  curve  when,  if  the  straight 
line  is  indefinitely  produced,  it  approaches  nearer  and  nearer  to  the  curvilinear  arc  which 
is  also  prolonged  indefinitely  in  such  manner  that  the  distance  between  them  becomes 
indefinitely  diminished,  but  never  altogether  vanishes,  so  that  the  straight  line  &  the  curve 
never  really  meet. 

A  careful  consideration  of  the  curve  given  in  Fig.  I,  &  of  the  way  in  which  the  relation 

13 


i4  AD  LECTOREM  EX  EDITIONE  VIENNENSI 

nexus  inter  vires,  &  distantias,  est  utique  ad.rn.odum  necessaria  ad  intelligendam  Theoriam  ipsam, 
cujus  ea  est  prescipua  qucsdam  veluti  clavis,  sine  qua  omnino  incassum  tentarentur  cetera  ;  sea 
y  ejusmodi  est,  ut  tironum,  &  sane  etiam  mediocrium,  immo  etiam  longe  infra  mediocritatem 
collocatorum,  captum  non  excedat,  potissimum  si  viva  accedat  Professoris  vox  mediocriter  etiam 
versati  in  Mechanica,  cujus  ope,  pro  certo  habeo,  rem  ita  patentem  omnibus  reddi  posse,  ut 
ii  etiam,  qui  Geometries  penitus  ignari  sunt,  paucorum  admodum  explicatione  vocabulorum 
accidente,  earn  ipsis  oculis  intueantur  omnino  perspicuam. 

In  tertia  parte  supponuntur  utique  nonnulla,  ques  demonstrantur  in  secunda  ;  sed  ea  ipsa 
sunt  admodum  pauca,  &  Us,  qui  geometricas  demonstrationes  fastidiunt,  facile  admodum  exponi 
possunt  res  ipsez  ita,  ut  penitus  etiam  sine  ullo  Geometries  adjumento  percipiantur,  quanquam 
sine  Us  ipsa  demonstratio  haberi  non  poterit ;  ut  idcirco  in  eo  differre  debeat  is,  qui  secundam 
partem  attente  legerit,  y  Geometriam  calleat,  ab  eo,  qui  earn  omittat,  quod  ille  primus  veritates 
in  tertia  parte  adbibitis,  ac  ex  secunda  erutas,  ad  explicationem  Physices,  intuebitur  per  evi- 
dentiam  ex  ipsis  demonstrationibus  haustam,  hie  secundus  easdem  quodammodo  per  fidem  Geo- 
metris  adhibitam  credet.  Hujusmodi  inprimis  est  illud,  particulam  compositam  ex  punctis 
etiam  homogeneis,  presditis  lege  virium  proposita,  posse  per  solam  diversam  ipsorum  punctorum 
dispositionem  aliam  particulam  per  certum  intervallum  vel  perpetuo  attrahere,  vel  perpetuo 
repellere,  vel  nihil  in  earn  agere,  atque  id  ipsum  viribus  admodum  diversis,  y  qua  respectu  diver- 
sarum  particularum  diver  see  sint,  &  diver  see  respectu  partium  diver sarum  ejusdem  particules, 
ac  aliam  particulam  alicubi  etiam  urgeant  in  latus,  unde  plurium  phesnomenorum  explicatio  in 
Physica  sponte  ftuit. 


Verum  qui  omnem  Theories,  y  deductionum  compagem  aliquanto  altius  inspexerit,  ac 
diligentius  perpenderit,  videbit,  ut  spero,  me  in  hoc  perquisitionis  genere  multo  ulterius 
progressum  esse,  quam  olim  Newtonus  ipse  desideravit.  Is  enim  in  postremo  Optices  questione 
prolatis  Us,  ques  per  vim  attractivam,  &  vim  repulsivam,  mutata  distantia  ipsi  attractives  suc- 
cedentem,  explicari  poterant,  hesc  addidit :  "  Atque  h<sc  quidem  omnia  si  ita  sint,  jam  Natura 
universa  valde  erit  simplex,  y  consimilis  sui,  perficiens  nimirum  magnos  omnes  corporum 
ceslestium  motus  attractione  gravitatis,  qucs  est  mutua  inter  corpora  ilia  omnia,  &  minores  fere 
omnes  particularum  suarum  motus  alia  aliqua  vi  attrahente,  y  repellente,  ques  est  inter  particulas 
illas  mutua."  Aliquanto  autem  inferius  de  primigeniis  particulis  agens  sic  habet :  "  Porro 
videntur  mihi  hce  particules  primigenics  non  modo  in  se  vim  inerties  habere,  motusque  leges  passivas 
illas,  ques  ex  vi  ista  necessario  oriuntur  ;  verum  etiam  motum  perpetuo  accipere  a  certis  principiis 
actuosis,  qualia  nimirum  sunt  gravitas,  y  causa  fermentationis,  y  cohesrentia  corporum.  Atque 
hesc  quidem  principia  considero  non  ut  occultas  qualitates,  ques  ex  specificis  rerum  formis  oriri 
fingantur,  sed  ut  universales  Natures  leges,  quibus  res  ipscs  sunt  formates.  Nam  principia 
quidem  talia  revera  existere  ostendunt  phesnomena  Natures,  licet  ipsorum  causes  ques  sint, 
nondum  fuerit  explicatum.  Affirmare,  singulas  rerum  species  specificis  presditas  esse  qualita- 
tibus  occultis,  per  quas  eae  vim  certam  in  agenda  habent,  hoc  utique  est  nihil  dicere :  at  ex 
phesnomenis  Natures  duo,  vel  tria  derivare  generalia  motus  principia,  y  deinde  explicare, 
quemadmodum  proprietates,  y  actiones  rerum  corporearum  omnium  ex  istis  principiis  conse- 
quantur,  id  vero  magnus  esset  factus  in  Philosophia  progressus,  etiamsi  principiorum  istorum 
causes  nondum  essent  cognites.  Quare  motus  principia  supradicta  proponere  non  dubito,  cum 
per  Naturam  universam  latissime  pateant." 


Hcsc  ibi  Newtonus,  ubi  is  quidem  magnos  in  Philosophia  progressus  facturum  arbitratus 
est  eum,  qui  ad  duo,  vel  tria  generalia  motus  principia  ex  Natures  phesnomenis  derivata  phesno- 
menorum  explicationem  reduxerit,  y  sua  principia  protulit,  ex  quibus  inter  se  diversis  eorum 
aliqua  tantummodo  explicari  posse  censuit.  Quid  igitur,  ubi  y  ea  ipsa  tria,  y  alia  prcscipua 
quesque,  ut  ipsa  etiam  impenetrabilitas,  y  impulsio  reducantur  ad  principium  unicum  legitima 
ratiocinatione  deductum  ?  At  id  per  meam  unicam,  y  simplicem  virium  legem  presstari,  patebit 
sane  consideranti  operis  totius  Synopsim  quandam,  quam  hie  subjicio  ;  sed  multo  magis  opus 
ipsum  diligentius  pervolventi. 


PREFACE  TO  READER  THAT  APPEARED   IN  THE  VIENNA  EDITION  15 

between  the  forces  &  the  distances  is  represented  by  it,  is  absolutely  necessary  for  the  under- 
standing of  the  Theory  itself,  to  which  it  is  as  it  were  the  chief  key,  without  which  it  would 
be  quite  useless  to  try  to  pass  on  to  the  rest.  But  it  is  of  such  a  nature  that  it  does  not  go 
beyond  the  capacity  of  beginners,  not  even  of  those  of  very  moderate  ability,  or  of  classes 
even  far  below  the  level  of  mediocrity ;  especially  if  they  have  the  additional  assistance  of 
a  teacher's  voice,  even  though  he  is  only  moderately  familiar  with  Mechanics.  By  his  help, 
I  am  sure,  the  subject  can  be  made  clear  to  every  one,  so  that  those  of  them  that  are  quite 
ignorant  of  geometry,  given  the  explanation  of  but  a  few  terms,  may  get  a  perfectly  good 
idea  of  the  subject  by  ocular  demonstration. 

In  the  third  part,  some  of  the  theorems  that  have  been  proved  in  the  second  part  are 
certainly  assumed,  but  there  are  very  few  such ;  &,  for  those  who  do  not  care  for  geo- 
metrical proofs,  the  facts  in  question  can  be  quite  easily  stated  in  such  a  manner  that  they 
can  be  completely  understood  without  any  assistance  from  geometry,  although  no  real 
demonstration  is  possible  without  them.  There  is  thus  bound  to  be  a  difference  between 
the  reader  who  has  gone  carefully  through  the  second  part,  &  who  is  well  versed  in  geo- 
metry, &  him  who  omits  the  second  part ;  in  that  the  former  will  regard  the  facts,  that 
have  been  proved  in  the  second  part,  &  are  now  employed  in  the  third  part  for  the  ex- 
planation of  Physics,  through  the  evidence  derived  from  the  demonstrations  of  these  facts, 
whilst  the  second  will  credit  these  same  facts  through  the  mere  faith  that  he  has  in  geome- 
tricians. A  specially  good  instance  of  this  is  the  fact,  that  a  particle  composed  of  points 
quite  homogeneous,  subject  to  a  law  of  forces  as  stated,  may,  merely  by  altering  the  arrange- 
ment of  those  points,  either  continually  attract,  or  continually  repel,  or  have  no  effect  at 
all  upon,  another  particle  situated  at  a  known  distance  from  it ;  &  this  too,  with  forces  that 
differ  widely,  both  in  respect  of  different  particles  &  in  respect  of  different  parts  of  the  same 
particle  ;  &  may  even  urge  another  particle  in  a  direction  at  right  angles  to  the  line  join- 
ing the  two,  a  fact  that  readily  gives  a  perfectly  natural  explanation  of  many  physical 
phenomena. 

Anyone  who  shall  have  studied  somewhat  closely  the  whole  system  of  my  Theory,  & 
what  I  deduce  from  it,  will  see,  I  hope,  that  I  have  advanced  in  this  kind  of  investigation 
much  further  than  Newton  himself  even  thought  open  to  his  desires.  For  he,  in  the  last 
of  his  "  Questions  "  in  his  Opticks,  after  stating  the  facts  that  could  be  explained  by  means 
of  an  attractive  force,  &  a  repulsive  force  that  takes  the  place  of  the  attractive  force  when 
the  distance  is  altered,  has  added  these  words  : — "  Now  if  all  these  things  are  as  stated,  then 
the  whole  of  Nature  must  be  exceedingly  simple  in  design,  &  similar  in  all  its  parts,  accom- 
plishing all  the  mighty  motions  of  the  heavenly  bodies,  as  it  does,  by  the  attraction  of 
gravity,  which  is  a  mutual  force  between  any  two  bodies  of  the  whole  system  ;  and  Nature 
accomplishes  nearly  all  the  smaller  motions  of  their  particles  by  some  other  force  of  attrac- 
tion or  repulsion,  which  is  mutual  between  any  two  of  those  particles."  Farther  on,  when 
he  is  speaking  about  elementary  particles,  he  says  : — "  Moreover,  it  appears  to  me  that  these 
elementary  particles  not  only  possess  an  essential  property  of  inertia,  &  laws  of  motion, 
though  only  passive,  which  are  the  necessary  consequences  of  this  property ;  but  they  also 
constantly  acquire  motion  from  the  influence  of  certain  active  principles  such  as,  for 
instance,  gravity,  the  cause  of  fermentation,  &  the  cohesion  of  solids.  I  do  not  consider  these 
principles  to  be  certain  mysterious  qualities  feigned  as  arising  from  characteristic  forms  of 
things,  but  as  universal  laws  of  Nature,  by  the  influence  of  which  these  very  things  have 
been  created.  For  the  phenomena  of  Nature  show  that  these  principles  do  indeed  exist, 
although  their  nature  has  not  yet  been  elucidated.  To  assert  that  each  &  every  species  is 
endowed  with  a  mysterious  property  characteristic  to  it,  due  to  which  it  has  a  definite  mode 
in  action,  is  really  equivalent  to  saying  nothing  at  all.  On  the  other  hand,  to  derive  from 
the  phenomena  of  Nature  two  or  three  general  principles,  &  then  to  explain  how  the  pro- 
perties &  actions  of  all  corporate  things  follow  from  those  principles,  this  would  indeed  be 
a  mighty  advance  in  philosophy,  even  if  the  causes  of  those  principles  had  not  at  the  time 
been  discovered.  For  these  reasons  I  do  not  hesitate  in  bringing  forward  the  principles  of 
motion  given  above,  since  they  are  clearly  to  be  perceived  throughout  the  whole  range  of 
Nature." 

These  are  the  words  of  Newton,  &  therein  he  states  his  opinion  that  he  indeed  will 
have  made  great  strides  in  philosophy  who  shall  have  reduced  the  explanation  of  phenomena 
to  two  or  three  general  principles  derived  from  the  phenomena  of  Nature ;  &  he 
brought  forward  his  own  principles,  themselves  differing  from  one  another,  by  which  he 
thought  that  some  only  of  the  phenomena  could  be  explained.  What  then  if  not  only  the 
three  he  mentions,  but  also  other  important  principles,  such  as  impenetrability  &  impul- 
sive force,  be  reduced  to  a  single  principle,  deduced  by  a  process  of  rigorous  argument !  It 
will  be  quite  clear  that  this  is  exactly  what  is  done  by  my  single  simple  law  of  forces,  to 
anyone  who  studies  a  kind  of  synopsis  of  the  whole  work,  which  I  add  below  ;  but  it  will  be 
iar  more  clear  to  him  who  studies  the  whole  work  with  some  earnestness, 


SYNOPSIS   TOTIUS   OPERIS 

EX  EDITIONE  VIENNENSI 

PARS  I 

sex  numeris  exhibeo,  quando,  &  qua  occasione  Theoriam  meam 
invenerim,  ac  ubi  hucusque  de  ea  egerim  in  dissertationibus  jam  editis,  quid 
ea  commune  habeat  cum  Leibnitiana,  quid  cum  Newtoniana  Theoria,  in 
quo  ab  utraque  discrepet,  &  vero  etiam  utrique  praestet  :  addo,  quid 
alibi  promiserim  pertinens  ad  aequilibrium,  &  oscillationis  centrum,  & 
quemadmodum  iis  nunc  inventis,  ac  ex  unico  simplicissimo,  ac  elegant- 
issimo  theoremate  profluentibus  omnino  sponte,  cum  dissertatiunculam 

brevem  meditarer,  jam  eo  consilio  rem  aggressus ;  repente  mihi  in  opus  integrum  justse 

molis  evaserit  tractatio. 

7  Turn  usque  ad  num.    II   expono  Theoriam  ipsam  :    materiam  constantem  punctis 

prorsus  simplicibus,  indivisibilibus,  &  inextensis,  ac  a  se  invicem  distantibus,  quae  puncta 
habeant  singula  vim  inertiae,  &  praeterea  vim  activam  mutuam  pendentem  a  distantiis,  ut 
nimirum,  data  distantia,  detur  &  magnitude,  &  directio  vis  ipsius,  mutata  autem  distantia, 
mutetur  vis  ipsa,  quae,  imminuta  distantia  in  infinitum,  sit  repulsiva,  &  quidem 
excrescens  in  infinitum  :  aucta  autem  distantia,  minuatur,  evanescat,  mutetur  in  attrac- 
tivam  crescentem  primo,  turn  decrescentem,  evanescentem,  abeuntem  iterum  in  repul- 
sivam,  idque  per  multas  vices,  donee  demum  in  majoribus  distantiis  abeat  in  attractivam 
decrescentem  ad  sensum  in  ratione  reciproca  duplicata  distantiarum  ;  quern  nexum  virium 
cum  distantiis,  &  vero  etiam  earum  transitum  a  positivis  ad  negativas,  sive  a  repulsivis  ad 
attractivas,  vel  vice  versa,  oculis  ipsis  propono  in  vi,  qua  binae  elastri  cuspides  conantur  ad 
es  invicem  accedere,  vel  a  se  invicem  recedere,  prout  sunt  plus  justo  distractae,  vel  con- 
tractae. 

II  Inde  ad  num.  16  ostendo,  quo  pacto  id  non  sit  aggregatum  quoddam  virium  temere 

coalescentium,  sed  per  unicam  curvam  continuam  exponatur  ope  abscissarum  exprimentium 
distantias,  &  ordinatarum  exprimentium  vires,  cujus  curvae  ductum,  &  naturam  expono, 
ac  ostendo,  in  quo  differat  ab  hyperbola  ilia  gradus  tertii,  quae  Newtonianum  gravitatem 
exprimit  :  ac  demum  ibidem  &  argumentum,  &  divisionem  propono  operis  totius. 

1 6  Hisce  expositis  gradum  facio  ad  exponendam  totam  illam  analysim,  qua  ego  ad  ejusmodi 

Theoriam  deveni,  &  ex  qua  ipsam  arbitror  directa,  &  solidissima  ratiocinatione  deduci 
totam.  Contendo  nimirum  usque  ad  numerum  19  illud,  in  collisione  corporum  debere  vel 
haberi  compenetrationem,  vel  violari  legem  continuitatis,  velocitate  mutata  per  saltum,  si 
cum  inaequalibus  velocitatibus  deveniant  ad  immediatum  contactum,  quae  continuitatis  lex 
cum  (ut  evinco)  debeat  omnino  observari,  illud  infero,  antequam  ad  contactum  deveniant 
corpora,  debere  mutari  eorum  velocitates  per  vim  quandam,  quae  sit  par  extinguendse 
velocitati,  vel  velocitatum  differentiae,  cuivis  utcunque  magnae. 

19  A  num.  19  ad  28  expendo  effugium,  quo  ad  eludendam  argumenti  mei  vim  utuntur  ii, 

qui  negant  corpora  dura,  qua  quidem  responsione  uti  non  possunt  Newtoniani,  &  Corpus- 
culares  generaliter,  qui  elementares  corporum  particulas  assumunt  prorsus  duras  :  qui  autem 
omnes  utcunque  parvas  corporum  particulas  molles  admittunt,  vel  elasticas,  difficultatem 
non  effugiunt,  sed  transferunt  ad  primas  superficies,  vel  puncta,  in  quibus  committeretur 
omnino  saltus,  &  lex  continuitatis  violaretur  :  ibidem  quendam  verborum  lusum  evolvo, 
frustra  adhibitum  ad  eludendam  argumenti  mei  vim. 


*  Series  numerorum,  quibus  tractari  incipiunt,  quae  sunt  in  textu, 

16 


SYNOPSIS   OF  THE   WHOLE   WORK 

(FROM  THE  VIENNA   EDITION) 

PART  I 

N  the  first  six  articles,  I  state  the  time  at  which  I  evolved  my  Theory,  what  i  * 
led  me  to  it,  &  where  I  have  discussed  it  hitherto  in  essays  already  pub- 
lished :  also  what  it  has  in  common  with  the  theories  of  Leibniz  and 
Newton  ;  in  what  it  differs  from  either  of  these,  &  in  what  it  is  really 
superior  to  them  both.  In  addition  I  state  what  I  have  published  else- 
where about  equilibrium  &  the  centre  of  oscillation  ;  &  how,  having  found 
out  that  these  matters  followed  quite  easily  from  a  single  theorem  of  the 
most  simple  &  elegant  kind,  I  proposed  to  write  a  short  essay  thereon  ;  but  when  I  set  to 
work  to  deduce  the  matter  from  this  principle,  the  discussion,  quite  unexpectedly  to  me, 
developed  into  a  whole  work  of  considerable  magnitude. 

From  this  .until  Art.  II,  I  explain  the  Theory  itself  :  that  matter  is  unchangeable,  7 
and  consists  of  points  that  are  perfectly  simple,  indivisible,  of  no  extent,  &  separated  from 
one  another ;  that  each  of  these  points  has  a  property  of  inertia,  &  in  addition  a  mutual 
active  force  depending  on  the  distance  in  such  a  way  that,  if  the  distance  is  given,  both  the 
magnitude  &  the  direction  of  this  force  are  given  ;  but  if  the  distance  is  altered,  so  also  is 
the  force  altered  ;  &  if  the  distance  is  diminished  indefinitely,  the  force  is  repulsive,  &  in 
fact  also  increases  indefinitely ;  whilst  if  the  distance  is  increased,  the  force  will  be  dimin- 
ished, vanish,  be  changed  to  an  attractive  force  that  first  of  all  increases,  then  decreases, 
vanishes,  is  again  turned  into  a  repulsive  force,  &  so  on  many  times  over  ;  until  at  greater 
distances  it  finally  becomes  an  attractive  force  that  decreases  approximately  in  the  inverse 
ratio  of  the  squares  of  the  distances.  This  connection  between  the  forces  &  the  distances, 
&  their  passing  from  positive  to  negative,  or  from  repulsive  to  attractive,  &  conversely,  I 
illustrate  by  the  force  with  which  the  two  ends  of  a  spring  strive  to  approach  towards,  or 
recede  from,  one  another,  according  as  they  are  pulled  apart,  or  drawn  together,  by  more 
than  the  natural  amount. 

From  here  on  to  Art.  1 6  I  show  that  it  is  not  merely  an  aggregate  of  forces  combined  n 
haphazard,  but  that  it  is  represented  by  a  single  continuous  curve,  by  means  of  abscissse 
representing  the  distances  &  ordinates  representing  the  forces.  I  expound  the  construction 
&  nature  of  this  curve ;  &  I  show  how  it  differs  from  the  hyperbola  of  the  third  degree 
which  represents  Newtonian  gravitation.  Finally,  here  too  I  set  forth  the  scope  of  the 
whole  work  &  the  nature  of  the  parts  into  which  it  is  divided. 

These  statements  having  been  made,  I  start  to  expound  the  whole  of  the  analysis,  by  16 
which  I  came  upon  a  Theory  of  this  kind,  &  from  which  I  believe  I  have  deduced  the  whole 
of  it  by  a  straightforward  &  perfectly  rigorous  chain  of  reasoning.  I  contend  indeed,  from 
here  on  until  Art.  19,  that,  in  the  collision  of  solid  bodies,  either  there  must  be  compene- 
tration,  or  the  Law  of  Continuity  must  be  violated  by  a  sudden  change  of  velocity,  if 
the  bodies  come  into  immediate  contact  with  unequal  velocities.  Now  since  the  Law  of 
Continuity  must  (as  I  prove  that  it  must)  be  observed  in  every  case,  I  infer  that,  before 
the  bodies  reach  the  point  of  actual  contact,  their  velocities  must  be  altered  by  some  force 
which  is  capable  of  destroying  the  velocity,  or  the  difference  of  the  velocities,  no  matter  how 
great  that  may  be. 

From  Art.  19  to  Art.  28  I  consider  the  artifice,  adopted  for  the  purpose  of  evading  the  19 
strength  of  my  argument  by  those  who  deny  the  existence  of  hard  bodies ;  as  a  matter  of 
fact  this  cannot  be  used  as  an  argument  against  me  by  the  Newtonians,  or  the  Corpuscular- 
ians  in  general,  for  they  assume  that  the  elementary  particles  of  solids  are  perfectly  hard. 
Moreover,  those  who  admit  that  all  the  particles  of  solids,  however  small  they  may  be,  are 
soft  or  elastic,  yet  do  not  escape  the  difficulty,  but  transfer  it  to  prime  surfaces,  or  points ; 
&  here  a  sudden  change  would  be  made  &  the  Law  of  Continuity  violated.  In  the  same 
connection  I  consider  a  certain  verbal  quibble,  used  in  a  vain  attempt  to  foil  the  force  of 
my  reasoning. 

*  These  numbers  are  the  numbers  of  the  articles,  in  which  the  matters  given  in  the  text  are  first  discussed. 

17  C 


1 8  SYNOPSIS  TOTIUS  OPERIS 

28  Sequentibus  num.  28  &  29  binas  alias  responsiones  rejicio  aliorum,  quarum  altera,  ut 

mei  argument!  vis  elidatur,  affirmat  quispiam,  prima  materiae  elementa  compenetrari,  alter 
dicuntur  materiae  puncta  adhuc  moveri  ad  se  invicem,  ubi  localiter  omnino  quiescunt,  & 
contra  primum  effugium  evinco  impenetrabilitatem  ex  inductione ;  contra  secundum 
expono  aequivocationem  quandam  in  significatione  vocis  motus,  cui  aequivocationi  totum 
innititur. 

30  Hinc  num.  30,  &  31  ostendo,  in  quo  a  Mac-Laurino  dissentiam,  qui  considerata  eadem, 

quam  ego  contemplatus  sum,  collisione  corporum,  conclusit,  continuitatis  legem  violari, 
cum  ego  eandem  illaesam  esse  debere  ratus  ad  totam  devenerim  Theoriam  meam. 

32  Hie  igitur,  ut  meae  deductionis  vim  exponam,  in  ipsam  continuitatis  legem  inquire,  ac 

a  num.  32  ad  38  expono,  quid  ipsa  sit,  quid  mutatio  continua  per  gradus  omnes  intermedios, 
quae  nimirum  excludat  omnem  saltum  ab  una  magnitudine  ad  aliam  sine  transitu  per 

39  intermedias,  ac  Geometriam  etiam  ad  explicationem  rei  in  subsidium  advoco  :  turn  earn 
probo  primum  ex  inductione,  ac  in  ipsum  inductionis  principium  inquirens  usque  ad  num. 
44,  exhibeo,  unde  habeatur  ejusdem  principii  vis,  ac  ubi  id  adhiberi  possit,  rem  ipsam 
illustrans  exemplo  impenetrabilitatis  erutae  passim  per  inductionem,  donee  demum  ejus  vim 

45  applicem  ad  legem  continuitatis  demonstrandam  :  ac  sequentibus  numeris  casus  evolvo 
quosdam  binarum  classium,  in  quibus  continuitatis  lex  videtur  laedi  nee  tamen  laeditur. 

48  Post  probationem  principii  continuitatis  petitam  ab  inductione,  aliam  num.  48  ejus 

probationem  aggredior  metaphysicam  quandam,  ex  necessitate  utriusque  limitis  in  quanti- 
tatibus  realibus,  vel  seriebus  quantitatum  realium  finitis,  quae  nimirum  nee  suo  principio, 
nee  suo  fine  carere  possunt.  Ejus  rationis  vim  ostendo  in  motu  locali,  &  in  Geometria 

52  sequentibus  duobus  numeris  :  turn  num.  52  expono  difficultatem  quandam,  quas  petitur 
ex  eo,  quod  in  momento  temporis,  in  quo  transitur  a  non  esse  ad  esse,  videatur  juxta  ejusmodi 
Theoriam  debere  simul  haberi  ipsum  esse,  &  non  esse,  quorum  alterum  ad  finem  praecedentis 
seriei  statuum  pertinet,  alterum  ad  sequentis  initium,  ac  solutionem  ipsius  fuse  evolvo, 
Geometria  etiam  ad  rem  oculo  ipsi  sistendam  vocata  in  auxilium. 


63  Num.  63,  post  epilogum  eorum  omnium,  quae  de  lege  continuitatis  sunt  dicta,  id 

principium  applico  ad  excludendum  saltum  immediatum  ab  una  velocitate  ad  aliam,  sine 
transitu  per  intermedias,  quod  &  inductionem  laederet  pro  continuitate  amplissimam,  & 
induceret  pro  ipso  momento  temporis,  in  quo  fieret  saltus,  binas  velocitates,  ultimam 
nimirum  seriei  praecedentis,  &  primam  novas,  cum  tamen  duas  simul  velocitates  idem  mobile 
habere  omnino  non  possit.  Id  autem  ut  illustrem,  &  evincam,  usque  ad  num.  72  considero 
velocitatem  ipsam,  ubi  potentialem  quandam,  ut  appello,  velocitatem  ab  actuali  secerno, 
&  multa,  quae  ad  ipsarum  naturam,  ac  mutationes  pertinent,  diligenter  evolvo,  nonnullis 
etiam,  quae  inde  contra  meae  Theoriae  probationem  objici  possunt,  dissolutis. 


His  expositis  conclude  jam  illud  ex  ipsa  continuitate,  ubi  corpus  quodpiam  velocius 
movetur  post  aliud  lentius,  ad  contactum  immediatum  cum  ilia  velocitatum  inaequalitate 
deveniri  non  posse,  in  quo  scilicet  contactu  primo  mutaretur  vel  utriusque  velocitas,  vel 
alterius,  per  saltum,  sed  debere  mutationem  velocitatis  incipere  ante  contactum  ipsum. 

73  Hinc  num.  73  infero,  debere  haberi  mutationis  causam,  quae  appelletur  vis  :   turn  num.  74 

74  hanc  vim  debere  esse  mutuam,  &  agere  in  partes  contrarias,  quod  per  inductionem  evinco, 

75  &  inde  infero  num.  75,  appellari  posse  repulsivam  ejusmodi  vim  mutuam,  ac  ejus  legem 
exquirendam  propono.     In  ejusmodi  autem  perquisitione  usque  ad  num.  80  invenio  illud, 
debere  vim  ipsam  imminutis  distantiis  crescere  in  infinitum  ita  ut  par  sit  extinguendae 
velocitati  utcunque  magnse  ;    turn  &  illud,  imminutis  in  infinitum  etiam  distantiis,  debere 
in  infinitum  augeri,  in  maximis  autem  debere  esse  e  contrario  attractivam,  uti  est  gravitas  : 
inde  vero  colligo  limitem  inter  attractionem,  &  repulsionem  :   turn  sensim  plures,  ac  etiam 
plurimos  ejusmodi   limites  invenio,  sive  transitus  ab  attractione  ad  repulsionem,  &  vice 
versa,  ac  formam  totius  curvae  per  ordinatas  suas  exprimentis  virium  legem  determino. 


SYNOPSIS  OF  THE  WHOLE  WORK  19 

In  the  next  articles,  28  &  29,  I  refute  a  further  pair  of  arguments  advanced  by  others ;  28 
in  the  first  of  these,  in  order  to  evade  my  reasoning,  someone  states  that  there  is  compene- 
tration  of  the  primary  elements  of  matter  ;  in  the  second,  the  points  of  matter  are  said  to 
be  moved  with  regard  to  one  another,  even  when  they  are  absolutely  at  rest  as  regards 
position.  In  reply  to  the  first  artifice,  I  prove  the  principle  of  impenetrability  by  induc- 
tion ;  &  in  reply  to  the  second,  I  expose  an  equivocation  in  the  meaning  of  the  term  motion, 
an  equivocation  upon  which  the  whole  thing  depends. 

Then,  in  Art.  30,  31,  I  show  in  what  respect  I  differ  from  Maclaurin,  who,  having     30 
considered  the  same  point  as  myself,  came  to  the  conclusion  that  in  the  collision  of  bodies 
the  Law  of  Continuity  was  violated  ;  whereas  I  obtained  the  whole  of  my  Theory  from  the 
assumption  that  this  law  must  be  unassailable. 

At  this  point  therefore,  in  order  that  the  strength  of  my  deductive  reasoning  might     32 
be  shown,  I  investigate  the  Law  of  Continuity  ;  and  from  Art.  32  to  Art.  38,  I  set  forth  its 
nature,  &  what  is  meant  by  a  continuous  change  through  all  intermediate  stages,  such  as 
to  exclude  any  sudden  change  from  any  one  magnitude  to  another  except  by  a  passage 
through  intermediate  stages  ;    &  I  call  in  geometry  as  well  to  help  my  explanation  of  the 
matter.     Then  I  investigate  its  truth  first  of  all  by  induction  ;    &,  investigating  the  prin-     39 
ciple  of  induction  itself,  as  far  as  Art.  44, 1  show  whence  the  force  of  this  principle  is  derived, 
&  where  it  can  be  used.     I  give  by  way  of  illustration  an  example  in  which  impenetrability 
is  derived  entirely  by  induction  ;  &  lastly  I  apply  the  force  of  the  principle  to  demonstrate 
the  Law  of  Continuity.     In  the  articles  that  follow  I  consider  certain  cases  of  two  kinds,     45 
in  which  the  Law  of  Continuity  appears  to  be  violated,  but  is  not  however  really  violated. 

After  this  proof  of  the  principle  of  continuity  procured  through  induction,  in  Art.  48,  48 
I  undertake  another  proof  of  a  metaphysical  kind,  depending  upon  the  necessity  of  a  limit 
on  either  side  for  either  real  quantities  or  for  a  finite  series  of  real  quantities ;  &  indeed  it 
is  impossible  that  these  limits  should  be  lacking,  either  at  the  beginning  or  the  end.  I 
demonstrate  the  force  of  this  reasoning  in  the  case  of  local  motion,  &  also  in  geometry,  in  the 
next  two  articles.  Then  in  Art.  52  I  explain  a  certain  difficulty,  which  is  derived  from  the  S2 
fact  that,  at  the  instant  at  which  there  is  a  passage  from  non-existence  to  existence,  it  appears 
according  to  a  theory  of  this  kind  that  we  must  have  at  the  same  time  both  existence  and 
non-existence.  For  one  of  these  belongs  to  the  end  of  the  antecedent  series  of  states,  &  the 
other  to  the  beginning  of  the  consequent  series.  I  consider  fairly  fully'  the  solution  of  this 
problem  ;  and  I  call  in  geometry  as  well  to  assist  in  giving  a  visual  representation  of  the 
matter. 

In  Art.  63,  after  summing  up  all  that  has  been  said  about  the  Law  of  Continuity,  I  63 
apply  the  principle  to  exclude  the  possibility  of  any  sudden  change  from  one  velocity  to 
another,  except  by  passing  through  intermediate  velocities ;  this  would  be  contrary  to  the 
very  full  proof  that  I  give  for  continuity,  as  it  would  lead  to  our  having  two  velocities  at 
the  instant  at  which  the  change  occurred.  That  is  to  say,  there  would  be  the  final  velocity 
of  the  antecedent  series,  &  the  initial  velocity  of  the  consequent  series ;  in  spite  of  the  fact 
that  it  is  quite  impossible  for  a  moving  body  to  have  two  different  velocities  at  the  same 
time.  Moreover,  in  order  to  illustrate  &  prove  the  point,  from  here  on  to  Art.  72,  I 
consider  velocity  itself ;  and  I  distinguish  between  a  potential  velocity,  as  I  call  it,  &  an 
actual  velocity ;  I  also  investigate  carefully  many  matters  that  relate  to  the  nature  of  these 
velocities  &  to  their  changes.  Further,  I  settle  several  difficulties  that  can  be  brought 
up  in  opposition  to  the  proof  of  my  Theory,  in  consequence. 

This  done,  I  then  conclude  from  the  principle  of  continuity  that,  when  one  body  with 
a  greater  velocity  follows  after  another  body  having  a  less  velocity,  it  is  impossible  that 
there  should  ever  be  absolute  contact  with  such  an  inequality  of  velocities  ;  that  is  to  say, 
a  case  of  the  velocity  of  each,  or  of  one  or  the  other,  of  them  being  changed  suddenly  at 
the  instant  of  contact.  I  assert  on  the  other  hand  that  the  change  in  the  velocities  must 
begin  before  contact.  Hence,  in  Art.  73,  I  infer  that  there  must  be  a  cause  for  this  change  :  73 
which  is  to  be  called  "  force."  Then,  in  Art.  74,  I  prove  that  this  force  is  a  mutual  one,  &  74 
that  it  acts  in  opposite  directions ;  the  proof  is  by  induction.  From  this,  in  Art.  75,  I  75 
infer  that  such  a  mutual  force  may  be  said  to  be  repulsive  ;  &  I  undertake  the  investigation 
of  the  law  that  governs  it.  Carrying  on  this  investigation  as  far  as  Art.  80,  I  find  that  this 
force  must  increase  indefinitely  as  the  distance  is  diminished,  in  order  that  it  may  be  capable 
of  destroying  any  velocity,  however  great  that  velocity  may  be.  Moreover,  I  find  that, 
whilst  the  force  must  be  indefinitely  increased  as  the  distance  is  indefinitely  decreased,  it 
must  be  on  the  contrary  attractive  at  very  great  distances,  as  is  the  case  for  gravitation. 
Hence  I  infer  that  there  must  be  a  limit-point  forming  a  boundary  between  attraction  & 
repulsion  ;  &  then  by  degrees  I  find  more,  indeed  very  many  more,  of  such  limit-points, 
or  points  of  transition  from  attraction  to  repulsion,  &  from  repulsion  to  attraction ;  &  I 
determine  the  form  of  the  entire  curve,  that  expresses  by  its  ordinates  the  law  of  these  forces. 


20  SYNOPSIS  TOTIUS  OPERIS 

8 1  Eo  usque  virium  legem  deduce,  ac  definio ;  turn  num.  81  eruo  ex  ipsa  lege  consti- 

tutionem  elementorum  materiae,  quae  debent  esse  simplicia,  ob  repulsionem  in  minimis 
distantiis  in  immensum  auctam  ;  nam  ea,  si  forte  ipsa  elementa  partibus  constarent,  nexum 
omnem  dissolveret.  Usque  ad  num.  88  inquire  in  illud,  an  hasc  elementa,  ut  simplicia  esse 
debent,  ita  etiam  inextensa  esse  debeant,  ac  exposita  ilia,  quam  virtualem  extensionem 
appellant,  eandem  exclude  inductionis  principio,  &  difficultatem  evolvo  turn  earn,  quae  peti 
possit  ab  exemplo  ejus  generis  extensionis,  quam  in  anima  indivisibili,  &  simplice  per  aliquam 
corporis  partem  divisibilem,  &  extensam  passim  admittunt  :  vel  omnipraesentiae  Dei  :  turn 
earn,  quae  peti  possit  ab  analogia  cum  quiete,  in  qua  nimirum  conjungi  debeat  unicum 
spatii  punctum  cum  serie  continua  momentorum  temporis,  uti  in  extensione  virtuali  unicum 
momentum  temporis  cum  serie  continua  punctorum  spatii  conjungeretur,  ubi  ostendo,  nee 
quietem  omnimodam  in  Natura  haberi  usquam,  nee  adesse  semper  omnimodam  inter 

88  tempus,  &  spatium  analogiam.  Hie  autem  ingentem  colligo  ejusmodi  determinationis 
fructum,  ostendens  usque  ad  num.  91,  quantum  prosit  simplicitas,  indivisibilitas,  inextensio 
elementorum  materiae,  ob  summotum  transitum  a  vacuo  continue  per  saltum  ad  materiam 
continuam,  ac  ob  sublatum  limitem  densitatis,  quae  in  ejusmodi  Theoria  ut  minui  in 
infinitum  potest,  ita  potest  in  infinitum  etiam  augeri,  dum  in  communi,  ubi  ad  contactum 
deventum  est,  augeri  ultra  densitas  nequaquam  potest,  potissimum  vero  ob  sublatum  omne 
continuum  coexistens,  quo  sublato  &  gravissimae  difficultates  plurimse  evanescunt,  & 
infinitum  actu  existens  habetur  nullum,  sed  in  possibilibus  tantummodo  remanet  series 
finitorum  in  infinitum  producta. 


91  His  definitis,  inquire  usque  ad  num.  99  in  illud,  an  ejusmodi  elementa  sint  censenda 

homogenea,  an  heterogenea  :  ac  primo  quidem  argumentum  pro  homogeneitate  saltern  in 
eo,  quod  pertinet  ad  totam  virium  legem,  invenio  in  homogenietate  tanta  primi  cruris 
repulsivi  in  minimis  distantiis,  ex  quo  pendet  impenetrabilitas,  &  postremi  attractivi,  quo 
gravitas  exhibetur,  in  quibus  omnis  materia  est  penitus  homogenea.  Ostendo  autem,  nihil 
contra  ejusmodi  homogenietatem  evinci  ex  principio  Leibnitiano  indiscernibilium,  nihil  ex 
inductione,  &  ostendo,  unde  tantum  proveniat  discrimen  in  compositis  massulis,  ut  in 
frondibus,  &  foliis ;  ac  per  inductionem,  &  analogiam  demonstro,  naturam  nos  ad  homo- 
geneitatem  elementorum,  non  ad  heterogeneitatem  deducere. 

100  Ea  ad  probationem  Theoriae  pertinent ;    qua  absoluta,  antequam  inde  fructus  colli- 
gantur  multiplices,  gradum  hie  facio  ad  evolvendas  difficultates,  quae  vel  objectae  jam  sunt, 
vel  objici  posse  videntur  mihi,  primo  quidem  contra  vires  in  genere,  turn  contra  meam 
hanc  expositam,  comprobatamque  virium  legem,  ac  demum  contra  puncta  ilia  indivisibilia, 
&  inextensa,  quae  ex  ipsa  ejusmodi  virium  lege  deducuntur. 

101  Primo  quidem,  ut  iis  etiam  faciam  satis,  qui  inani  vocabulorum  quorundam  sono 
perturbantur,  a  num.  101  ad  104  ostendo,  vires    hasce    non    esse   quoddam  occultarum 
qualitatum  genus,  sed  patentem  sane  Mechanismum,  cum  &  idea  earum  sit   admodum 
distincta,  &  existentia,  ac  lex  positive  comprobata  ;   ad  Mechanicam  vero  pertineat  omnis 

104  tractatio  de  Motibus,  qui  a  datis  viribus  etiam  sine  immediate  impulsu  oriuntur.  A  num. 
104  ad  106  ostendo,  nullum  committi  saltum  in  transitu  a  repulsionibus  ad  attractiones, 

1 06  &  vice  versa,  cum  nimirum  per  omnes  inter medias  quantitates  is  transitus  fiat.  Inde  vero 
ad  objectiones  gradum  facio,  quae  totam  curvas  formam  impetunt.  Ostendo  nimirum  usque 
ad  num.  116,  non  posse  omnes  repulsiones  a  minore  attractione  desumi ;  repulsiones  ejusdem 
esse  seriei  cum  attractionibus,  a  quibus  differant  tantummodo  ut  minus  a  majore,  sive  ut 
negativum  a  positivo ;  ex  ipsa  curvarum  natura,  quae,  quo  altioris  sunt  gradus,  eo  in 
pluribus  punctis  rectam  secare  possunt,  &  eo  in  immensum  plures  sunt  numero ;  haberi 
potius,  ubi  curva  quaeritur,  quae  vires  exprimat,  indicium  pro  curva  ejus  naturae,  ut  rectam 
in  plurimis  punctis  secet,  adeoque  plurimos  secum  afferat  virium  transitus  a  repulsivis  ad 
attractivas,  quam  pro  curva,  quae  nusquam  axem  secans  attractiones  solas,  vel  solas  pro 
distantiis  omnibus  repulsiones  exhibeat  :  sed  vires  repulsivas,  &  multiplicitatem  transituum 
esse  positive  probatam,  &  deductam  totam  curvas  formam,  quam  itidem  ostendo,  non  esse 
ex  arcubus  natura  diversis  temere  coalescentem,  sed  omnino  simplicem,  atque  earn  ipsam 


SYNOPSIS  OF  THE  WHOLE  WORK  21 

So  far  I  have  been  occupied  in  deducing  and  settling  the  law  of  these  forces.  Next, 
in  Art.  8r,  I  derive  from  this  law  the  constitution  of  the  elements  of  matter.  These  must  be  81 
quite  simple,  on  account  of  the  repulsion  at  very  small  distances  being  immensely  great ; 
for  if  by  chance  those  elements  were  made  up  of  parts,  the  repulsion  would  destroy  all 
connections  between  them.  Then,  as  far  as  Art.  88,  I  consider  the  point,  as  to  whether 
these  elements,  as  they  must  be  simple,  must  therefore  be  also  of  no  extent ;  &,  having  ex- 
plained what  is  called  "  virtual  extension,"  I  reject  it  by  the  principle  of  induction.  I 
then  consider  the  difficulty  which  may  be  brought  forward  from  an  example  of  this  kind  of 
extension  ;  such  as  is  generally  admitted  in  the  case  of  the  indivisible  and  one-fold  soul 
pervading  a  divisible  &  extended  portion  of  the  body,  or  in  the  case  of  the  omnipresence 
of  GOD.  Next  I  consider  the  difficulty  that  may  be  brought  forward  from  an  analogy  with 
rest ;  for  here  in  truth  one  point  of  space  must  be  connected  with  a  continuous  series  of 
instants  of  time,  just  as  in  virtual  extension  a  single  instant  of  time  would  be  connected  with 
a  continuous  series  of  points  of  space.  I  show  that  there  can  neither  be  perfect  rest  any-  gg 
where  in  Nature,  nor  can  there  be  at  all  times  a  perfect  analogy  between  time  and  space. 
In  this  connection,  I  also  gather  a  large  harvest  from  such  a  conclusion  as  this ;  showing, 
as  far  as  Art.  91,  the  great  advantage  of  simplicity,  indivisibility,  &  non-extension  in  the 
elements  of  matter.  For  they  do  away  with  the  idea  of  a  passage  from  a  continuous  vacuum 
to  continuous  matter  through  a  sudden  change.  Also  they  render  unnecessary  any  limit 
to  density  :  this,  in  a  Theory  like  mine,  can  be  just  as  well  increased  to  an  indefinite  extent, 
as  it  can  be  indefinitely  decreased  :  whilst  in  the  ordinary  theory,  as  soon  as  contact  takes 
place,  the  density  cannot  in  any  way  be  further  increased.  But,  most  especially,  they  do 
away  with  the  idea  of  everything  continuous  coexisting ;  &  when  this  is  done  away  with, 
the  majority  of  the  greatest  difficulties  vanish.  Further,  nothing  infinite  is  found  actually 
existing  ;  the  only  thing  possible  that  remains  is  a  series  of  finite  things  produced  inde- 
finitely. 

These  things  being  settled,  I  investigate,  as  far  as  Art.  99,  the  point  as  to  whether  QJ 
elements  of  this  kind  are  to  be  considered  as  being  homogeneous  or  heterogeneous.  I  find 
my  first  evidence  in  favour  of  homogeneity — at  least  as  far  as  the  complete  law  of  forces 
is  concerned — in  the  equally  great  homogeneity  of  the  first  repulsive  branch  of  my  curve 
of  forces  for  very  small  distances,  upon  which  depends  impenetrability,  &  of  the  last  attrac- 
tive branch,  by  which  gravity  is  represented.  Moreover  I  show  that  there  is  nothing  that 
can  be  proved  in  opposition  to  homogeneity  such  as  this,  that  can  be  derived  from  either 
the  Leibnizian  principle  of  "  indiscernibles,"  or  by  induction.  I  also  show  whence  arise 
those  differences,  that  are  so  great  amongst  small  composite  bodies,  such  as  we  see  in  boughs 
&  leaves ;  &  I  prove,  by  induction  &  analogy,  that  the  very  nature  of  things  leads  us  to 
homogeneity,  &  not  to  heterogeneity,  for  the  elements  of  matter. 

These  matters  are  all  connected  with  the  proof  of  my  Theory.  Having  accomplished  IOo 
this,  before  I  start  to  gather  the  manifold  fruits  to  be  derived  from  it,  I  proceed  to  consider 
the  objections  to  my  theory,  such  as  either  have  been  already  raised  or  seem  to  me  capable 
of  being  raised  ;  first  against  forces  in  general,  secondly  against  the  law  of  forces  that  I 
have  enunciated  &  proved,  &  finally  against  those  indivisible,  non-extended  points  that 
are  deduced  from  a  law  of  forces  of  this  kind. 

First  of  all  then,  in  order  that  I  may  satisfy  even  those  who  are  confused  over  the  101 
empty  sound  of  certain  terms,  I  show,  in  Art.  101  to  104,  that  these  forces  are  not  some 
sort  of  mysterious  qualities ;  but  that  they  form  a  readily  intelligible  mechanism,  since 
both  the  idea  of  them  is  perfectly  distinct,  as  well  as  their  existence,  &  in  addition  the  law 
that  governs  them  is  demonstrated  in  a  direct  manner.  To  Mechanics  belongs  every  dis- 
cussion concerning  motions  that  arise  from  given  forces  without  any  direct  impulse.  In 
Art.  104  to  106,  I  show  that  no  sudden  change  takes  place  in  passing  from  repulsions  to  104 
attractions  or  vice  versa  ;  for  this  transition  is  made  through  every  intermediate  quantity. 
Then  I  pass  on  to  consider  the  objections  that  are  made  against  the  whole  form  of  my  106 
curve.  I  show  indeed,  from  here  on  to  Art.  116,  that  all  repulsions  cannot  be  taken  to 
come  from  a  decreased  attraction  ;  that  repulsions  belong  to  the  self-same  series  as  attrac- 
tions, differing  from  them  only  as  less  does  from  more,  or  negative  from  positive.  From 
the  very  nature  of  the  curves  (for  which,  the  higher  the  degree,  the  more  points  there  are 
in  which  they  can  intersect  a  right  line,  &  vastly  more  such  curves  there  are),  I  deduce 
that  there  is  more  reason  for  assuming  a  curve  of  the  nature  of  mine  (so  that  it  may  cut  a 
right  line  in  a  large  number  of  points,  &  thus  give  a  large  number  of  transitions  of  the  forces 
from  repulsions  to  attractions),  than  for  assuming  a  curve  that,  since  it  does  not  cut 
the  axis  anywhere,  will  represent  attractions  alone,  or  repulsions  alone,  at  all  distances. 
Further,  I  point  out  that  repulsive  forces,  and  a  multiplicity  of  transitions  are  directly 
demonstrated,  &  the  whole  form  of  the  curve  is  a  matter  of  deduction  ;  &  I  also  show  that 
it  is  not  formed  of  a  number  of  arcs  differing  in  nature  connected  together  haphazard ; 


22  SYNOPSIS  TOTIUS  OPERIS 

simplicitatem  in  Supplementis  cvidentissime  demonstro,  exhibens  methodum,  qua  deveniri 
possit  ad  aequationem  ejusmodi  curvse  simplicem,  &  uniformem  ;  licet,  ut  hie  ostendo,  ipsa 
ilia  lex  virium  possit  mente  resolvi  in  plures,  quae  per  plures  curvas  exponantur,  a  quibus 
tamen  omnibus  ilia  reapse  unica  lex,  per  unicam  illam  continuant,  &  in  se  simplicem  curvam 
componatur. 

121  A  num.  121  refello,  quae  objici  possunt  a  lege  gravitatis  decrescentis  in  ratione  reciproca 

duplicata  distantiarum,  quae  nimirum  in  minimis  distantiis  attractionem  requirit  crescentem 
in  infinitum.  Ostendo  autem,  ipsam  non  esse  uspiam  accurate  in  ejusmodi  ratione,  nisi 
imaginarias  resolutiones  exhibeamus ;  nee  vero  ex  Astronomia  deduci  ejusmodi  legem 
prorsus  accurate  servatam  in  ipsis  Planetarum,  &  Cometarum  distantiis,  sed  ad  summum  ita 

124  proxime,  ut  differentia  ab  ea  lege  sit  perquam  exigua  :  ac  a  num.  124  expendo  argumentum, 
quod  pro  ejusmodi  lege  desumi  possit  ex  eo,  quod  cuipiam  visa  sit  omnium  optima,  & 
idcirco  electa  ab  Auctore  Naturae,  ubi  ipsum  Optimismi  principium  ad  trutinam  revoco,  ac 
exclude,  &  vero  illud  etiam  evinco,  non  esse,  cur  omnium  optima  ejusmodi  lex  censeatur  : 
in  Supplementis  vero  ostendo,  ad  qua;  potius  absurda  deducet  ejusmodi  lex,  &  vero  etiam 
aliae  plures  attractionis,  quae  imminutis  in  infinitum  distantiis  excrescat  in  infinitum. 


131  Num.  131  a  viribus  transeo  ad  elementa,  &  primum  ostendo,  cur  punctorum  inexten- 

sorum  ideam  non  habeamus,  quod  nimirum  earn  haurire  non  possumus  per  sensus,  quos 
solae  massae,  &  quidem  grandiores,  afficiunt,  atque  idcirco  eandem  nos  ipsi  debemus  per 
reflexionem  efformare,  quod  quidem  facile  possumus.  Ceterum  illud  ostendo,  me  non 
inducere  primum  in  Physicam  puncta  indivisibilia,  &  inextensa,  cum  eo  etiam  Leibnitianae 
monades  recidant,  sed  sublata  extensione  continua  difficultatem  auferre  illam  omnem,  quae 
jam  olim  contra  Zenonicos  objecta,  nunquam  est  satis  soluta,  qua  fit,  ut  extensio  continua 
ab  inextensis  effici  omnino  non  possit. 


140  Num.  140  ostendo,  inductionis  principium  contra  ipsa  nullam  habere  vim,  ipsorum 

autem  existentiam  vel  inde  probari,  quod  continuitas  se  se  ipsam  destruat,  &  ex  ea  assumpta 
probetur  argumentis  a  me  institutis  hoc  ipsum,  prima  elementa  esse  indivisibilia,  &  inextensa, 

143  nee  ullum  haberi  extensum  continuum.  A  num.  143  ostendo,  ubi  continuitatem  admittam, 
nimirum  in  solis  motibus ;  ac  illud  explico,  quid  mihi  sit  spatium,  quid  tempus,  quorum 
naturam  in  Supplementis  multo  uberius  expono.  Porro  continuitatem  ipsam  ostendo  a 
natura  in  solis  motibus  obtineri  accurate,  in  reliquis  affectari  quodammodo  ;  ubi  &  exempla 
quaedam  evolvo  continuitatis  primo  aspectu  violatae,  in  quibusdam  proprietatibus  luminis, 
ac  in  aliis  quibusdam  casibus,  in  quibus  quaedam  crescunt  per  additionem  partium,  non  (ut 
ajunt)  per  intussumptionem. 

\ 

153  A  num.  153  ostendo,  quantum  haec  mea  puncta  a  spiritibus  differant ;  ac  illud  etiam 

evolvo,  unde  fiat,  ut  in  ipsa  idea  corporis  videatur  includi  extensio  continua,  ubi  in  ipsam 
idearum  nostrarum  originem  inquire,  &  quae  inde  praejudicia  profluant,  expono.  Postremo 

165  autem  loco  num.  165  innuo,  qui  fieri  possit,  ut  puncta  inextensa,  &  a  se  invicem  distantia, 
in  massam  coalescant,  quantum  libet,  cohaerentem,  &  iis  proprietatibus  praeditam,  quas  in 
corporibus  experimur,  quod  tamen  ad  tertiam  partem  pertinet,  ibi  multo  uberius  pertrac- 
tandum  ;  ac  ibi  quidem  primam  hanc  partem  absolve. 


PARS   II 

166  Num.  166  hujus  partis  argumentum  propono  ;  sequenti  vero  167,  quae  potissimum  in 

curva  virium  consideranda  sint,  enuncio.  Eorum  considerationem  aggressus,  primo  quidem 

1 68  usque  ad  num.  172  in  ipsos  arcus  inquire,  quorum  alii  attractivi,  alii  repulsivi,  alii  asym- 
ptotici,  ubi  casuum  occurrit  mira  multitudo,  &  in  quibusdam  consectaria  notatu  digna,  ut 
&  illud,  cum  ejus  formae  curva  plurium  asymptotorum  esse  possit,  Mundorum  prorsus 
similium  seriem  posse  oriri,  quorum  alter  respectu  alterius  vices  agat  unius,  &  indissolubilis 


SYNOPSIS  OF  THE  WHOLE  WORK  23 

but  that  it  is  absolutely  one-fold.  This  one-fold  character  I  demonstrate  in  the  Supple- 
ments in  a  very  evident  manner,  giving  a  method  by  which  a  simple  and  uniform  equation 
may  be  obtained  for  a  curve  of  this  kind.  Although,  as  I  there  point  out,  this  law  of  forces 
may  be  mentally  resolved  into  several,  and  these  may  be  represented  by  several  correspond- 
ing curves,  yet  that  law,  actually  unique,  may  be  compounded  from  all  of  these  together 
by  means  of  the  unique,  continuous  &  one-fold  curve  that  I  give. 

In  Art.  121,  I  start  to  give  a  refutation  of  those  objections  that  may  be  raised  from  I2i 
a  consideration  of  the  fact  that  the  law  of  gravitation,  decreasing  in  the  inverse  duplicate 
ratio  of  the  distances,  demands  that  there  should  be  an  attraction  at  very  small  distances, 
&  that  it  should  increase  indefinitely.  However,  I  show  that  the  law  is  nowhere  exactly  in 
conformity  with  a  ratio  of  this  sort,  unless  we  add  explanations  that  are  merely  imaginative ; 
nor,  I  assert,  can  a  law  of  this  kind  be  deduced  from  astronomy,  that  is  followed  with  per- 
fect accuracy  even  at  the  distances  of  the  planets  &  the  comets,  but  one  merely  that  is  at 
most  so  very  nearly  correct,  that  the  difference  from  the  law  of  inverse  squares  is  very 
slight.  From  Art.  124  onwards,!  examine  the  value  of  the  argument  that  can  be  drawn  124 
in  favour  of  a  law  of  this  sort  from  the  view  that,  as  some  have  thought,  it  is  the  best  of 
all,  &  that  on  that  account  it  was  selected  by  the  Founder  of  Nature.  In  connection  with 
this  I  examine  the  principle  of  Optimism,  &  I  reject  it ;  moreover  I  prove  conclusively 
that  there  is  no  reason  why  this  sort  of  law  should  be  supposed  to  be  the  best  of  all.  Fur- 
ther in  the  Supplements,  I  show  to  what  absurdities  a  law  of  this  sort  is  more  likely  to  lead  ; 
&  the  same  thing  for  other  laws  of  an  attraction  that  increases  indefinitely  as  the  distance 
is  diminished  indefinitely. 

In  Art.  131  I  pass  from  forces  to  elements.  I  first  of  all  show  the  reason  why  we  may  1*1 
not  appreciate  the  idea  of  non-extended  points ;  it  is  because  we  are  unable  to  perceive 
them  by  means  of  the  senses,  which  are  only  affected  by  masses,  &  these  too  must  be  of 
considerable  size.  Consequently  we  have  to  build  up  the  idea  by  a  process  of  reasoning  ; 
&  this  we  can  do  without  any  difficulty.  In  addition,  I  point  out  that  I  am  not  the  first 
to  introduce  indivisible  &  non-extended  points  into  physical  science  ;  for  the  "  monads  " 
of  Leibniz  practically  come  to  the  same  thing.  But  I  show  that,  by  rejecting  the  idea  of 
continuous  extension,  I  remove  the  whole  of  the  difficulty,  which  was  raised  against  the 
disciples  of  Zeno  in  years  gone  by,  &  has  never  been  answered  satisfactorily ;  namely,  the 
difficulty  arising  from  the  fact  that  by  no  possible  means  can  continuous  extension  be 
made  up  from  things  of  no  extent. 

In  Art.  140  I  show  that  the  principle  of  induction  yields  no  argument  against  these  140 
indivisibles ;  rather  their  existence  is  demonstrated  by  that  principle,  for  continuity  is 
self-contradictory.  On  this  assumption  it  may  be  proved,  by  arguments  originated  by 
myself,  that  the  primary  elements  are  indivisible  &  non-extended,  &  that  there  does  not 
exist  anything  possessing  the  property  of  continuous  extension.  From  Art.  143  onwards,  j ., 
I  point  out  the  only  connection  in  which  I  shall  admit  continuity,  &  that  is  in  motion. 
I  state  the  idea  that  I  have  with  regard  to  space,  &  also  time  :  the  nature  of  these  I  explain 
much  more  fully  in  the  Supplements.  Further,  I  show  that  continuity  itself  is  really  a 
property  of  motions  only,  &  that  in  all  other  things  it  is  more  or  less  a  false  assumption. 
Here  I  also  consider  some  examples  in  which  continuity  at  first  sight  appears  to  be 
violated,  such  as  in  some  of  the  properties  of  light,  &  in  certain  other  cases  where  things 
increase  by  addition  of  parts,  and  not  by  intussumption,  as  it  is  termed. 

From  Art.  153  onwards,  I  show  how  greatly  these  points  of  mine  differ  from  object-     153 
souls.     I  consider  how  it  comes  about  that  continuous  extension  seems  to  be  included 
in  the  very  idea  of  a  body ;   &  in  this  connection,  I  investigate  the  origin  of  our  ideas 
&  I  explain  the  prejudgments    that    arise    therefrom.     Finally,    in   Art.    165,    I   lightly     165 
sketch  what  might  happen  to  enable  points  that  are  of  no  extent,  &  at  a  distance  from 
one  another,  to  coalesce  into  a  coherent  mass  of  any  size,  endowed  with  those  properties 
that  we  experience  in  bodies.    This,  however,  belongs  to  the  third  part ;  &  there  it  will  be 
much  more  fully  developed.     This  finishes  the  first  part. 

PART  II 

In  Art.  1 66  I  state  the  theme  of  this  second  part ;    and  in  Art.  167  I  declare  what     166 
matters  are  to  be  considered  more  especially  in  connection  with  the  curve  of  forces.     Com- 
ing to  the  consideration  of  these  matters,  I  first  of  all,  as  far  as  Art.  172,  investigate  the     168 
arcs  of  the  curve,  some  of  which  are  attractive,  some  repulsive  and  some  asymptotic.     Here 
a  marvellous  number  of  different  cases  present  themselves,  &  to  some  of  them  there  are 
noteworthy  corollaries ;  such  as  that,  since  a  curve  of  this  kind  is  capable  of  possessing  a 
considerable  number  of  asymptotes,  there  can  arise  a  series  of  perfectly  similar  cosmi,  each 
of  which  will  act  upon  all  the  others  as  a  single  inviolate  elementary  system.     From  Art.  172 


24  SYNOPSIS  TOTIUS  OPERIS 

172  element!.  Ad.  num.  179  areas  contemplor  arcubus  clausas,  quae  respondentes  segmento  axis 
cuicunque,  esse  possunt  magnitudine  utcunque  magnae,  vel  parvae,  sunt  autem  mensura 

179  incrementi,  vel  decrement!  quadrat!  velocitatum.  Ad  num.  189  inquire  in  appulsus  curvse 
ad  axem,  sive  is  ibi  secetur  ab  eadem  (quo  casu  habentur  transitus  vel  a  repulsione  ad 
attractionem,  vel  ab  attractione  ad  repulsionem,  quos  dico  limites,  &  quorum  maximus  est 
in  tota  mea  Theoria  usus),  sive  tangatur,  &  curva  retro  redeat,  ubi  etiam  pro  appulsibus 
considero  recessus  in  infinitum  per  arcus  asymptoticos,  &  qui  transitus,  sive  limites,  oriantur 
inde,  vel  in  Natura  admitti  possint,  evolvo. 

189  Num.   189  a  consideratione  curvae  ad  punctorum  combinationem  gradum  facio,  ac 

primo  quidem  usque  ad  num.  204  ago  de  systemate  duorum  punctorum,  ea  pertractans, 
quas  pertinent  ad  eorum  vires  mutuas,  &  motus,  sive  sibi  relinquantur,  sive  projiciantur 
utcunque,  ubi  &  conjunctione  ipsorum  exposita  in  distantiis  limitum,  &  oscillationibus 
variis,  sive  nullam  externam  punctorum  aliorum  actionem  sentiant,  sive  perturbentur  ab 
eadem,  illud  innuo  in  antecessum,  quanto  id  usui  futurum  sit  in  parte  tertia  ad  exponenda 
cohaesionis  varia  genera,  fermentationes,  conflagrationes,  emissiones  vaporum,  proprietates 
luminis,  elasticitatem,  mollitiem. 

204  Succedit  a  Num.  204  ad  239  multo  uberior  consideratio  trium  punctorum,  quorum 

vires  generaliter  facile  definiuntur  data  ipsorum  positione  quacunque  :  verum  utcunque 
data  positione,  &  celeritate  nondum  a  Geometris  inventi  sunt  motus  ita,  ut  generaliter  pro 
casibus  omnibus  absolvi  calculus  possit.  Vires  igitur,  &  variationem  ingentem,  quam 
diversae  pariunt  combinationes  punctorum,  utut  tantummodo  numero  trium,  persequor 

209  usque  ad  num.  209.  Hinc  usque  ad  num.  214  quaedam  evolvo,  quae  pertinent  ad  vires 
ortas  in  singulis  ex  actione  composita  reliquorum  duorum,  &  quae  tertium  punctum  non  ad 
accessum  urgeant,  vel  recessum  tantummodo  respectu  eorundem,  sed  &  in  latus,  ubi  & 
soliditatis  imago  prodit,  &  ingens  sane  discrimen  in  distantiis  particularum  perquam  exiguis 
ac  summa  in  maximis,  in  quibus  gravitas  agit,  conformitas,  quod  quanto  itidem  ad  Naturae 

214  explicationem  futurum  sit  usui,  significo.  Usque  ad  num.  221  ipsis  etiam  oculis  contem- 
plandum  propono  ingens  discrimen  in  legibus  virium,  quibus  bina  puncta  agunt  in  tertium, 
sive  id  jaceat  in  recta,  qua  junguntur,  sive  in  recta  ipsi  perpendiculari,  &  eorum  intervallum 
secante  bifariam,  constructis  ex  data  primigenia  curva  curvis  vires  compositas  exhibentibus  : 

221  turn  sequentibus  binis  numeris  casum  evolvo  notatu  dignissimum,  in  quo  mutata  sola 
positione  binorum  punctorum,  punctum  tertium  per  idem  quoddam  intervallum,  situm  in 
eadem  distantia  a  medio  eorum  intervallo,  vel  perpetuo  attrahitur,  vel  perpetuo  repellitur, 
vel  nee  attrahitur,  nee  repellitur  ;   cujusmodi  discrimen  cum  in  massis  haberi  debeat  multo 

222  majus,  illud  indico,  num.  222,  quantus  inde  itidem  in  Physicam  usus  proveniat. 


223  Hie  jam  num.  223  a  viribus  binorum  punctorum  transeo  ad  considerandum  totum 

ipsorum  systema,  &  usque  ad  num.  228  contemplor  tria  puncta  in  directum  sita,  ex  quorum 
mutuis  viribus  relationes  quaedam  exurgunt,  quas  multo  generaliores  redduntur  inferius,  ubi 
in  tribus  etiam  punctis  tantummodo  adumbrantur,  quae  pertinent  ad  virgas  rigidas,  flexiles, 
elasticas,  ac  ad  vectem,  &  ad  alia  plura,  quae  itidem  inferius,  ubi  de  massis,  multo  generaliora 

228  fiunt.  Demum  usque  ad  num.  238  contemplor  tria  puncta  posita  non  in  directum,  sive  in 
aequilibrio  sint,  sive  in  perimetro  ellipsium  quarundam,  vel  curvarum  aliarum  ;  in  quibus 
mira  occurrit  analogia  limitum  quorundam  cum  limitibus,  quos  habent  bina  puncta  in  axe 
curvae  primigeniae  ad  se  invicem,  atque  ibidem  multo  major  varietas  casuum  indicatur  pro 
massis,  &  specimen  applicationis  exhibetur  ad  soliditatem,  &  liquationem  per  celerem 

238  intestinum  motum  punctis  impressum.  Sequentibus  autem  binis  numeris  generalia  quaedam 
expono  de  systemate  punctorum  quatuor  cum  applicatione  ad  virgas  solidas,  rigidas,  flexiles, 
ac  ordines  particularum  varies  exhibeo  per  pyramides,  quarum  infimae  ex  punctis  quatuor, 
superiores  ex  quatuor  pyramidibus  singulae  coalescant. 


24°  A  num.  240  ad  massas  gradu  facto  usque  a  num.  264  considero,  quae  ad  centrum  gravi- 

tatis  pertinent,  ac  demonstro  generaliter,  in  quavis  massa  esse  aliquod,  &  esse  unicum  : 
ostendo,  quo  pacto  determinari  generaliter  possit,  &  quid  in  methodo,  quae  communiter 
adhibetur,  desit  ad  habendam  demonstrationis  vim,  luculenter  expono,  &  suppleo,  ac 


SYNOPSIS  OF  THE  WHOLE  WORK  25 

to  Art.  179,  I  consider  the  areas  included  by  the  arcs;  these,  corresponding  to  different     172 
segments  of  the  axis,  may  be  of  any  magnitude  whatever,  either  great  or  small ;   moreover 
they  measure  the  increment  or  decrement  in  the  squares  of  the  velocities.     Then,  on  as     179 
far  as  Art.  189, 1  investigate  the  approach  of  the  curve  to  the  axis ;  both  when  the  former 
is  cut  by  the  latter,  in  which  case  there  are  transitions  from  repulsion  to  attraction  and 
from  attraction  to  repulsion,  which  I  call  '  limits,'  &  use  very  largely  in  every  part  of  my 
Theory ;   &  also  when  the  former  is  touched  by  the  latter,  &  the  curve  once  again  recedes 
from  the  axis.     I  consider,  too,  as  a  case  of  approach,  recession  to  infinity  along  an  asymp- 
totic arc  ;    and   I   investigate  what  transitions,  or  limits,  may  arise   from  such  a  case,  & 
whether  such  are  admissible  in  Nature. 

In  Art.  189,  I  pass  on  from  the  consideration  of  the  curve  to  combinations  of  points.  l%9 
First,  as  far  as  Art.  204,  I  deal  with  a  system  of  two  points.  I  work  out  those  things  that 
concern  their  mutual  forces,  and  motions,  whether  they  are  left  to  themselves  or  pro- 
jected in  any  manner  whatever.  Here  also,  having  explained  the  connection  between 
these  motions  &  the  distances  of  the  limits,  &  different  cases  of  oscillations,  whether  they 
are  affected  by  external  action  of  other  points,  or  are  not  so  disturbed,  I  make  an  antici- 
patory note  of  the  great  use  to  which  this  will  be  put  in  the  third  part,  for  the  purpose 
of  explaining  various  kinds  of  cohesion,  fermentations,  conflagrations,  emissions  of  vapours, 
the  properties  of  light,  elasticity  and  flexibility. 

There  follows,  from  Art.  204  to  Art.  239,  the  much  more  fruitful  consideration  of  a  204 
system  of  three  points.  The  forces  connected  with  them  can  in  general  be  easily  deter- 
mined for  any  given  positions  of  the  points ;  but,  when  any  position  &  velocity  are  given, 
the  motions  have  not  yet  been  obtained  by  geometricians  in  such  a  form  that  the  general 
calculation  can  be  performed  for  every  possible  case.  So  I  proceed  to  consider  the  forces, 
&  the  huge  variation  that  different  combinations  of  the  points  beget,  although  they  are 
only  three  in  number,  as  far  as  Art.  209.  From  that,  on  to  Art.  214,  I  consider  certain  209 
things  that  have  to  do  with  the  forces  that  arise  from  the  action,  on  each  of  the  points,  of 
the  other  two  together,  &  how  these  urge  the  third  point  not  only  to  approach,  or  recede 
from,  themselves,  but  also  in  a  direction  at  right  angles ;  in  this  connection  there  comes 
forth  an  analogy  with  solidity,  &  a  truly  immense  difference  between  the  several  cases  when 
the  distances  are  very  small,  &  the  greatest  conformity  possible  at  very  great  distances 
such  as  those  at  which  gravity  acts ;  &  I  point  out  what  great  use  will  be  made  of  this  also 
in  explaining  the  constitution  of  Nature.  Then  up  to  Art.  221,  I  give  ocular  demonstra-  214 
tions  of  the  huge  differences  that  there  are  in  the  laws  of  forces  with  which  two  points  act 
upon  a  third,  whether  it  lies  in  the  right  line  joining  them,  or  in  the  right  line  that  is  the 
perpendicular  which  bisects  the  interval  between  them  ;  this  I  do  by  constructing,  from 
the  primary  curve,  curves  representing  the  composite  forces.  Then  in  the  two  articles  221 
that  follow,  I  consider  the  case,  a  really  important  one,  in  which,  by  merely  changing  the 
position  of  the  two  points,  the  third  point,  at  any  and  the  same  definite  interval  situated 
at  the  same  distance  from  the  middle  point  of  the  interval  between  the  two  points,  will 
be  either  continually  attracted,  or  continually  repelled,  or  neither  attracted  nor  repelled  ; 
&  since  a  difference  of  this  kind  should  hold  to  a  much  greater  degree  in  masses,  I  point 
out,  in  Art.  222,  the  great  use  that  will  be  made  of  this  also  in  Physics.  222 

At  this  point  then,  in  Art.  223,  I  pass  from  the  forces  derived  from  two  points  to  the  223 
consideration  of  a  whole  system  of  them ;  and,  as  far  as  Art.  228,  I  study  three  points 
situated  in  a  right  line,  from  the  mutual  forces  of  which  there  arise  certain  relations,  which 
I  return  to  later  in  much  greater  generality  ;  in  this  connection  also  are  outlined,  for  three 
points  only,  matters  that  have  to  do  with  rods,  either  rigid,  flexible  or  elastic,  and  with 
the  lever,  as  well  as  many  other  things ;  these,  too,  are  treated  much  more  generally  later 
on,  when  I  consider  masses.  Then  right  on  to  Art.  238,  I  consider  three  points  that  do 
not  lie  in  a  right  line,  whether  they  are  in  equilibrium,  or  moving  in  the  perimeters  of 
certain  ellipses  or  other  curves.  Here  we  come  across  a  marvellous  analogy  between  certain 
limits  and  the  limits  which  two  points  lying  on  the  axis  of  the  primary  curve  have  with 
respect  to  each  other  ;  &  here  also  a  much  greater  variety  of  cases  for  masses  is  shown, 
&  an  example  is  given  of  the  application  to  solidity,  &  liquefaction,  on  account  of  a  quick 
internal  motion  being  impressed  on  the  points  of  the  body.  Moreover,  in  the  two  articles 
that  then  follow,  I  state  some  general  propositions  with  regard  to  a  system  of  four  points, 
together  with  their  application  to  solid  rods,  both  rigid  and  flexible ;  I  also  give  an  illus- 
tration of  various  classes  of  particles  by  means  of  pyramids,  each  of  which  is  formed  of  four 
points  in  the  most  simple  case,  &  of  four  of  such  pyramids  in  the  more  complicated  cases. 

From  Art.  240  as  far  as  Art.  264,  I  pass  on  to  masses  &  consider  matters  pertaining  to     24° 
the  centre  of  gravity  ;  &  I  prove  that  in  general  there  is  one,  &  only  one,  in  any  given  mass. 
I  show  how  it  can  in  general  be  determined,  &  I  set  forth  in  clear  terms  the  point  that  is 
lacking  in  the  usual  method,  when  it  comes  to  a  question  of  rigorous  proof ;  this  deficiency 


26  SYNOPSIS  TOTIUS  OPERIS 

exemplum  profero  quoddam  ejusdem  generis,  quod  ad  numerorum  pertinet  multiplica- 
tionem,  &  ad  virium  compositionem  per  parallelogramma,  quam  alia  methodo  generaliore 
exhibeo  analoga  illi  ipsi,  qua  generaliter  in  centrum  gravitatis  inquire  :  turn  vero  ejusdem 
ope  demonstro  admodum  expedite,  &  accuratissime  celebre  illud  Newtoni  theorema  de 
statu  centri  gravitatis  per  mutuas  internas  vires  numquam  turbato. 

264  Ejus  tractionis  fructus  colligo  plures  :  conservationem  ejusdem  quantitatis  motuum  in 

265  Mundo  in  eandem  plagam  num.  264,  sequalitatem  actionis,  &  reactionis  in  massis  num.  265, 

266  collisionem  corporum,  &  communicationem  motus  in   congressibus  directis  cum  eorum 
276     legibus,  inde  num.  276  congressus  obliques,  quorum  Theoriam  a  resolutione  motuum  reduce 

277,  278     ad  compositionem  num.  277,  quod  sequent!  numero  278  transfero  ad  incursum  etiam  in 
270     planum  immobile ;   ac  a  num.  279  ad  289  ostendo  nullam  haberi  in  Natura  veram  virium, 
aut   motuum  resolutionem,  sed  imaginariam  tantummodo,  ubi  omnia  evolvo,  &  explico 
casuum  genera,  quae  prima  fronte  virium  resolutionem  requirere  videntur. 

289  A  num.  289  ad  297  leges  expono    compositionis  virium,  &  resolutionis,  ubi  &  illud 

notissimum,  quo  pacto  in  compositione  decrescat  vis,  in  resolutione  crescat,  sed  in  ilia  priore 
conspirantium  summa  semper  maneat,  contrariis  elisis ;  in  hac  posteriore  concipiantur 
tantummodo  binae  vires  contrarise  adjectas,  quse  consideratio  nihil  turbet  phenomena  ; 
unde  fiat,  ut  nihil  inde  pro  virium  vivarum  Theoria  deduci  possit,  cum  sine  iis  explicentur 
omnia,  ubi  plura  itidem  explico  ex  iis  phsenomenis,  quse  pro  ipsis  viribus  vivis  afferri  solent. 


2Q7  A  num.  297  occasione  inde  arrepta  aggredior  qusedam,  quae  ad  legem  continuitatis 

pertinent,  ubique  in  motibus  sancte  servatam,  ac  ostendo  illud,  idcirco  in  collisionibus 
corporum,  ac  in  motu  reflexo,  leges  vulgo  definitas,  non  nisi  proxime  tantummodo  observari, 
&  usque  ad  num.  307  relationes  varias  persequor  angulorum  incidentisa,  &  reflexionis,  sive 
vires  constanter  in  accessu  attrahant,  vel  repellant  constanter,  sive  jam  attrahant,  jam 
repellant :  ubi  &  illud  considero,  quid  accidat,  si  scabrities  superficiei  agentis  exigua  sit, 
quid,  si  ingens,  ac  elementa  profero,  quae  ad  luminis  reflexionem,  &  refractionem  explican- 
dam,  definiendamque  ex  Mechanica  requiritur,  relationem  itidem  vis  absolutae  ad  relativam 
in  obliquo  gravium  descensu,  &  nonnulla,  quae  ad  oscillationum  accuratiorem  Theoriam 
necessaria  sunt,  prorsus  elementaria,  diligenter  expono. 

307  A  num.  307  inquire  in  trium  massarum  systema,  ubi  usque  ad  num.  313  theoremata 

evolvo  plura,  quae  pertinent  ad  directionem  virium  in  singulis  compositarum  e  binis 
reliquarum  actionibus,  ut  illud,  eas  directiones  vel  esse  inter  se  parallelas,  vel,  si  utrinque 

313  indefinite  producantur,  per  quoddam  commune  punctum  transire  omnes  :  turn  usque  ad 
321  theoremata  alia  plura,  quae  pertinent  ad  earumdem  compositarum  virium  rationem  ad 
se  invicem,  ut  illud  &  simplex,  &  elegans,  binarum  massarum  vires  acceleratrices  esse  semper 
in  ratione  composita  ex  tribus  reciprocis  rationibus,  distantise  ipsarum  a  massa  tertia,  sinus 
anguli,  quern  singularum  directio  continet  cum  sua  ejusmodi  distantia,  &  massae  ipsius  earn 
habentis  compositam  vim,  ad  distantiam,  sinum,  massam  alteram  ;  vires  autem  motrices 
habere  tantummodo  priores  rationes  duas  elisa  tertia. 


321  Eorum  theorematum  fructum  colligo  deducens  inde  usque  ad  num.   328,  quae  ad 

aequilibrium  pertinent  divergentium  utcumque  virium,  &  ipsius  aequilibrii  centrum,  ac 
nisum  centri  in  fulcrum,  &  quae  ad  prseponderantiam,  Theoriam  extendens  ad  casum  etiam, 
quo  massae  non  in  se  invicem  agant  mutuo  immediate,  sed  per  intermedias  alias,  quse  nexum 
concilient,  &  virgarum  nectentium  suppleant  vices,  ac  ad  massas  etiam  quotcunque,  quarum 
singulas  cum  centro  conversionis,  &  alia  quavis  assumpta  massa  connexas  concipio,  unde 
principium  momenti  deduce  pro  machinis  omnibus  :  turn  omnium  vectium  genera  evolvo, 
ut  &  illud,  facta  suspensione  per  centrum  gravitatis  haberi  aequilibrium,  sed  in  ipso  centro 
debere  sentiri  vim  a  fulcro,  vel  sustinente  puncto,  sequalem  summae  ponderum  totius 
systematis,  unde  demum  pateat  ejus  ratio,  quod  passim  sine  demonstratione  assumitur, 
nimirum  systemate  quiescente,  &  impedito  omni  partium  motu  per  aequilibrium,  totam 
massam  concipi  posse  ut  in  centro  gravitatis  collectam. 


SYNOPSIS  OF  THE  WHOLE  WORK  27 

I  supply,  &  I  bring  forward  a  certain  example  of  the  same  sort,  that  deals  with  the  multi- 
plication of  numbers,  &  to  the  composition  of  forces  by  the  parallelogram  law ;  the  latter 
I  prove  by  another  more  general  method,  analogous  to  that  which  I  use  in  the  general 
investigation  for  the  centre  of  gravity.  Then  by  its  help  I  prove  very  expeditiously  & 
with  extreme  rigour  that  well-known  theorem  of  Newton,  in  which  he  affirmed  that  the 
state  of  the  centre  of  gravity  is  in  no  way  altered  by  the  internal  mutual  forces. 

I  gather  several  good  results  from  this  method  of  treatment.  In  Art.  264,  the  con-  264 
servation  of  the  same  quantity  of  motion  in  the  Universe  in  one  plane  ;  in  Art.  265  the  265 
equality  of  action  and  reaction  amongst  masses ;  then  the  collision  of  solid  bodies,  and  the  266 
communication  of  motions  in  direct  impacts  &  the  laws  that  govern  them,  &  from  that,  276 
in  Art.  276,  oblique  impacts  ;  in  Art.  277  I  reduce  the  theory  of  these  from  resolution  of  277 
motions  to  compositions,  &  in  the  article  that  follows,  Art.  278,  I  pass  to  impact  on  to  a  278 
fixed  plane;  from  Art.  279  to  Art.  289  I  show  that  there  can  be  no  real  resolution  of  forces  279 
or  of  motions  in  Nature,  but  only  a  hypothetical  one ;  &  in  this  connection  I  consider  & 
explain  all  sorts  of  cases,  in  which  at  first  sight  it  would  seem  that  there  must  be  resolution. 

From  Art.  289  to  Art.  297, 1  state  the  laws  for  the  composition  &  resolution  of  forces ;  289 
here  also  I  give  the  explanation  of  that  well-known  fact,  that  force  decreases  in  composition, 
increases  in  resolution,  but  always  remains  equal  to  the  sum  of  the  parts  acting  in  the  same 
direction  as  itself  in  the  first,  the  rest  being  equal  &  opposite  cancel  one  another ;  whilst 
in  the  second,  all  that  is  done  is  to  suppose  that  two  equal  &  opposite  forces  are  added  on, 
which  supposition  has  no  effect  on  the  phenomena.  Thus  it  comes  about  that  nothing 
can  be  deduced  from  this  in  favour  of  the  Theory  of  living  forces,  since  everything  can  be 
explained  without  them  ;  in  the  same  connection,  I  explain  also  many  of  the  phenomena, 
which  are  usually  brought  forward  as  evidence  in  favour  of  these  '  living  forces.' 

In  Art.  297,  I  seize  the  opportunity  offered  by  the  results  just  mentioned  to  attack  207 
certain  matters  that  relate  to  the  law  of  continuity,  which  in  all  cases  of  motion  is  strictly 
observed  ;  &  I  show  that,  in  the  collision  of  solid  bodies,  &  in  reflected  motion,  the  laws, 
as  usually  stated,  are  therefore  only  approximately  followed.  From  this,  as  far  as  Art.  307, 
I  make  out  the  various  relations  between  the  angles  of  incidence  &  reflection,  whether  the 
forces,  as  the  bodies  approach  one  another,  continually  attract,  or  continually  repel,  or 
attract  at  one  time  &  repel  at  another.  I  also  consider  what  will  happen  if  the  roughness 
of  the  acting  surface  is  very  slight,  &  what  if  it  is  very  great.  I  also  state  the  first  principles, 
derived  from  mechanics,  that  are  required  for  the  explanation  &  determination  of  the 
reflection  &  refraction  of  light ;  also  the  relation  of  the  absolute  to  the  relative  force  in 
the  oblique  descent  of  heavy  bodies ;  &  some  theorems  that  are  requisite  for  the  more 
accurate  theory  of  oscillations  ;  these,  though  quite  elementary,  I  explain  with  great  care. 

From  Art.  307  onwards,  I  investigate  the  system  of  three  bodies ;  in  this  connection, 
as  far  as  Art.  313,  I  evolve  several  theorems  dealing  with  the  direction  of  the  forces  on  each 
one  of  the  three  compounded  from  the  combined  actions  of  the  other  two  ;  such  as  the 
theorem,  that  these  directions  are  either  all  parallel  to  one  another,  or  all  pass  through 
some  one  common  point,  when  they  are  produced  indefinitely  on  both  sides.  Then,  as  ^j, 
far  as  Art.  321,  I  make  out  several  other  theorems  dealing  with  the  ratios  of  these  same 
resultant  forces  to  one  another  ;  such  as  the  following  very  simple  &  elegant  theorem,  that 
the  accelerating  forces  of  two  of  the  masses  will  always  be  in  a  ratio  compounded  of  three 
reciprocal  ratios ;  namely,  that  of  the  distance  of  either  one  of  them  from  the  third  mass, 
that  of  the  sine  of  the  angle  which  the  direction  of  each  force  makes  with  the  corresponding 
distance  of  this  kind,  &  that  of  the  mass  itself  on  which  the  force  is  acting,  to  the  corre- 
sponding distance,  sine  and  mass  for  the  other  :  also  that  the  motive  forces  only  have  the 
first  two  ratios,  that  of  the  masses  being  omitted. 

I  then  collect  the  results  to  be  derived  from  these  theorems,  deriving  from  them,  as  far  ,2I 
as  Art.  328,  theorems  relating  to  the  equilibrium  of  forces  diverging  in  any  manner,  &  the 
centre  of  equilibrium,  &  the  pressure  of  the  centre  on  a  fulcrum.  I  extend  the  theorem 
relating  to  preponderance  to  the  case  also,  in  which  the  masses  do  not  mutually  act  upon 
one  another  in  a  direct  manner,  but  through  others  intermediate  between  them,  which 
connect  them  together,  &  supply  the  place  of  rods  joining  them  ;  and  also  to  any  number  of 
masses,  each  of  which  I  suppose  to  be  connected  with  the  centre  of  rotation  &  some  other 
assumed  mass,  &  from  this  I  derive  the  principles  of  moments  for  all  machines.  Then  I 
consider  all  the  different  kinds  of  levers ;  one  of  the  theorems  that  I  obtain  is,  that,  if  a 
lever  is  suspended  from  the  centre  of  gravity,  then  there  is  equilibrium  ;  but  a  force  should 
be  felt  in  this  centre  from  the  fulcrum  or  sustaining  point,  equal  to  the  sum  of  the  weights 
of  the  whole  system ;  from  which  there  follows  most  clearly  the  reason,  which  is  every- 
where assumed  without  proof,  why  the  whole  mass  can  be  supposed  to  be  collected  at  its 
centre  of  gravity,  so  long  as  the  system  is  in  a  state  of  rest  &  all  motions  of  its  parts  are  pro- 
hibited by  equilibrium. 


28  SYNOPSIS  TOTIUS  OPERIS 

328  A  num.  328  ad  347  deduce  ex  iisdem  theorematis,  quae  pertinent  ad  centrum  oscilla- 

tionis  quotcunque  massarum,  sive  sint  in  eadem  recta,  sive  in  piano  perpendiculari  ad  axem 
rotationis  ubicunque,  quse  Theoria  per  systema  quatuor  massarum,  excolendum  aliquanto 
diligentius,  uberius  promoveri  deberet  &  extendi  ad  generalem  habendum  solidorum  nexum, 

344  qua  re  indicata,  centrum  itidem  percussionis  inde  evolve,  &  ejus  analogiam  cum  centre 
oscillationis  exhibeo. 

347  Collecto  ejusmodi  fructu  ex  theorematis  pertinentibus  ad  massas  tres,  innuo  num.  347, 
quae  mihi  communia  sint  cum  ceteris  omnibus,  &  cum  Newtonianis  potissimum,  pertinentia 
ad  summas  virium,  quas  habet  punctum,  vel  massa  attracta,  vel  repulsa  a  punctis  singulis 

348  alterius  massae  ;  turn  a  num.  348  ad  finem  hujus  partis,  sive  ad  num.  358,  expono  quasdam, 
quae  pertinent  ad  fluidorum  Theoriam,  &  primo  quidem  ad  pressionem,  ubi  illud  innuo 
demonstratum  a  Newtono,  si  compressio  fluidi  sit  proportionalis  vi  comprimenti,  vires 
repulsivas  punctorum  esse  in  ratione  reciproca  distantiarum,  ac  vice  versa  :   ostendo  autem 
illud,  si  eadem  vis  sit  insensibilis,  rem,  praeter  alias  curvas,  exponi  posse  per  Logisticam, 
&  in  fluidis  gravitate  nostra  terrestri  prseditis  pressiones  haberi  debere  ut  altitudines ; 
deinde  vero  attingo  ilia  etiam,  quae  pertinent  ad  velocitatem  fluidi   erumpentis  e  vase,  & 
expono,  quid  requiratur,  ut  ea  sit  sequalis  velocitati,  quae  acquiretur  cadendo  per  altitudinem 
ipsam,  quemadmodum  videtur  res  obtingere  in  aquae  efHuxu  :    quibus  partim  expositis, 
partim  indicatis,  hanc  secundam  partem  conclude. 


PARS  III 

358  Num.  358  propono  argumentum  hujus  tertise  partis,  in  qua  omnes  e  Theoria  mea 

360  generales  materis  proprietates  deduce,  &  particulares  plerasque  :  turn  usque  ad  num.  371 
ago  aliquanto  fusius  de  impenetrabilitate,  quam  duplicis  generis  agnosco  in  meis  punctorum 
inextensorum  massis,  ubi  etiam  de  ea  apparenti  quadam  compenetratione  ago,  ac  de  luminis 
trarlsitu  per  substantias  intimas  sine  vera  compenetratione,  &  mira  quaedam  phenomena 

371  hue  pertinentia  explico  admodum  expedite.  Inde  ad  num.  375  de  extensione  ago,  quae 
mihi  quidem  in  materia,  &  corporibus  non  est  continua,  sed  adhuc  eadem  praebet  phaeno- 
menae  sensibus,  ac  in  communi  sententia  ;  ubi  etiam  de  Geometria  ago,  quae  vim  suam  in 

375  mea  Theoria  retinet  omnem  :  turn  ad  num.  383  figurabilitatem  perseqUor,  ac  molem, 
massam,  densitatem  singillatim,  in  quibus  omnibus  sunt  quaedam  Theoriae  meae  propria 

383     scitu  non  indigna.     De  Mobilitate,  &  Motuum  Continuitate,  usque  ad  num.  388  notatu 

388  digna  continentur  :  turn  usque  ad  num.  391  ago  de  aequalitate  actionis,  &  reactionis,  cujus 
consectaria  vires  ipsas,  quibus  Theoria  mea  innititur,  mirum  in  modum  conformant. 
Succedit  usque  ad  num.  398  divisibilitas,  quam  ego  ita  admitto,  ut  quaevis  massa  existens 
numerum  punctorum  realium  habeat  finitum  tantummodo,  sed  qui  in  data  quavis  mole 
possit  esse  utcunque  magnus ;  quamobrem  divisibilitati  in  infinitum  vulgo  admissae  sub- 
stituo  componibilitatem  in  infinitum,  ipsi,  quod  ad  Naturae  phenomena  explicanda 

398  pertinet,  prorsus  aequivalentem.  His  evolutis  addo  num.  398  immutabilitatem  primorum 
materiae  elementorum,  quse  cum  mihi  sint  simplicia  prorsus,  &  inextensa,  sunt  utique 
immutabilia,  &  ad  exhibendam  perennem  phasnomenorum  seriem  aptissima. 


399  A  num.  399  ad  406  gravitatem  deduco  ex  mea  virium  Theoria,  tanquam  ramum 

quendam  e  communi  trunco,  ubi  &  illud  expono,  qui  fieri  possit,  ut  fixae  in  unicam  massam 

406  non  coalescant,  quod  gravitas  generalis  requirere  videretur.  Inde  ad  num.  419  ago  de 
cohaesione,  qui  est  itidem  veluti  alter  quidam  ramus,  quam  ostendo,  nee  in  quiete  con- 
sistere,  nee  in  motu  conspirante,  nee  in  pressione  fluidi  cujuspiam,  nee  in  attractione 
maxima  in  contactu,  sed  in  limitibus  inter  repulsionem,  &  attractionem  ;  ubi  &  problema 
generale  propono  quoddam  hue  pertinens,  &  illud  explico,  cur  massa  fracta  non  iterum 
coalescat,  cur  fibrae  ante  fractionem  distendantur,  vel  contrahantur,  &  innuo,  quae  ad 
cohaesionem  pertinentia  mihi  cum  reliquis  Philosophis  communia  sint. 

419  A  cohacsione  gradum  facio  num.  419  ad  particulas,  quae  ex  punctis  cohaerentibus 

efformantur,  de   quibus  .ago   usque  ad  num.  426.  &  varia  persequor    earum   discrimina  : 


SYNOPSIS  OF  THE  WHOLE   WORK  29 

From  Art.  328  to  Art.  347,  I  deduce  from  these  same  theorems,  others  that  relate  to     328 
the  centre  of  oscillation  of  any  number  of  masses,  whether  they  are  in  the  same  right  line, 
or  anywhere  in  a  plane  perpendicular  to  the  axis  of  rotation  ;  this  theory  wants  to  be  worked 
somewhat  more  carefully  with  a  system  of  four  bodies,  to  be  gone  into  more  fully,  &  to 
be  extended  so  as  to  include  the  general  case  of  a  system  of  solid  bodies ;   having  stated 
this,  I  evolve  from  it  the  centre  of  percussion,  &  I  show  the  analogy  between  it  &  the  centre     344 
of  oscillation. 

I  obtain  all  such  results  from  theorems  relating  to  three  masses.  After  that,  in  Art.  347 
347,  I  intimate  the  matters  in  which  I  agree  with  all  others,  &  especially  with  the  followers 
of  Newton,  concerning  sums  of  forces,  acting  on  a  point,  or  an  attracted  or  repelled  mass, 
due  to  the  separate  points  of  another  mass.  Then,  from  Art.  348  to  the  end  of  this  part,  348 
i.e.,  as  far  as  Art.  359,  I  expound  certain  theorems  that  belong  to  the  theory  of  fluids ;  & 
first  of  all,  theorems  with  regard  to  pressure,  in  connection  with  which  I  mention  that  one 
which  was  proved  by  Newton,  namely,  that,  if  the  compression  of  a  fluid  is  proportional  to 
the  compressing  force,  then  the  repulsive  forces  between  the  points  are  in  the  reciprocal 
ratio  of  the  distances,  &  conversely.  Moreover,  I  show  that,  if  the  same  force  is  insen- 
sible, then  the  matter  can  be  represented  by  the  logistic  &  other  curves ;  also  that  in  fluids 
subject  to  our  terrestrial  gravity  pressures  should  be  found  proportional  to  the  depths. 
After  that,  I  touch  upon  those  things  that  relate  to  the  velocity  of  a  fluid  issuing  from  a 
vessel ;  &  I  show  what  is  necessary  in  order  that  this  should  be  equal  to  the  velocity  which 
would  be  acquired  by  falling  through  the  depth  itself,  just  as  it  is  seen  to  happen  in  the 
case  of  an  efflux  of  water.  These  things  in  some  part  being  explained,  &  in  some  part 
merely  indicated,  I  bring  this  second  part  to  an  end. 

PART  III 

In  Art.  358,  I  state  the  theme  of  this  third  part ;  in  it  I  derive  all  the  general  &  most     358 
of  the  special,  properties  of  matter  from  my  Theory.     Then,  as  far  as  Art.  371,  I  deal  some-     360 
what  more  at  length  with  the  subject  of  impenetrability,  which  I  remark  is  of  a  twofold 
kind  in  my  masses  of  non-extended  points ;  in  this  connection  also,  I  deal  with  a  certain 
apparent  case  of  compenetrability,  &  the  passage  of  light  through  the  innermost  parts  of 
bodies  without  real  compenetration ;    I  also  explain  in  a  very  summary  manner  several 
striking  phenomena  relating  to  the  above.     From  here  on  to  Art.  375,  I  deal  with  exten-     371 
sion  ;    this  in  my  opinion  is  not  continuous  either  in  matter  or  in  solid  bodies,  &  yet  it 
yields  the  same  phenomena  to  the  senses  as  does  the  usually  accepted  idea  of  it ;    here  I 
also  deal  with  geometry,  which  conserves  all  its  power  under  my  Theory.     Then,  as  far     375 
as  Art.  383,  I  discuss  figurability,  volume,  mass  &  density,  each  in  turn  ;    in  all  of  these 
subjects  there  are  certain  special  points  of  my  Theory  that  are  not  unworthy  of  investi- 
gation.    Important  theorems  on  mobility  &  continuity  of  motions  are  to  be  found  from 
here  on  to  Art.  388  ;   then,  as  far  as  Art.  391,  I  deal  with  the  equality  of  action  &  reaction, 
&  my  conclusions  with  regard  to  the  subject  corroborate  in  a  wonderful  way  the  hypothesis 
of  those  forces,  upon  which  my  Theory  depends.     Then  follows  divisibility,  as  far  as  Art.     39 1 
398  ;   this  principle  I  admit  only  to  the  extent  that  any  existing  mass  may  be  made  up  of 
a  number  of  real  points  that  are  finite  only,  although  in  any  given  mass  this  finite  number 
may  be  as  great  as  you  please.     Hence  for  infinite  divisibility,  as  commonly  accepted,  I 
substitute  infinite  multiplicity ;  which  comes  to  exactly  the  same  thing,  as  far  as  it  is 
concerned  with  the  explanation  of  the  phenomena  of  Nature.     Having  considered  these 
subjects  I  add,  in  Art.  398,  that  of  the  immutability  of  the  primary  elements  of  matter ;     398 
according  to  my  idea,  these  are  quite  simple  in  composition,  of  no  extent,  they  are  every- 
where unchangeable,  &  hence  are  splendidly  adapted  for  explaining  a  continually  recurring 
set  of  phenomena. 

From  Art.  399  to  Art.  406, 1  derive  gravity  from  my  Theory  of  forces,  as  if  it  were  a  399 
particular  branch  on  a  common  trunk  ;  in  this  connection  also  I  explain  how  it  can  happen 
that  the  fixed  stars  do  not  all  coalesce  into  one  mass,  as  would  seem  to  be  required  under  406 
universal  gravitation.  Then,  as  far  as  Art.  419,  I  deal  with  cohesion,  which  is  also  as  it 
were  another  branch ;  I  show  that  this  is  not  dependent  upon  quiescence,  nor  on  motion 
that  is  the  same  for  all  parts,  nor  on  the  pressure  of  some  fluid,  nor  on  the  idea  that  the 
attraction  is  greatest  at  actual  contact,  but  on  the  limits  between  repulsion  and  attraction. 
I  propose,  &  solve,  a  general  problem  relating  to  this,  namely,  why  masses,  once  broken, 
do  not  again  stick  together,  why  the  fibres  are  stretched  or  contracted  before  fracture 
takes  place  ;  &  I  intimate  which  of  my  ideas  relative  to  cohesion  are  the  same  as  those 
held  by  other  philosophers. 

In  Art.  419, 1  pass  on  from  cohesion  to  particles  which  are  formed  from  a  number  of    4J9 
cohering  points ;  &  I  consider  these  as  far  as  Art.  426,  &  investigate  the  various  distinctions 


30  SYNOPSIS  TOTIUS  OPERIS 

ostendo  nimirum,  quo  pacto  varias  induere  possint  figuras  quascunque,  quarum  tenacissime 
sint ;  possint  autem  data  quavis  figura  discrepare  plurimum  in  numero,  &  distributione 
punctorum,  unde  &  oriantur  admodum  inter  se  diversae  vires  unius  particulae  in  aliam,  ac 
itidem  diversae  in  diversis  partibus  ejusdem  particulae  respectu  diversarum  partium,  vel 
etiam  respectu  ejusdem  partis  particulse  alterius,  cum  a  solo  numero,  &  distributione 
punctorum  pendeat  illud,  ut  data  particula  datam  aliam  in  datis  earum  distantiis,  & 
superficierum  locis,  vel  attrahat,  vel  repellat,  vel  respectu  ipsius  sit  prorsus  iners  :  turn  illud 
addo,  particulas  eo  dimcilius  dissolubiles  esse,  quo  minores  sint ;  debere  autem  in  gravitate 
esse  penitus  uniformes,  quaecunque  punctorum  dispositio  habeatur,  &  in  aliis  proprietatibus 
plerisque  debere  esse  admodum  (uti  observamus)  diversas,  quae  diversitas  multo  major  in 
majoribus  massis  esse  debeat. 

426  A  num.  426  ad  446  de  solidis,  &   fluidis,  quod   discrimen   itidem   pertinet   ad  varia 

cohaesionum  genera  ;  &  discrimen  inter  solida,  &  fluida  diligenter  expono,  horum  naturam 
potissimum  repetens  ex  motu  faciliori  particularum  in  gyrum  circa  alias,  atque  id  ipsum  ex 
viribus  circumquaque  aequalibus  ;  illorum  vero  ex  inaequalitate  virium,  &  viribus  quibusdam 
in  latus,  quibus  certam  positionem  ad  se  invicem  servare  debeant.  Varia  autem  distinguo 
fluidorum  genera,  &  discrimen  profero  inter  virgas  rigidas,  flexiles,  elasticas,  fragiles,  ut  & 
de  viscositate,  &  humiditate  ago,  ac  de  organicis,  &  ad  certas  figuras  determinatis  corporibus, 
quorum  efformatio  nullam  habet  difficultatem,  ubi  una  particula  unam  aliam  possit  in 
certis  tantummodo  superficiei  partibus  attrahere,  &  proinde  cogere  ad  certam  quandam 
positionem  acquirendam  respectu  ipsius,  &  retinendam.  Demonstro  autem  &  illud,  posse 
admodum  facile  ex  certis  particularum  figuris,  quarum  ipsae  tenacissimae  sint,  totum  etiam 
Atomistarum,  &  Corpuscularium  systema  a  mea  Theoria  repeti  ita,  ut  id  nihil  sit  aliud, 
nisi  unicus  itidem  hujus  veluti  trunci  foecundissimi  ramus  e  diversa  cohaesionis  ratione 
prorumpens.  Demum  ostendo,  cur  non  quaevis  massa,  utut  constans  ex  homogeneis 
punctis,  &  circa  se  maxime  in  gyrum  mobilibus,  fluida  sit ;  &  fluidorum  resistentiam  quoque 
attingo,  in  ejus  leges  inquirens. 


446  A  num.  446  ad  450  ago  de  iis,  quae  itidem  ad  diversa  pertinent  soliditatis  genera,  nimirum 

de  elasticis,  &  mollibus,  ilia  repetens  a  magna  inter  limites  proximos  distantia,  qua  fiat,  ut 
puncta  longe  dimota  a  locis  suis,  idem  ubique  genus  virium  sentiant,  &  proinde  se  ad 
priorem  restituant  locum  ;  hasc  a  limitum  frequentia,  atque  ingenti  vicinia,  qua  fiat,  ut  ex 
uno  ad  alium  delata  limitem  puncta,  ibi  quiescant  itidem  respective,  ut  prius.  Turn  vero 
de  ductilibus,  &  malleabilibus  ago,  ostendens,  in  quo  a  fragilibus  discrepent  :  ostendo  autem, 
haec  omnia  discrimina  a  densitate  nullo  modo  pendere,  ut  nimirum  corpus,  quod 
multo  sit  altero  densius,  possit  tarn  multo  majorem,  quam  multo  minorem  soliditatem,  & 
cohaesionem  habere,  &  quaevis  ex  proprietatibus  expositis  aeque  possit  cum  quavis  vel  majore, 
vel  minore  densitate  componi. 


450  Num.  450  inquire  in  vulgaria  quatuor  elementa  ;  turn  a  num.  451  ad  num.  467  persequor 

452  chemicas  operationes ;  num.  452  explicans  dissolutionem,  453  praecipitationem,  454,  &  455 
commixtionem  plurium  substantiarum  in  unam  :  turn  num.  456,  &  457  liquationem  binis 
methodis,  458  volatilizationem,  &  effervescentiam,  461  emissionem  efHuviorum,  quae  e  massa 
constanti  debeat  esse  ad  sensum  constans,  462  ebullitionem  cum  variis  evaporationum 
generibus  ;  463  deflagrationem,  &  generationem  aeris  ;  464  crystallizationem  cum  certis 
figuris ;  ac  demum  ostendo  illud  num.  465,  quo  pacto  possit  fermentatio  desinere  ;  &  num. 
466,  quo  pacto  non  omnia  fermentescant  cum  omnibus. 

467  A  fermentatione  num.  467  gradum  facio  ad  ignem,  qui  mihi  est  fermentatio  quaedam 

substantiae  lucis  cum  sulphurea  quadam  substantia,  ac  plura  inde  consectaria  deduce  usque 

471  ad  num.  471  ;    turn  ab  igne  ad  lumen  ibidem  transeo,  cujus  proprietates  praecipuas,  ex 

472  quibus  omnia  lucis  phaenomena  oriuntur,  propono  num.  472,  ac  singulas  a  Theoria  mea 
deduce,  &  fuse  explico  usque  ad  num.  503,  nimirum  emissionem  num.  473,  celeritatem  474, 
propagationem  rectilineam  per  media  homogenea,  &  apparentem  tantummodo  compene- 
trationem  a  num.  475  ad  483,  pellucidatem,  &  opacitatem  num.  483,  reflexionem  ad  angulos 
aequales  inde  ad  484,  refractionem   ad   487,  tenuitatem   num.    487,  calorem,  &   ingentes 
intestines  motus  allapsu  tenuissimae  lucis  genitos,  num.  488,  actionem  majorem  corporum 
eleosorum,  &  sulphurosorum  in  lumen  num.  489  :  turn  num.  490  ostendo,  nullam  resist- 


SYNOPSIS  OF  THE  WHOLE  WORK  31 

between  them.  I  show  how  it  is  possible  for  various  shapes  of  all  sorts  to  be  assumed, 
which  offer  great  resistance  to  rupture  ;  &  how  in  a  given  shape  they  may  differ  very  greatly 
in  the  number  &  disposition  of  the  points  forming  them.  Also  that  from  this  fact  there 
arise  very  different  forces  for  the  action  of  one  particle  upon  another,  &  also  for  the  action 
of  different  parts  of  this  particle  upon  other  different  parts  of  it,  or  on  the  same  part  of 
another  particle.  For  that  depends  solely  on  the  number  &  distribution  of  the  points, 
so  that  one  given  particle  either  attracts,  or  repels,  or  is  perfectly  inert  with  regard  to 
another  given  particle,  the  distances  between  them  and  the  positions  of  their  surfaces  being 
also  given.  Then  I  state  in  addition  that  the  smaller  the  particles,  the  greater  is  the  diffi- 
culty in  dissociating  them  ;  moreover,  that  they  ought  to  be  quite  uniform  as  regards 
gravitation,  no  matter  what  the  disposition  of  the  points  may  be  ;  but  in  most  other 
properties  they  should  be  quite  different  from  one  another  (which  we  observe  to  be  the 
case) ;  &  that  this  difference  ought  to  be  much  greater  in  larger  masses. 

From  Art.  426  to  Art.  446, 1  consider  solids  &  fluids,  the  difference  between  which  is  426 
also  a  matter  of  different  kinds  of  cohesion.  I  explain  with  great  care  the  difference 
between  solids  &  fluids ;  deriving  the  nature  of  the  latter  from  the  greater  freedom  of  motion 
of  the  particles  in  the  matter  of  rotation  about  one  another,  this  being  due  to  the  forces 
being  nearly  equal ;  &  that  of  the  former  from  the  inequality  of  the  forces,  and  from  certain 
lateral  forces  which  help  them  to  keep  a  definite  position  with  regard  to  one  another.  I 
distinguish  between  various  kinds  of  fluids  also,  &  I  cite  the  distinction  between  rigid, 
flexible,  elastic  &  fragile  rods,  when  I  deal  with  viscosity  &  humidity  ;  &  also  in  dealing  with 
organic  bodies  &  those  solids  bounded  by  certain  fixed  figures,  of  which  the  formation 
presents  no  difficulty ;  in  these  one  particle  can  only  attract  another  particle  in  certain 
parts  of  the  surface,  &  thus  urge  it  to  take  up  some  definite  position  with  regard  to  itself, 
&  keep  it  there.  I  also  show  that  the  whole  system  of  the  Atomists,  &  also  of  the  Corpus- 
cularians,  can  be  quite  easily  derived  by  my  Theory,  from  the  idea  of  particles  of  definite 
shape,  offering  a  high  resistance  to  deformation  ;  so  that  it  comes  to  nothing  else  than 
another  single  branch  of  this  so  to  speak  most  fertile  trunk,  breaking  forth  from  it 
on  account  of  a  different  manner  of  cohesion.  Lastly,  I  show  the  reason  why  it  is  that 
not  every  mass,  in  spite  of  its  being  constantly  made  up  of  homogeneous  points,  &  even 
these  in  a  high  degree  capable  of  rotary  motion  about  one  another,  is  a  fluid.  I  also  touch 
upon  the  resistance  of  fluids,  &  investigate  the  laws  that  govern  it. 

From  Art.  446  to  Art.  450,  I  deal  with  those  things  that  relate  to  the  different  kinds  446 
of  solidity,  that  is  to  say,  with  elastic  bodies,  &  those  that  are  soft.  I  attribute  the  nature 
of  the  former  to  the  existence  of  a  large  interval  between  the  consecutive  limits,  on  account 
of  which  it  comes  about  that  points  that  are  far  removed  from  their  natural  positions  still 
feel  the  effects  of  the  same  kind  of  forces,  &  therefore  return  to  their  natural  positions ; 
&  that  of  the  latter  to  the  frequency  &  great  closeness  of  the  limits,  on  account  of  which  it 
comes  about  that  points  that  have  been  moved  from  one  limit  to  another,  remain  there 
in  relative  rest  as  they  were  to  start  with.  Then  I  deal  with  ductile  and  malleable  solids, 
pointing  out  how  they  differ  from  fragile  solids.  Moreover  I  show  that  all  these  differ- 
ences are  in  no  way  dependent  on  density ;  so  that,  for  instance,  a  body  that  is  much  more 
dense  than  another  body  may  have  either  a  much  greater  or  a  much  less  solidity  and 
cohesion  than  another  ;  in  fact,  any  of  the  properties  set  forth  may  just  as  well  be  combined 
with  any  density  either  greater  or  less. 

In  Art.  450  I  consider  what  are  commonly  called  the  "  four  elements  "  ;    then  from     450 
Art.  451  to  Art.  467,  I  treat  of  chemical  operations ;  I  explain  solution  in  Art.  452,  preci-     452 
pitation  in  Art.  453,  the  mixture  of  several  substances  to  form  a  single  mass  in  Art.  454, 
455,  liquefaction  by  two  methods  in  Art.  456,  457,  volatilization  &  effervescence  in  Art. 
458,  emission  of  effluvia  (which  from  a  constant  mass  ought  to  be  approximately  constant) 
in. Art.  461,  ebullition  &  various  kinds  of  evaporation  in  Art.  462,  deflagration  &  generation 
of  gas  in  Art.  463,  crystallization  with  definite  forms  of  crystals  in  Art.  464  ;  &  lastly,  I  show, 
in  Art.  465,  how  it  is  possible  for  fermentation  to  cease,  &  in  Art.  466,  how  it  is  that  any 
one  thing  does  not  ferment  when  mixed  with  any  other  thing. 

From  fermentation  I  pass  on,  in  Art.  467,  to  fire,  which  I  look  upon  as  a  fermentation     467 
of  some  substance  in  light  with  some  sulphureal  substance ;   &  from  this  I  deduce  several 
propositions,  up  to  Art.  471.     There  I  pass  on  from  fire  to  light,  the  chief  properties  of     471 
which,  from  which  all  the  phenomena  of  light  arise,  I  set  forth  in  Art.  472  ;  &  I  deduce     472 
&  fully  explain  each  of  them  in  turn  as  far  as  Art.  503.     Thus,  emission  in  Art.  473,  velo- 
city in  Art.  474,  rectilinear  propagation  in  homogeneous  media,  &  a  compenetration  that 
is  merely  apparent,  from  Art.  475  on  to  Art.  483,  pellucidity  &  opacity  in  Art.  483,  reflec- 
tion at  equal  angles  to  Art.  484,  &  refraction  to  Art.  487,  tenuity  in  Art.  487,  heat  &  the 
great  internal  motions  arising  from  the  smooth  passage  of  the  extremely  tenuous  light  in 
Art.  488,  the  greater  action  of  oleose  &  sulphurous  bodies  on  light  in  Art.  489.     Then  I 


32  SYNOPSIS  TOTIUS  OPERIS 

entiam  veram  pati,  ac  num.  491  explico,  unde  sint  phosphora,  num.  492  cur  lumen  cum 
majo  e  obliquitate  incidens  reflectatur  magis,  num.  493  &  494  unde  diversa  refrangibilitas 
ortum  ducat,  ac  num.  495,  &  496  deduce  duas  diversas  dispositiones  ad  asqualia  redeuntes 
intervalla,  unde  num.  497  vices  illas  a  Newtono  detectas  facilioris  reflexionis,  &  facilioris 
transmissus  eruo,  &  num.  498  illud,  radios  alios  debere  reflecti,  alios  transmitti  in  appulsu 
ad  novum  medium,  &  eo  plures  reflecti,  quo  obliquitas  incidentise  sit  major,  ac  num. 
499  &  500  expono,  unde  discrimen  in  intervallis  vicium,  ex  quo  uno  omnis  naturalium 
colorum  pendet  Newtoniana  Theoria.  Demum  num.  501  miram  attingo  crystalli 
Islandicse  proprietatem,  &  ejusdem  causam,  ac  num.  502  diffractionem  expono,  quse  est 
quaedam  inchoata  refractio,  sive  reflexio. 

503  Post  lucem  ex  igne  derivatam,  quse  ad  oculos  pertinet,  ago  brevissime  num.  503  de 

504  sapore,  &  odore,  ac  sequentibus  tribus  numeris  de  sono  :    turn  aliis  quator  de  tactu,  ubi 
507     etiam  de  frigore,  &  calore  :    deinde  vero  usque  ad  num.  514  de  electricitate,  ubi  totam 
511     Franklinianam  Theoriam  ex  meis  principiis  explico,  eandem  ad  bina  tantummodo  reducens 

principia,  quse  ex  mea  generali  virium  Theoria  eodem  fere  pacto  deducuntur,  quo  prsecipi- 
514     tationes,  atque  dissolutiones.     Demum  num.  514,  ac  515  magnetismum  persequor,  tam 
directionem  explicans,  quam  attractionem  magneticam. 

516  Hisce  expositis,  quas  ad  particulares  .etiam  proprietates  pertinent,  iterum  a  num.  516 

ad  finem  usque  generalem  corporum  complector  naturam,  &  quid  materia  sit,  quid  forma, 
quse  censeri  debeant  essentialia,  quse  accidentialia  attributa,  adeoque  quid  transformatio 
sit,  quid  alteratio,  singillatim  persequor,  &  partem  hanc  tertiam  Theorise  mesa  absolve. 


De  Appendice  ad  Metaphysicam  pertinente  innuam  hie  illud  tantummodo,  me  ibi 
exponere  de  anima  illud  inprimis,  quantum  spiritus  a  materia  differat,  quern  nexum  anima 
habeat  cum  corpore,  &  quomodo  in  ipsum  agat  :  turn  de  DEO,  ipsius  &  existentiam  me 
pluribus  evincere,  quae  nexum  habeant  cum  ipsa  Theoria  mea,  &  Sapientiam  inprimis,  ac 
Providentiam,  ex  qua  gradum  ad  revelationem  faciendum  innuo  tantummodo.  Sed  hsec 
in  antecessum  veluti  delibasse  sit  satis. 


SYNOPSIS  OF  THE   WHOLE  WORK  33 

show,  in  Art.  490,  that  it  suffers  no  real  resistance,  &  in  Art.  491  I  explain  the  origin  of 
bodies  emitting  light,  in  Art.  492  the  reason  why  light  that  falls  with  greater  obliquity 
is  reflected  more  strongly,  in  Art.  493,  494  the  origin  of  different  degrees  of  refrangibility, 
&  in  Art.  495,  496  I  deduce  that  there  are  two  different  dispositions  recurring  at  equal 
intervals ;  hence,  in  Art.  497,  I  bring  out  those  alternations,  discovered  by  Newton,  of 
easier  reflection  &  easier  transmission,  &  in  Art.  498  I  deduce  that  some  rays  should  be 
reflected  &  others  transmitted  in  the  passage  to  a  fresh  medium,  &  that  the  greater  the  obli- 
quity of  incidence,  the  greater  the  number  of  reflected  rays.  In  Art.  499,  500  I  state  the 
origin  of  the  difference  between  the  lengths  of  the  intervals  of  the  alternations ;  upon  this 
alone  depends  the  whole  of  the  Newtonian  theory  of  natural  colours.  Finally,  in  Art.  501, 
I  touch  upon  the  wonderful  property  of  Iceland  spar  &  its  cause,  &  in  Art.  502  I  explain 
diffraction,  which  is  a  kind  of  imperfect  refraction  or  reflection. 

After  light  derived  from  fire,  which  has  to  do  with  vision,  I  very  briefly  deal  with 
taste  &  smell  in  Art.  503,  £  of  sound  in  the  three  articles  that  follow  next.     Then,  in  the     S°3 
next  four  articles,  I  consider  touch,  &  in  connection  with  it,  cold  &  heat  also.     After  that,     5°4 
as  far  as  Art.  514,  I  deal  with  electricity ;   here  I  explain  the  whole  of  the  Franklin  theory     5°7 
by  means  of  my  principles ;    I  reduce  this  theory  to  two  principles  only,  &  these  are     5 1 1 
derived  from  my  general  Theory  of  forces  in  almost  the  same  manner  as  I  have  already  derived 
precipitations  &  solutions.     Finally,  in  Art.  514,  515,  I  investigate  magnetism,  explaining     5H 
both  magnetic  direction  £  attraction. 

These  things  being  expounded,  all  of  which  relate  to  special  properties,  I  once  more 
consider,  in  the  articles  from  516  to  the  end,  the  general  nature  of  bodies,  what  matter  is,     516 
its  form,  what  things  ought  to  be  considered  as  essential,  &  what  as  accidental,  attributes ; 
and  also  the  nature  of  transformation  and  alteration  are  investigated,  each  in  turn ;    & 
thus  I  bring  to  a  close  the  third  part  of  my  Theory. 

I  will  mention  here  but  this  one  thing  with  regard  to  the  appendix  on  Metaphysics  ; 
namely,  that  I  there  expound  more  especially  how  greatly  different  is  the  soul  from  matter, 
the  connection  between  the  soul  &  the  body,  &  the  manner  of  its  action  upon  it.  Then 
with  regard  to  GOD,  I  prove  that  He  must  exist  by  many  arguments  that  have  a  close  con- 
nection with  this  Theory  of  mine ;  I  especially  mention,  though  but  slightly,  His  Wisdom 
and  Providence,  from  which  there  is  but  a  step  to  be  made  towards  revelation.  But  I  think 
that  I  have,  so  to  speak,  given  my  preliminary  foretaste  quite  sufficiently. 


[I]    PHILOSOPHIC    NATURALIS    THEORIA 


In  quo  conveniat 
cum  systemate 
Newtoniano,  & 
Leibnitiano. 


Cujusmodi  systema> 
Theoria  exhibeat. 


PARS  I 

Theorice  expositio,  analytica  deductio^  &  vindicatio. 

lRIUM  mutuarum  Theoria,  in  quam  incidi  jam  ab  Anno  1745,  dum  e 
notissimis  principiis  alia  ex  aliis  consectaria  eruerem,  &  ex  qua  ipsam 
simplicium  materise  elementorum  constitutionem  deduxi,  systema 
exhibet  medium  inter  Leibnitianum,  &  Newtonianum,  quod  nimirum 
&  ex  utroque  habet  plurimum,  &  ab  utroque  plurimum  dissidet ;  at 
utroque  in  immensum  simplicius,  proprietatibus  corporum  generalibus 
sane  omnibus,  &  [2]  peculiaribus  quibusque  praecipuis  per  accuratissimas 
demonstrationes  deducendis  est  profecto  mirum  in  modum  idoneum. 

2.  Habet  id  quidem  ex  Leibnitii  Theoria  elementa  prima  simplicia,  ac  prorsus  inex- 
tensa  :  habet  ex  Newtoniano  systemate  vires  mutuas,  quae  pro  aliis  punctorum  distantiis  a 
se  invicem  aliae  sint ;  &  quidem  ex  ipso  itidem  Newtono  non  ejusmodi  vires  tantummodo, 
quse  ipsa  puncta  determinent  ad  accessum,  quas  vulgo  attractiones  nominant ;  sed  etiam 
ejusmodi,  quae  determinent  ad  recessum,  &  appellantur  repulsiones  :  atque  id  ipsum  ita, 
ut,  ubi  attractio  desinat,  ibi,  mutata  distantia,  incipiat  repulsio,  &  vice  versa,  quod  nimirum 
Newtonus  idem  in  postrema  Opticse  Quaestione  proposuit,  ac  exemplo  transitus  a  positivis 
ad  negativa,  qui  habetur  in  algebraicis  formulis,  illustravit.  Illud  autem  utrique  systemati 
commune  est  cum  hoc  meo,  quod  quaevis  particula  materiae  cum  aliis  quibusvis,  utcunque 
remotis,  ita  connectitur,  ut  ad  mutationem  utcunque  exiguam  in  positione  unius  cujusvis, 
determinationes  ad  motum  in  omnibus  reliquis  immutentur,  &  nisi  forte  elidantur  omnes 
oppositas,  qui  casus  est  infinities  improbabilis,  motus  in  iis  omnibus  aliquis  inde  ortus 
habeatur. 


In  quo  differat  a 
Leibnitiano  &  ipsi 
praestet. 


3.  Distat  autem  a  Leibnitiana  Theoria  longissime,  turn  quia  nullam  extensionem 
continuam  admittit,  quae  ex  contiguis,  &  se  contingentibus  inextensis  oriatur  :  in  quo 
quidem  dirficultas  jam  olim  contra  Zenonem  proposita,  &  nunquam  sane  aut  soluta  satis, 
aut  solvenda,  de  compenetratione  omnimoda  inextensorum  contiguorum,  eandem  vim 
adhuc  habet  contra  Leibnitianum  systema  :  turn  quia  homogeneitatem  admittit  in  elementis, 
omni  massarum  discrimine  a  sola  dispositione,  &  diversa  combinatione  derivato,  ad  quam 
homogeneitatem  in  elementis,  &  discriminis  rationem  in  massis,  ipsa  nos  Naturae  analogia 
ducit,  ac  chemicae  resolutiones  inprimis,  in  quibus  cum  ad  adeo  pauciora  numero,  &  adeo 
minus  inter  se  diversa  principiorum  genera,  in  compositorum  corporum  analysi  deveniatur, 
id  ipsum  indicio  est,  quo  ulterius  promoveri  possit  analysis,  eo  ad  majorem  simplicitatem, 
&  homogeneitatem  devenire  debere,  adeoque  in  ultima  demum  resolutione  ad  homogenei- 
tatem, &  simplicitatem  summam,  contra  quam  quidem  indiscernibilium  principium,  & 
principium  rationis  sufficients  usque  adeo  a  Leibnitianis  depraedicata,  meo  quidem  judicio, 
nihil  omnino  possunt. 


in  quo  differat  a  A    Distat  itidem  a  Newtoniano  systemate  quamplunmum,  turn  in  eo,  quod  ea,  quae 

Newtoniano  &  ipsi     XT  .      .  r\          •  r\      •  r 

praestet.  Newtonus  in  ipsa  postremo  (Juaestione  (Jpticae  conatus  est  expncare  per  tna  pnncipia, 

gravitatis,  cohsesionis,  fermentationis,  immo  &  reliqua  quamplurima,  quae  ab  iis  tribus 
principiis  omnino  non  pendent,  per  unicam  explicat  legem  virium,  expressam  unica,  &  ex 
pluribus  inter  se  commixtis  non  composita  algebraica  formula,  vel  unica  continua  geometrica 
curva  :  turn  in  eo,  quod  in  mi-[3]-nimis  distantiis  vires  admittat  non  positivas,  sive 
attractivas,  uti  Newtonus,  sed  negativas,  sive  repulsivas,  quamvis  itidem  eo  majores  in 


34 


A   THEORY   OF  NATURAL   PHILOSOPHY 

PART  I 

Exposition  ^   ^Analytical  Derivation    &   Proof  of  the    Theory 

I.  '    ^i    ^^     HE  following  Theory  of  mutual  forces,  which  I  lit  upon  as  far  back  as  the  year  The  kind  of  sys- 
1745,  whilst  I  was  studying  various  propositions  arising  from  other  very  p^ents.6 
well-known  principles,  &  from  which  I  have  derived  the  very  constitu- 
tion of  the  simple  elements  of  matter,  presents  a  system  that  is  midway 
between  that  of  Leibniz  &  that  of  Newton  ;  it  has  very  much  in  common 
with  both,  &  differs  very  much  from  either  ;  &,  as  it  is  immensely  more 
simple  than  either,  it  is  undoubtedly  suitable  in  a  marvellous  degree  for 

deriving  all  the  general  properties  of  bodies,  &  certain  of  the  special  properties  also,  by 

means  of  the  most  rigorous  demonstrations. 

2.  It  indeed  holds  to  those  simple  &  perfectly  non-extended  primary  elements  upon  what  there  is  in 
which  is  founded  the  theory  of  Leibniz  ;   &  also  to  the  mutual  forces,  which  vary  as  the  *  s£^"0"{  to$^ 
distances  of  the  points  from  one  another  vary,  the  characteristic  of  the  theory  of  Newton  ;  ton  *&  Leibniz. 

in  addition,  it  deals  not  only  with  the  kind  of  forces,  employed  by  Newton,  which  oblige 
the  points  to  approach  one  another,  &  are  commonly  called  attractions ;  but  also  it 
considers  forces  of  a  kind  that  engender  recession,  &  are  called  repulsions.  Further,  the 
idea  is  introduced  in  such  a  manner  that,  where  attraction  ends,  there,  with  a  change  of 
distance,  repulsion  begins ;  this  idea,  as  a  matter  of  fact,  was  suggested  by  Newton  in  the 
last  of  his  '  Questions  on  Optics ',  &  he  illustrated  it  by  the  example  of  the  passage  from 
positive  to  negative,  as  used  in  algebraical  formulas.  Moreover  there  is  this  common  point 
between  either  of  the  theories  of  Newton  &  Leibniz  &  my  own  ;  namely,  that  any  particle 
of  matter  is  connected  with  every  other  particle,  no  matter  how  great  is  the  distance 
between  them,  in  such  a  way  that,  in  accordance  with  a  change  in  the  position,  no  matter 
how  slight,  of  any  one  of  them,  the  factors  that  determine  the  motions  of  all  the  rest  are 
altered  ;  &,  unless  it  happens  that  they  all  cancel  one  another  (&  this  is  infinitely  impro- 
bable), some  motion,  due  to  the  change  of  position  in  question,  will  take  place  in  every  one 
of  them. 

3.  But  my  Theory  differs  in  a  marked  degree  from  that  of  Leibniz.     For  one  thing,  How  it  differs  from, 
because  it  does  not  admit  the  continuous  extension  that  arises  from  the  idea  of  consecutive, 
non-extended  points  touching  one  another  ;  here,  the  difficulty  raised  in  times  gone  by  in 

opposition  to  Zeno,  &  never  really  or  satisfactorily  answered  (nor  can  it  be  answered),  with 
regard  to  compenetration  of  all  kinds  with  non-extended  consecutive  points,  still  holds  the 
same  force  against  the  system  of  Leibniz.  For  another  thing,  it  admits  homogeneity 
amongst  the  elements,  all  distinction  between  masses  depending  on  relative  position  only, 
&  different  combinations  of  the  elements ;  for  this  homogeneity  amongst  the  elements,  & 
the  reason  for  the  difference  amongst  masses,  Nature  herself  provides  us  with  the  analogy. 
Chemical  operations  especially  do  so ;  for,  since  the  result  of  the  analysis  of  compound 
substances  leads  to  classes  of  elementary  substances  that  are  so  comparatively  few  in  num- 
ber, &  still  less  different  from  one  another  in  nature  ;  it  strongly  suggests  that,  the  further 
analysis  can  be  pushed,  the  greater  the  simplicity,  &  homogeneity,  that  ought  to  be  attained  ; 
thus,  at  length,  we  should  have,  as  the  result  of  a  final  decomposition,  homogeneity  & 
simplicity  of  the  highest  degree.  Against  this  homogeneity  &  simplicity,  the  principle  of 
indiscernibles,  &  the  doctrine  of  sufficient  reason,  so  long  &  strongly  advocated  by  the 
followers  of  Leibniz,  can,  in  my  opinion  at  least,  avail  in  not  the  slightest  degree. 

4.  My  Theory  also  differs  as  widely  as  possible  from  that  of  Newton.     For  one  thing,  HOW  it  differs  from, 
because  it  explains  by  means  of  a  single  law  of  forces  all  those  things  that  Newton  himself,  *    surpasses,    the 

i       i  i   i.     .  X          •  f-\      •      ,  i  •    i        theory  of  Newton. 

in  the  last  of  his  Questions  on  Uptics  ,  endeavoured  to  explain  by  the  three  principles 
of  gravity,  cohesion  &  fermentation  ;  nay,  &  very  many  other  things  as  well,  which  do  not 
altogether  follow  from  those  three  principles.  Further,  this  law  is  expressed  by  a  single 
algebraical  formula,  &  not  by  one  composed  of  several  formulae  compounded  together  ;  or 
by  a  single  continuous  geometrical  curve.  For  another  thing,  it  admits  forces  that  at  very 
small  distances  are  not  positive  or  attractive,  as  Newton  supposed,  but  negative  or  repul- 

35 


missum. 


36  PHILOSOPHIC   NATURALIS  THEORIA 

infinitum,  quo  distantise  in  infinitum  decrescant.  Unde  illud  necessario  consequitur,  ut  nee 
cohaesio  a  contactu  immediate  oriatur,  quam  ego  quidem  longe  aliunde  desumo  ;  nee  ullus 
immediatus,  &,  ut  ilium  appellare  soleo,  mathematicus  materiae  contactus  habeatur,  quod 
simplicitatem,  &  inextensionem  inducit  elementorum,  quae  ipse  variarum  figurarum  voluit, 
&  partibus  a  se  invicem  distinctis  composita,  quamvis  ita  cohasrentia,  ut  nulla  Naturae  vi 
dissolvi  possit  compages,  &  adhaesio  labefactari,  quas  adhaesio  ipsi,  respectu  virium  nobis 
cognitarum,  est  absolute  infinita. 

Ubi  de  ipsa    ctum  5.  Quae  ad  ejusmodi  Theoriam  pertinentia  hucusque  sunt  edita,  continentur  disserta- 

ante ;  &  quid  pro-  tionibus  meis,  De  viribus  vivis,  edita  Anno  1741;,  De  Lumine  A.  1748,  De  Leee  Continuitatis 

ml«<mm  "  .  T  r^  ...      .     •  .  rj  .         ... 

A.  1754,  De  Lege  virium  in  natura  existentium  A.  1755,  De  divisibihtate  materite,  C5  principiis 
corporum  A.  1757,  ac  in  meis  Supplementis  Stayanae  Philosophiae  versibus  traditae,  cujus  primus 
Tomus  prodiit  A.  1755  :  eadem  autem  satis  dilucide  proposuit,  &  amplissimum  ipsius  per 
omnem  Physicam  demonstravit  usum  vir  e  nostra  Societate  doctissimus  Carolus  Benvenutus 
in  sua  Physics  Generalis  Synopsi  edita  Anno  1754.  In  ea  Synopsi  proposuit  idem  &  meam 
deductionem  aequilibrii  binarum  massarum,  viribus  parallelis  animatarum,  quas  ex  ipsa  mea 
Theoria  per  notissimam  legem  compositionis  virium,  &  aequalitatis  inter  actionem,  &  reac- 
tionem,  fere  sponte  consequitur,  cujus  quidem  in  supplementis  illis  §  4.  ad  lib.  3.  mentionem 
feci,  ubi  &  quae  in  dissertatione  De  centra  Gravitatis  edideram,  paucis  proposui ;  &  de  centre 
oscillationis  agens,  protuli  aliorum  methodos  praecipuas  quasque,  quae  ipsius  determinationem 
a  subsidiariis  tantummodo  principiis  quibusdam  repetunt.  Ibidem  autem  de  sequilibrii 
centre  agens  illud  affirmavi  :  In  Natura  nullce  sunt  rigidce  virgce,  infiexiles,  &  omni  gravitate, 
ac  inertia  carentes,  adeoque  nee  revera  ullce  leges  pro  Us  conditcz  ;  &  si  ad  genuina,  &  simpli- 
cissima  natures  principia,  res  exigatur,  invenietur,  omnia  pendere  a  compositione  virium,  quibus  in 
se  invicem  agunt  particula  materice  ;  a  quibus  nimirum  viribus  omnia  Natures  pb&nomena 
proficiscuntur.  Ibidem  autem  exhibitis  aliorum  methodis  ad  centrum  oscillationis  perti- 
nentibus,  promisi,  me  in  quarto  ejusdem  Philosophiae  tomo  ex  genuinis  principiis  investiga- 
turum,  ut  aequilibrii,  sic  itidem  oscillationis  centrum. 


Qua  occasione  hoc  6.  Porro  cum  nuper  occasio  se  mihi  praebuisset  inquirendi  in  ipsum  oscillationis  centrum 
turn 'opus."  Cnp  ex  meis  principiis,  urgente  Scherffero  nostro  viro  doctissimo,  qui  in  eodem  hoc  Academico 
Societatis  Collegio  nostros  Mathesim  docet ;  casu  incidi  in  theorema  simplicisimum  sane,  & 
admodum  elegans,  quo  trium  massarum  in  se  mutuo  agentium  comparantur  vires,  [4]  quod 
quidem  ipsa  fortasse  tanta  sua  simplicitate  effugit  hucusque  Mechanicorum  oculos ;  nisi 
forte  ne  effugerit  quidem,  sed  alicubi  jam  ab  alio  quopiam  inventum,  &  editum,  me,  quod 
admodum  facile  fieri  potest,  adhuc  latuerit,  ex  quo  theoremate  &  asquilibrium,  ac  omne 
vectium  genus,  &  momentorum  mensura  pro  machinis,  &  oscillationis  centrum  etiam  pro 
casu,  quo  oscillatio  fit  in  latus  in  piano  ad  axem  oscillationis  perpendiculari,  &  centrum 
percussionis  sponte  fluunt,  &  quod  ad  sublimiores  alias  perquisitiones  viam  aperit  admodum 
patentem.  Cogitaveram  ego  quidem  initio  brevi  dissertatiuncula  hoc  theorema  tantummodo 
edere  cum  consectariis,  ac  breve  Theoriae  meae  specimen  quoddam  exponere  ;  sed  paullatim 
excrevit  opusculum,  ut  demum  &  Theoriam  omnem  exposuerim  ordine  suo,  &  vindicarim, 
&  ad  Mechanicam  prius,  turn  ad  Physicam  fere  universam  applicaverim,  ubi  &  quae  maxima 
notatu  digna  erant,  in  memoratis  dissertationibus  ordine  suo  digessi  omnia,  &  alia  adjeci 
quamplurima,  quae  vel  olim  animo  conceperam,  vel  modo  sese  obtulerunt  scribenti,  &  omnem 
hanc  rerum  farraginem  animo  pervolventi. 


eiementa  in-  7.  Prima  elementa  materiae  mihi  sunt  puncta  prorsus  indivisibilia,  &  inextensa,  quae  in 

imrftenso  vacuo  ita  dispersa  sunt,  ut  bina  quaevis  a  se  invicem  distent  per  aliquod  intervallum, 
quod  quidem  indefinite  augeri  potest,  &  minui,  sed  penitus  evanescere  non  potest,  sine 
conpenetratione  ipsorum  punctorum  :  eorum  enim  contiguitatem  nullam  admitto  possi- 
bilem  ;  sed  illud  arbitror  omnino  certum,  si  distantia  duorum  materiae  punctorum  sit  nulla, 
idem  prorsus  spatii  vulgo  concept!  punctum  indivisibile  occupari  ab  utroque  debere,  & 


A  THEORY  OF  NATURAL  PHILOSOPHY  37 

sive ;  although  these  also  become  greater  &  greater  indefinitely,  as  the  distances  decrease 
indefinitely.  From  this  it  follows  of  necessity  that  cohesion  is  not  a  consequence  of  imme- 
diate contact,  as  I  indeed  deduce  from  totally  different  considerations ;  nor  is  it  possible 
to  get  any  immediate  or,  as  I  usually  term  it,  mathematical  contact  between  the  parts  of 
matter.  This  idea  naturally  leads  to  simplicity  &  non-extension  of  the  elements,  such  as 
Newton  himself  postulated  for  various  figures ;  &  to  bodies  composed  of  parts  perfectly 
distinct  from  one  another,  although  bound  together  so  closely  that  the  ties  could  not  be 
broken  or  the  adherence  weakened  by  any  force  in  Nature ;  this  adherence,  as  far  as  the 
forces  known  to  us  are  concerned,  is  in  his  opinion  unlimited. 

5.  What  has  already  been  published  relating  to  this  kind  of  Theory  is  contained  in  my  when  &  where  I 
dissertations,  De  Viribus   vivis,  issued  in  1745,  De  Lumine,  1748,  De  Lege  Continuitatis,  *£££  th^theory'* 
1754,  De  Lege  virium  in  natura  existentium,  1755,  De  divisibilitate  materia,  y  principiis  &  a  promise  that  i 
corporum,  1757,  &  in  my  Supplements  to  the  philosophy  of  Benedictus  Stay,  issued  in  verse,  made> 

of  which  the  first  volume  was  published  in  1755.  The  same  theory  was  set  forth  with 
considerable  lucidity,  &  its  extremely  wide  utility  in  the  matter  of  the  whole  of  Physics 
was  demonstrated,  by  a  learned  member  of  our  Society,  Carolus  Benvenutus,  in  his  Physics 
Generalis  Synopsis  published  in  1754.  In  this  synopsis  he  also  at  the  same  time  gave  my 
deduction  of  the  equilibrium  of  a  pair  of  masses  actuated  by  parallel  forces,  which  follows 
quite  naturally  from  my  Theory  by  the  well-known  law  for  the  composition  of  forces,  & 
the  equality  between  action  &  reaction  ;  this  I  mentioned  in  those  Supplements,  section 
4  of  book  3,  &  there  also  I  set  forth  briefly  what  I  had  published  in  my  dissertation  De 
centra  Gravitatis.  Further,  dealing  with  the  centre  of  oscillation,  I  stated  the  most  note- 
worthy methods  of  others  who  sought  to  derive  the  determination  of  this  centre  from 
merely  subsidiary  principles.  Here  also,  dealing  with  the  centre  of  equilibrium,  I  asserted  : — 
"  In  Nature  there  are  no  rods  that  are  rigid,  inflexible,  totally  devoid  of  weight  &  inertia  ; 
y  so,  neither  are  there  really  any  laws  founded  on  them.  If  the  matter  is  worked  back  to  the 
genuine  W  simplest  natural  principles,  it  will  be  found  that  everything  depends  on  the  com- 
position of  the  forces  with  which  the  particles  of  matter  act  upon  one  another  ;  y  from  these 
very  forces,  as  a  matter  of  fact,  all  phenomena  of  Nature  take  their  origin."  Moreover,  here 
too,  having  stated  the  methods  of  others  for  the  determination  of  the  centre  of  oscillation, 
I  promised  that,  in  the  fourth  volume  of  the  Philosophy,  I  would  investigate  by  means  of 
genuine  principles,  such  as  I  had  used  for  the  centre  of  equilibrium,  the  centre  of 
oscillation  as  well. 

6.  Now,  lately  I  had  occasion  to  investigate  this  centre  of  oscillation,  deriving  it  from  The  occasion  that 
my  own  principles,  at  the  request  of  Father  Scherffer,  a  man  of  much  learning,  who  teaches  |^ 
mathematics  in  this  College  of  the  Society.     Whilst  doing  this,  I  happened  to  hit  upon  a  matter. 

really  most  simple  &  truly  elegant  theorem,  from  which  the  forces  with  which  three 
masses  mutually  act  upon  one  another  are  easily  to  be  found  ;  this  theorem,  perchance 
owing  to  its  extreme  simplicity,  has  escaped  the  notice  of  mechanicians  up  till  now  (unless 
indeed  perhaps  it  has  not  escaped  notice,  but  has  at  some  time  previously  been  discovered 
&  published  by  some  other  person,  though,  as  may  very  easily  have  happened,  it  may  not 
have  come  to  my  notice).  From  this  theorem  there  come,  as  the  natural  consequences, 
the  equilibrium  &  all  the  different  kinds  of  levers,  the  measurement  of  moments  for 
machines,  the  centre  of  oscillation  for  the  case  in  which  the  oscillation  takes  place  sideways 
in  a  plane  perpendicular  to  the  axis  of  oscillation,  &  also  the  centre  of  percussion ;  it  opens 
up  also  a  beautifully  clear  road  to  other  and  more  sublime  investigations.  Initially,  my 
idea  was  to  publish  in  a  short  esssay  merely  this  theorem  &  some  deductions  from  it,  &  thus 
to  give  some  sort  of  brief  specimen  of  my  Theory.  But  little  by  little  the  essay  grew  in 
length,  until  it  ended  in  my  setting  forth  in  an  orderly  manner  the  whole  of  the  theory, 
giving  a  demonstration  of  its  truth,  &  showing  its  application  to  Mechanics  in  the  first  place, 
and  then  to  almost  the  whole  of  Physics.  To  it  I  also  added  not  only  those  matters  that 
seemed  to  me  to  be  more  especially  worth  mention,  which  had  all  been  already  set  forth 
in  an  orderly  manner  in  the  dissertations  mentioned  above,  but  also  a  large  number  of  other 
things,  some  of  which  had  entered  my  mind  previously,  whilst  others  in  some  sort  pb  truded 
themselves  on  my  notice  as  I  was  writing  &  turning  over  in  my  mind  all  this  conglomer- 
ation of  material. 

7.  The  primary  elements  of  matter  are  in  my  opinion  perfectly  indivisible  &  non-  The   primary   eie- 
extended  points ;  they  are  so  scattered  in  an  immense  vacuum  that  every  two  of  them  are  ^biVnon^xtended 
separated  from   one   another  by  a   definite   interval ;    this   interval   can   be   indefinitely  &    they    are    not 
increased  or  diminished,  but  can  never  vanish  altogether  without  compenetration  of  the  c 

points  themselves ;  for  I  do  not  admit  as  possible  any  immediate  contact  between  them. 
On  the  contrary  I  consider  that  it  is  a  certainty  that,  if  the  distance  between  two  points 
of  matter  should  become  absolutely  nothing,  then  the  very  same  indivisible  point  of  space, 
according  to  the  usual  idea  of  it,  must  be  occupied  by  both  together,  &  we  have  true 


38  PHILOSOPHIC  NATURALIS  THEORIA 

haberi  veram,  ac  omnimodam  conpenetrationem.  Quamobrem  non  vacuum  ego  quidem 
admitto  disseminatum  in  materia,  sed  materiam  in  vacuo  disseminatam,  atque  innatantem. 

Eorum  inertias  vis  g    jn  n;sce  punctis  admitto  determinationem  perseverandi  in  eodem  statu  quietis,  vel 

cujusmodi.  .  r          .  r.        ,.  ,   .   .  ,      .    J  .  .  •          i      *       XT  ' 

motus  umiormis  in  directum  l«)  m  quo  semel  sint  posita,  si  seorsum  smgula  in  JNatura 
existant ;  vel  si  alia  alibi  extant  puncta,  componendi  per  notam,  &  communem  metho- 
dum  compositionis  virium,  &  motuum,  parallelogrammorum  ope,  praecedentem  motum 
cum  mo-[5]-tu  quern  determinant  vires  mutuae,  quas  inter  bina  quaevis  puncta  agnosco 
a  distantiis  pendentes,  &  iis  mutatis  mutatas,  juxta  generalem  quandam  omnibus  com- 
munem legem.  In  ea  determinatione  stat  ilia,  quam  dicimus,  inertiae  vis,  quae,  an  a 
libera  pendeat  Supremi  Conditoris  lege,  an  ab  ipsa  punctorum  natura,  an  ab  aliquo  iis 
adjecto,  quodcunque,  istud  sit,  ego  quidem  non  quaere  ;  nee  vero,  si  velim  quasrere,  in- 
veniendi  spem  habeo  ;  quod  idem  sane  censeo  de  ea  virium  lege,  ad  quam  gradum  jam  facio. 

Eorundem    vires  g    Censeo  igitur  bina  quaecunque  materiae  puncta  determinari  asque  in  aliis  distantiis 

mutuae      in      alus       ,       y  •,..         ,       -1  .  .         . 

distantiis  attrac-  ad  mutuum  accessum,  in  alns  ad  recessum  mutuum,  quam  ipsam  determinationem  appello 
tivae,  in    aliis  re-  vim,  in  priore  casu  attractivam,  in  posteriore  repulsivam,  eo  nomine  non  agendi  modum,  sed 

pulsivae  :    v  i  n  u  m    .  ,r  .         .  .  ,  '.  .  . 

ejusmodi  exempia.  ipsam  determinationem  expnmens,  undecunque  provemat,  cujus  vero  magnitude  mutatis 
distantiis  mutetur  &  ipsa  secundum  certam  legem  quandam,  quae  per  geometricam  lineam 
curvam,  vel  algebraicam  formulam  exponi  possit,  &  oculis  ipsis,  uti  moris  est  apud  Mechanicos 
repraesentari.  Vis  mutuae  a  distantia  pendentis,  &  ea  variata  itidem  variatae,  atque  ad  omnes 
in  immensum  &  magnas,  &  parvas  distantias  pertinentis,  habemus  exemplum  in  ipsa 
Newtoniana  generali  gravitate  mutata  in  ratione  reciproca  duplicata  distantiarum,  qua; 
idcirco  numquam  e  positiva  in  negativam  migrare  potest,  adeoque  ab  attractiva  ad  repul- 
sivam, sive  a  determinatione  ad  accessum  ad  determinationem  ad  recessum  nusquam  migrat. 
Verum  in  elastris  inflexis  habemus  etiam  imaginem  ejusmodi  vis  mutuae  variatae  secundum 
distantias,  &  a  determinatione  ad  recessum  migrantis  in  determinationem  ad  accessum,  & 
vice  versa.  Ibi  enim  si  duae  cuspides,  compresso  elastro,  ad  se  invicem  accedant,  acquirunt 
determinationem  ad  recessum,  eo  majorem,  quo  magis,  compresso  elastro,  distantia 
decrescit ;  aucta  distantia  cuspidum,  vis  ad  recessum  minuitur,  donee  in  quadam  distantia 
evanescat,  &  fiat  prorsus  nulla  ;  turn  distantia  adhuc  aucta,  incipit  determinatio  ad  accessum, 
quae  perpetuo  eo  magis  crescit,  quo  magis  cuspides  a  se  invicem  recedunt  :  ac  si  e  contrario 
cuspidum  distantia  minuatur  perpetuo ;  determinatio  ad  accessum  itidem  minuetur, 
evanescet,  &  in  determinationem  ad  recessum  mutabitur.  Ea  determinatio  oritur  utique 
non  ab  immediata  cuspidum  actione  in  se  invicem,  sed  a  natura,  &  forma  totius  intermediae 
laminae  plicatae  ;  sed  hie  physicam  rei  causam  non  merer,  &  solum  persequor  exemplum 
determinationis  ad  accessum,  &  recessum,  quae  determinatio  in  aliis  distantiis  alium  habeat 
nisum,  &  migret  etiam  ab  altera  in  alteram. 


virium  earundero  10.  Lex  autem  virium  est  ejusmodi,  ut  in  minimis  distantiis  sint  repulsivae,  atque  eo 

majores  in  infmitum,  quo  distantiae  ipsae  minuuntur  in  infinitum,  ita,  ut  pares  sint  extinguen- 
[6]-dae  cuivis  velocitati  utcunque  magnae,  cum  qua  punctum  alterum  ad  alterum  possit 
accedere,  antequam  eorum  distantia  evanescat ;  distantiis  vero  auctis  minuuntur  ita,  ut  in 
quadam  distantia  perquam  exigua  evadat  vis  nulla  :  turn  adhuc,  aucta  distantia,  mutentur  in 
attractivas,  prime  quidem  crescentes,  turn  decrescentes,  evanescentes,  abeuntes  in  repulsivas, 
eodem  pacto  crescentes,  deinde  decrescentes,  evanescentes,  migrantes  iterum  in  attractivas, 
atque  id  per  vices  in  distantiis  plurimis,  sed  adhuc  perquam  exiguis,  donee,  ubi  ad  aliquanto 
majores  distantias  ventum  sit,  incipiant  esse  perpetuo  attractivae,  &  ad  sensum  reciproce 


(a)  Id  quidem  respectu  ejus  spatii,  in  quo  continemur  nos,  W  omnia  quis  nostris  observari  sensibus  possunt,  corpora  ; 
quod  quiddam  spatium  si  quiescat,  nihil  ego  in  ea  re  a  reliquis  differo  ;  si  forte  moveatur  motu  quopiam,  quern  motum 
ex  hujusmodi  determinatione  sequi  debeant  ipsa  materia  puncta  ;  turn  bcec  mea  erit  quiedam  non  absoluta,  sed  respectiva 
inertia:  vis,  quam  ego  quidem  exposui  W  in  dissertatione  De  Maris  aestu  fcf  in  Supplementis  Stayanis  Lib.  I.  §  13  ; 
ubi  etiam  illud  occurrit,  quam  oh  causam  ejusmodi  respectivam  inertiam  excogitarim,  &  quibus  rationihus  evinci  putem, 
absolutam  omnino  demonstrari  non  posse  ;  sed  ea  hue  non  pertinent. 


A  THEORY  OF  NATURAL  PHILOSOPHY  39 

compenetration  in  every  way.  Therefore  indeed  I  do  not  admit  the  idea  of  vacuum 
interspersed  amongst  matter,  but  I  consider  that  matter  is  interspersed  in  a  vacuum  & 
floats  in  it. 

8.  As  an  attribute  of  these  points  I  admit  an  inherent  propensity  to  remain  in  the  The  nat.ure  ?f  the 
same  state  of  rest,  or  of  uniform  motion  in  a  straight  line,  («)  in  which  they  are  initially  the"  possess.1* 
set,  if  each  exists  by  itself  in  Nature.     But  if  there  are  also  other  points  anywhere,  there 

is  an  inherent  propensity  to  compound  (according  to  the  usual  well-known  composition  of 
forces  &  motions  by  the  parallelogram  law),  the  preceding  motion  with  the  motion  which 
is  determined  by  the  mutual  forces  that  I  admit  to  act  between  any  two  of  them,  depending 
on  the  distances  &  changing,  as  the  distances  change,  according  to  a  certain  law  common 
to  them  all.  This  propensity  is  the  origin  of  what  we  call  the  '  force  of  inertia  ' ;  whether 
this  is  dependent  upon  an  arbitrary  law  of  the  Supreme  Architect,  or  on  the  nature  of  points 
itself,  or  on  some  attribute  of  them,  whatever  it  may  be,  I  do  not  seek  to  know ;  even  if  I 
did  wish  to  do  so,  I  see  no  hope  of  finding  the  answer  ;  and  I  truly  think  that  this  also 
applies  to  the  law  of  forces,  to  which  I  now  pass  on. 

9.  I  therefore  consider  that  any  two  points  of  matter  are  subject  to  a  determination  The  mutual  forces 
to  approach  one  another  at  some  distances,  &  in  an  equal  degree  recede  from  one  another  at  Stw^*^™!* 
other  distances.     This  determination  I  call  '  force ' ;  in  the  first  case  '  attractive ',  in  the  distances  &  repui- 
second  case  '  repulsive  ' ;   this  term  does  not  denote  the  mode  of  action,  but  the  propen-  ^mpies 

sity  itself,  whatever  its  origin,  of  which  the  magnitude  changes  as  the  distances  change ;  this  kind, 
this  is  in  accordance  with  a  certain  definite  law,  which  can  be  represented  by  a  geometrical 
curve  or  by  an  algebraical  formula,  &  visualized  in  the  manner  customary  with  Mechanicians. 
We  have  an  example  of  a  force  dependent  on  distance,  &  varying  with  varying  distance,  & 
pertaining  to  all  distances  either  great  or  small,  throughout  the  vastness  of  space,  in  the 
Newtonian  idea  of  general  gravitation  that  changes  according  to  the  inverse  squares  of  the 
distances  :  this,  on  account  of  the  law  governing  it,  can  never  pass  from  positive  to  nega- 
tive ;  &  thus  on  no  occasion  does  it  pass  from  being  attractive  to  being  repulsive,  i.e.,  from 
a  propensity  to  approach  to  a  propensity  to  recession.  Further,  in  bent  springs  we  have 
an  illustration  of  that  kind  of  mutual  force  that  varies  according  as  the  distance  varies,  & 
passes  from  a  propensity  to  recession  to  a  propensity  to  approach,  and  vice  versa.  For 
here,  if  the  two  ends  of  the  spring  approach  one  another  on  compressing  the  spring,  they 
acquire  a  propensity  for  recession  that  is  the  greater,  the  more  the  distance  diminishes 
between  them  as  the  spring  is  compressed.  But,  if  the  distance  between  the  ends  is 
increased,  the  force  of  recession  is  diminished,  until  at  a  certain  distance  it  vanishes  and 
becomes  absolutely  nothing.  Then,  if  the  distance  is  still  further  increased,  there  begins  a 
propensity  to  approach,  which  increases  more  &  more  as  the  ends  recede  further  &  further 
away  from  one  another.  If  now,  on  the  contrary,  the  distance  between  the  ends  is  con- 
tinually diminished,  the  propensity  to  approach  also  diminishes,  vanishes,  &  becomes  changed 
into  a  propensity  to  recession.  This  propensity  certainly  does  not  arise  from  the  imme- 
diate action  of  the  ends  upon  one  another,  but  from  the  nature  &  form  of  the  whole  of  the 
folded  plate  of  metal  intervening.  But  I  do  not  delay  over  the  physical  cause  of  the  thing 
at  this  juncture  ;  I  only  describe  it  as  an  example  of  a  propensity  to  approach  &  recession, 
this  propensity  being  characterized  by  one  endeavour  at  some  distances  &  another  at  other 
distances,  &  changing  from  one  propensity  to  another. 

10.  Now  the  law  of  forces  is  of  this  kind  ;   the  forces  are  repulsive  at  very  small  dis-  The  Iaw  .of  forces 
tances,  &  become  indefinitely  greater  &  greater,  as  the  distances  are  diminished  indefinitely,  for  the  pomts- 

in  such  a  manner  that  they  are  capable  of  destroying  any  velocity,  no  matter  how  large  it 
may  be,  with  which  one  point  may  approach  another,  before  ever  the  distance  between 
them  vanishes.  When  the  distance  between  them  is  increased,  they  are  diminished  in  such 
a  way  that  at  a  certain  distance,  which  is  extremely  small,  the  force  becomes  nothing. 
Then  as  the  distance  is  still  further  increased,  the  forces  are  change-d  to  attractive  forces ; 
these  at  first  increase,  then  diminish,  vanish,  &  become  repulsive  forces,  which  in  the  same 
way  first  increase,  then  diminish,  vanish,  &  become  once  more  attractive  ;  &  so  on,  in  turn, 
for  a  very  great  number  of  distances,  which1  are  all  still  very^  minute  :  until,  finally,  when 
we  get  to  comparatively  great  distances,  they  begin  to  be  continually  attractive  &  approxi- 

(a)  This  indeed  holds  true  for  that  space  in  which  we,  and  all  bodies  that  can  be  observed  by  our  senses,  are 
contained.  Now,  if  this  space  is  at  rest,  I  do  not  differ  from  other  philosophers  with  regard  to  the  matter  in  question  ; 
but  if  perchance  space  itself  moves  in  some  way  or  other,  what  motion  ought  these  points  of  matter  to  comply  with  owing 
to  this  kind  of  propensity  ?  In  that  case  Ms  force  of  inertia  that  I  postulate  is  not  absolute,  but  relative  ;  as  indeed 
I  explained  both  in  the  dissertation  De  Maris  Aestu,  and  also  in  the  Supplements  to  Stay's  Philosophy,  book  I,  section 
13.  Here  also  will  be  found  the  conclusions  at  which  I  arrived  with  regard  to  relative  inertia  of  this  sort,  and  the 
arguments  by  which  I  think  it  is  proved  that  it  is  impossible  to  show  that  it  is  generally  abxlute.  But  these  things  do 
not  concern  us  at  present. 


4° 


PHILOSOPHI/E  NATURALIS  THEORIA 


proportionales  quadratis  distantiarum,  atque  id  vel  utcunque  augeantur  distantiae  etiam  in 
infinitum,  vel  saltern  donee  ad  distantias  deveniatur  omnibus  Planetarum,  &  Cometarum 
distantiis  longe  majores. 

Leg  is  simpiicitas  ii.  Hujusmodi  lex  primo  aspectu  videtur  admodum  complicata,  &  ex  diversis  legibus 

exprimibihs     per  temere  jnter  se  coagmentatis  coalescens ;  at  simplicissima,  &  prorsus  incomposita  esse  potest, 

COIlLlIlUtllTl    CUf  VclIUi  •    t      i  •  •  •  1*1*  A  1  1  "  J"  1 

expressa  videlicet  per  unicam  contmuam  curvam,  vel  simphcem  Algebraicam  iormulam,  uti 
innui  superius.  Hujusmodi  curva  linea  est  admodum  apta  ad  sistendam  oculis  ipsis  ejusmodi 
legem,  nee  requirit  Geometram,  ut  id  praestare  possit  :  satis  est,  ut  quis  earn  intueatur 
tantummodo,  &  in  ipsa  ut  in  imagine  quadam  solemus  intueri  depictas  res  qualescunque, 
virium  illarum  indolem  contempletur.  In  ejusmodi  curva  eae,  quas  Geometrae  abscissas 
dicunt,  &  sunt  segmenta  axis,  ad  quern  ipsa  refertur  curva,  exprimunt  distantias  binorum 
punctorum  a  se  invicem  :  illae  vero,  quae  dicuntur  ordinatae,  ac  sunt  perpendiculares  lineee 
ab  axe  ad  curvam  ductae,  referunt  vires  :  quae  quidem,  ubi  ad  alteram  jacent  axis  partem, 
exhibent  vires  attractivas ;  ubi  jacent  ad  alteram,  rcpulsivas,  &  prout  curva  accedit  ad  axem, 
vel  recedit,  minuuntur  ipsae  etiam,  vel  augentur  :  ubi  curva  axem  secat,  &  ab  altera  ejus 
parte  transit  ad  alteram,  mutantibus  directionem  ordinatis,  abeunt  ex  positivis  in  negativas, 
vel  vice  versa  :  ubi  autem  arcus  curvae  aliquis  ad  rectam  quampiam  axi  perpendicularem 
in  infinitum  productam  semper  magis  accedit  ita  ultra  quoscumque  limites,  ut  nunquam  in 
earn  recidat,  quern  arcum  asymptoticum  appellant  Geometrae,  ibi  vires  ipsae  in  infinitum 
excrescunt. 


Forma  curvae  ips- 
ius. 


12.  Ejusmodi  curvam  exhibui,  &  exposui  in  dissertationibus  De  viribus  vivis  a  Num.  51, 
De  Lumine  Num.  5,  De  Lege  virium  in  Naturam  existentium  a  Num.  68,  &  in  sua  Synopsi 
Physics  Generalis  P.  Benvenutus  eandem  protulit  a  Num.  108.  En  brevem  quandemejus 
ideam.  In  Fig.  i,  Axis  C'AC  habet  in  puncto  A  asymptotum  curvae  rectilineam  AB 
indefinitam,  circa  quam  habentur  bini  curvae  rami  hinc,  &  inde  aequales,  prorsus  inter  se,  & 
similes,  quorum  alter  DEFGHIKLMNOPQRSTV  habet  inprimis  arcum  ED  [7]  asympto- 
ticum, qui  nimirum  ad  partes  BD,  si  indefinite  producatur  ultra  quoscunque  limites,  semper 
magis  accedit  ad  rectam  AB  productam  ultra  quoscunque  limites,  quin  unquam  ad  eandem 
deveniat ;  hinc  vero  versus  DE  perpetuo  recidit  ab  eadam  recta,  immo  etiam  perpetuo 
versus  V  ab  eadem  recedunt  arcus  reliqui  omnes,  quin  uspiam  recessus  mutetur  in  accessum. 
Ad  axem  C'C  perpetuo  primum  accedit,  donee  ad  ipsum  deveniat  alicubi  in  E  ;  turn  eodem 
ibi  secto  progreditur,  &  ab  ipso  perpetuo  recedit  usque  ad  quandam  distantiam  F,  postquam 
recessum  in  accessum  mutat,  &  iterum  ipsum  axem  secat  in  G,  ac  flexibus  continuis  contor- 
quetur  circa  ipsum,  quern  pariter  secat  in  punctis  quamplurimis,  sed  paucas  admodum 
ejusmodi  sectiones  figura  exhibet,  uti  I,  L,  N,  P,  R.  Demum  is  arcus  desinit  in  alterum 
crus  TpsV,  jacens  ex  parte  opposita  axis  respectu  primi  cruris,  quod  alterum  crus  ipsum 
habet  axem  pro  asymptoto,  &  ad  ipsum  accedit  ad  sensum  ita,  ut  distantiae  ab  ipso  sint  in 
ratione  reciproca  duplicata  distantiarum  a  recta  BA. 


Abscissae  exprimen- 

d!nateStaexprimen- 
tes  vires. 


13.  Si  ex  quovis  axis  puncto  a,  b,  d,  erigatur  usque  ad  curvam  recta  ipsi  perpendicularis 
aS>  ^r'  ^h  ,  segmentum  axis  Aa,  Ab,  Ad,  dicitur  abscissa,  &  refert  distantiam  duorum  materiae 
punctorum  quorumcunque  a  se  invicem  ;  perpendicularis  ag,  br,  db  ,  dicitur  ordinata,  & 
exhibet  vim  repulsivam,  vel  attractivam,  prout  jacet  respectu  axis  ad  partes  D,  vel  oppositas. 


Mutationes  ordina- 
tarum,  &  virium  iis 
expressarum. 


14.  Patet  autem,  in  ea  curvae  forma  ordinatam  ag  augeri  ultra  quoscunque  limites,  si 
abscissa  Aa,  minuatur  pariter  ultra  quoscunque  limites ;  quae  si  augeatur,  ut  abeat  in  Ab, 
ordinata  minuetur,  &  abibit  in  br,  perpetuo  imminutam  in  accessu  b  ad  E,  ubi  evanescet : 
turn  aucta  abscissa  in  Ad,  mutabit  ordinata  directionem  in  dh ,  ac  ex  parte  opposita  augebitur 
prius  usque  ad  F,  turn  decrescet  per  il  usque  ad  G,  ubi  evanescet,  &  iterum  mutabit 
directionem  regressa  in  mn  ad  illam  priorem,  donee  post  evanescentiam,  &  directionis 
mutationem  factam  in  omnibus  sectionibus  I,  L,  N,  P,  R,  fiant  ordinatas  op,  vs,  directionis 
constantis,  &  decrescentes  ad  sensum  in  ratione  reciproca  duplicata  abscissarum  Ao,  Av. 
Quamobrem  illud  est  manifestum,  per  ejusmodi  curvam  exprimi  eas  ipsas  vires,  initio 


A  THEORY  OF  NATURAL  PHILOSOPHY 


0 


PHILOSOPHIC  NATURALIS  THEORIA 


o 


A  THEORY  OF  NATURAL  PHILOSOPHY  43 

mately  inversely  proportional  to  the  squares  of  the  distances.  This  holds  good  as  the 
distances  are  increased  indefinitely  to  any  extent,  or  at  any  rate  until  we  get  to  distances 
that  are  far  greater  than  all  the  distances  of  the  planets  &  comets. 

11.  A  law  of  this  kind  will  seem  at  first  sight  to  be  very  complicated,  &  to  be  the  result  The   simplicity  of 

of  combining  together  several  different  laws  in  a  haphazard  sort  of  way  ;    but  it  can  be  of  the  law  can  ^  re~ 
^.t.        •        i     1   i  •    j    o  v          j    •        i         v    i  •  1  i    r        presented  by  means 

the  simplest  kind  &  not  complicated  in  the  slightest  degree ;  it  can  be  represented  for  of  a  continuous 
instance  by  a  single  continuous  curve,  or  by  an  algebraical  formula,  as  I  intimated  above.  curve- 
A  curve  of  this  sort  is  perfectly  adapted  to  the  .graphical  representation  of  this  sort  of  law, 
&  it  does  not  require  a  knowledge  of  geometry  to  set  it  forth.  It  is  sufficient  for  anyone 
merely  to  glance  at  it,  &  in  it,  just  as  in  a  picture  we  are  accustomed  to  view  all  manner  of 
things  depicted,  so  will  he  perceive  the  nature  of  these  forces.  In  a  curve  of  this  kind, 
those  lines,  that  geometricians  call  abscissae,  namely,  segments  of  the  axis  to  which  the 
curve  is  referred,  represent  the  distances  of  two  points  from  one  another  ;  &  those,  which 
we  called  ordinates,  namely,  lines  drawn  perpendicular  to  the  axis  to  meet  the  curve,  repre- 
sent forces.  These,  when  they  lie  on  one  side  of  the  axis  represent  attractive  forces,  and, 
when  they  lie  on  the  other  side,  repulsive  forces ;  &  according  as  the  curve  approaches  the 
axis  or  recedes  from  it,  they  too  are  diminished  or  increased.  When  the  curve  cuts  the 
axis  &  passes  from  one  side  of  it  to  the  other,  the  direction  of  the  ordinates  being  changed 
in  consequence,  the  forces  pass  from  positive  to  negative  or  vice  versa.  When  any  arc  of 
the  curve  approaches  ever  more  closely  to  some  straight  line  perpendicular  to  the  axis  and 
indefinitely  produced,  in  such  a  manner  that,  even  if  this  goes  on  beyond  all  limits,  yet 
the  curve  never  quite  reaches  the  line  (such  an  arc  is  called  asymptotic  by  geometricians), 
then  the  forces  themselves  will  increase  indefinitely. 

12.  I  set  forth  and  explained  a  curve  of  this  sort  in  my  dissertations  De  Firibus  vivis  The  form  of  the 
(Art.  51),  De  Lumine  (Art.  5),  De  lege  virium  in  Natura  existentium  (Art.  68)  ;   and  Father  curve- 
Benvenutus  published  the  same  thing  in  his  Synopsis  Physicce  Generalis  (Art.  108).     This 

will  give  you  some  idea  of  its  nature  in  a  few  words. 

In  Fig.  i  the  axis  C'AC  has  at  the  point  A  a  straight  line  AB  perpendicular  to  itself, 
which  is  an  asymptote  to  the  curve ;  there  are  two  branches  of  the  curve,  one  on  each  side 
of  AB,  which  are  equal  &  similar  to  one  another  in  every  way.  Of  these,  one,  namely 
DEFGHIKLMNOPQRSTV,  has  first  of  all  an  asymptotic  arc  ED  ;  this  indeed,  if  it  is 
produced  ever  so  far  in  the  direction  ED,  will  approach  nearer  &  nearer  to  the  straight  line 
AB  when  it  also  is  produced  indefinitely,  but  will  never  reach  it ;  then,  in  the  direction 
DE,  it  will  continually  recede  from  this  straight  line,  &  so  indeed  will  all  the  rest  of  the  arcs 
continually  recede  from  this  straight  line  towards  V.  The  first  arc  continually  approaches 
the  axis  C'C,  until  it  meets  it  in  some  point  E  ;  then  it  cuts  it  at  this  point  &  passes  on, 
continually  receding  from  the  axis  until  it  arrives  at  a  certain  distance  given  by  the  point 
F  ;  after  that  the  recession  changes  to  an  approach,  &  it  cuts  the  axis  once  more  in  G  ;  & 
so  on,  with  successive  changes  of  curvature,  the  curve  winds  about  the  axis,  &  at  the  same 
time  cuts  it  in  a  number  of  points  that  is  really  large,  although  only  a  very  few  of  the 
intersections  of  this  kind,  as  I,  L,  N,  P,  R,  are  shown  in  the  diagram.  Finally  the  arc  of  the 
curve  ends  up  with  the  other  branch  TpsV,  lying  on  the  opposite  side  of  the  axis  with 
respect  to  the  first  branch  ;  and  this  second  branch  has  the  axis  itself  as  its  asymptote, 
&  approaches  it  approximately  in  such  a  manner  that  the  distances  from  the  axis  are  in 
the  inverse  ratio  of  the  squares  of  the  distances  from  the  straight  line  AB. 

13.  If  from  any  point  of  the  axis,  such  as  a,  b,  or  d,  there  is  erected  a  straight  line  per-  The    abscissae    re- 
pendicular  to  it  to  meet  the  curve,  such  as  ag,  br,  or  db  then  the  segment  of  the  axis,  Aa,  £res^Jg 

Ab,  or  Ad,  is  called  the  abscissa,  &  represents  the  distance  of  any  two  points  of  matter  from  forces, 
one  another  ;   the  perpendicular,  ag,  br,  or  dh,  is  called  the  ordinate,  &  this  represents  the 
force,  which  is  repulsive  or  attractive,  according  as  the  ordinate  lies  with  regard  to  the 
axis  on  the  side  towards  D,  or  on  the  opposite  side. 

14.  Now  it  is  clear  that,  in  a  curve  of  this  form,  the  ordinate  ag  will  be  increased  Change  in  the  or- 
beyond  all  bounds,  if  the  abscissa  Aa  is  in  the  same  way  diminished  beyond  all  bounds ;  &  fbat  tlfey  reprSent! 
if  the  latter  is  increased  and  becomes  Ab,  the  ordinate  will  be  diminished,  &  it  will  become 

br,  which  will  continually  diminish  as  b  approaches  to  E,  at  which  point  it  will  vanish. 
Then  the  abscissa  being  increased  until  it  becomes  Ad,  the  ordinate  will  change  its  direction 
as  it  becomes  db,  &  will  be  increased  in  the  opposite  direction  at  first,  until  the  point  F  is 
reached,  when  it  will  be  decreased  through  the  value  il  until  the  point  G  is  attained,  at 
which  point  it  vanishes ;  at  the  point  G,  the  ordinate  will  once  more  change  its  direction 
as  it  returns  to  the  position  mn  on  the  same  side  of  the  axis  as  at  the  start.  Finally,  after 
vanishing  &  changing  direction  at  all  points  of  intersection  with  the  axis,  such  as  I,  L,  N, 
P,  R,  the  ordinates  take  the  several  positions  indicated  by  op,  vs  :  here  the  direction  remains 
unchanged,  &  the  ordinates  decrease  approximately  in  the  inverse  ratio  of  the  squares  of 
the  abscissae  Ao,  Av.  Hence  it  is  perfectly  evident  that,  by  a  curve  of  this  kind,  we  can 


44 


PHILOSOPHIC  NATURALIS  THEORIA 


Discrimen  hu  us 
legis  virium  a 
gravitate  N  e  w- 
toniana  :  ejus  usus 
in  Physica :  ordo 
pertractandorum. 


Occasio  inveniendae 
Theories  ex  consid- 
eraticine  impulsus. 


V 


repulsivas,  &  imminutis  in  infinitum  distantiis  auctas  in  infinitum,  auctis  imminutas,  turn 
evanescentes,  abeuntes,  mutata  directione,  in  attractivas,  ac  iterum  evenescentes,  mutatasque 
per  vices  :  donee  demum  in  satis  magna  distantia  evadant  attractive  ad  sensum  in  ratione 
reciproca  duplicata  distantiarum. 

15.  Haec  virium  lex  a  Newtoniana  gravitate  differt  in  ductu,  &  progressu  curvae  earn 
exprimentis  quse  nimirum,  ut  in  fig.  2,  apud  Newtonum  est  hyperbola  DV  gradus  tertii, 
jacens  tota  citra  axem,  quern  nuspiam 

secat,  jacentibus  omni-[8]-bus  ordinatis 
vs,  op,  bt,  ag  ex  parte  attractiva,  ut 
idcirco  nulla  habeatur  mutatio  e  positivo 
ad  negativum,  ex  attractione  in  repulsi- 
onem,  vel  vice  versa  ;  caeterum  utraque 
per  ductum  exponitur  curvae  continue 
habentis  duo  crura  infinita  asymptotica 
in  ramis  singulis  utrinque  in  infinitum 
productis.  Ex  hujusmodi  autem  virium 
lege,  &  ex  solis  principiis  Mechanicis 
notissimis,  nimirum  quod  ex  pluribus 
viribus,  vel  motibus  componatur  vis,  vel 
motus  quidam  ope  parallelogrammorum, 
quorum  latera  exprimant  vires,  vel  mo- 
tus componentes,  &  quod  vires  ejusmodi 

in  punctis  singulis,  tempusculis  singulis  aequalibus,  inducant  velocitates,  vel  motus  proportion- 
ales  sibi,  omnes  mihi  profluunt  generales,  &  praecipuae  quacque  particulars  proprietates  cor- 
porum,uti  etiam  superius  innui,  nee  ad  singulares  proprietates  derivandas  in  genere  afHrmo,  eas 
haberi  per  diversam  combinationem,  sed  combinationes  ipsas  evolvo,  &  geometrice  demon- 
stro,  quae  e  quibus  combinationibus  phasnomena,  &  corporum  species  oriri  debeant.  Verum 
antequam  ea  evolvo  in  parte  secunda,  &  tertia,  ostendam  in  hac  prima,  qua  via,  &  quibus 
positivis  rationibus  ad  earn  virium  legem  devenerim,  &  qua  ratione  illam  elementorum 
materiae  simplicitatem  eruerim,  turn  quas  difHcultatem  aliquam  videantur  habere  posse, 
dissolvam. 

1 6.  Cum  anno  1745  De  Viribus  vivis  dissertationem  conscriberem,  &  omnia,  quse   a 
viribus  vivis  repetunt,  qui  Leibnitianam  tuentur  sententiam,  &  vero  etiam  plerique  ex  iis, 
qui  per  solam  velocitatem  vires  vivas  metiuntur,  repeterem  immediate  a  sola    velocitate 
genita  per  potentiarum  vires,  quae  juxta  communem  omnium  Mechanicorum   sententiam 
velocitates  vel  generant,  vel  utcunque  inducunt  proportionales  sibi,  &  tempusculis,  quibus 
agunt,  uti  est  gravitas,  elasticitas,  atque  aliae  vires  ejusmodi ;    ccepi  aliquant:  o   diligentius 
inquirere  in  earn  productionem  velocitatis,  quae  per  impulsum  censetur  fieri,  ubi    tota 
velocitas  momento  temporis  produci  creditur  ab  iis,  qui  idcirco  percussionis  vim  infinities 
majorem  esse  censent  viribus  omnibus,  quae  pressionem  solam  momentis  singulis   exercent. 
Statim  illud  mihi  sese  obtulit,  alias  pro  percussionibus  ejusmodi,  quee  nimirum   momento 
temporis  finitam  velocitatem  inducant,  actionum  leges  haberi  debere. 


FIG 


origo  ejusdem  ex  17.  Verum  re  altius  considerata,  mihi  illud  incidit,  si  recta  utamur  ratiocinandi  methodo, 

susTmrnedUatTalin  eum  agendi  modum  submovendum  esse  a  Natura,  quae  nimirum  eandem  ubique   virium 

lege  Continuitatis.   legem,  ac  eandem  agendi  rationem  adhibeat :    impulsum  nimirum    immediatum  alterius 

corporis  in  alterum,  &  immediatam  percussionem  haberi  non  posse  sine  ilia    productione 

finitse  velocitatis  facta  momento  temporis  indivisibili,  &  hanc  sine  saltu  quodam,  &  Isesione 

illius,  quam  legem  Continuitatis  appellant,  quam  quidem  legem  in  Natura  existere,  &  quidem 

satis  [9]  valida  ratione  evinci  posse  existimabam.       En  autem  ratiocinationem  ipsam,   qua 

turn  quidem  primo  sum  usus,  ac  deinde  novis  aliis,  atque  aliis  meditationibus  illustravi,   ac 

confirmavi. 


minus  velox. 


Laesio  legis  Continu-  18.  Concipiantur  duo  corpora  aequalia,  quae  moveantur  in  directum  versus  eandem 

cOTpus^efocruTim-  plagam>  &  id,  quod  praecedit,  habeat  gradus  velocitatis  6,  id  vero,  quod  ipsum  persequitur 
mediate  incurrat  in  gradus  12.  Si  hoc  posterius  cum  sua  ilia  velocitate  illaesa  deveniat  ad  immediatum  contactum 
cum  illo  priore  ;  oportebit  utique,  ut  ipso  momento  temporis,  quo  ad  contactum  devenerint, 
illud  posterius  minuat  velocitatem  suam,  &  illud  primus  suam  augeat,  utrumque  per  saltum, 
abeunte  hoc  a  12  ad  9,  illo  a  6  ad  9,  sine  ullo  transitu  per  intermedios  gradus  n,  &  7  ;  10,  & 
8  ;  9^,  &  8i,  &c.  Neque  enim  fieri  potest,  ut  per  aliquam  utcunque  exiguam  continui 


A  THEORY  OF  NATURAL  PHILOSOPHY  45 

represent  the  forces  in  question,  which  are  initially  repulsive  &  increase  indefinitely  as  the 
distances  are  diminished  indefinitely,  but  which,  as  the  distances  increase,  are  first  of  all 
diminished,  then  vanish,  then  become  changed  in  direction  &  so  attractive,  again  vanish, 
&  change  their  direction,  &  so  on  alternately ;  until  at  length,  at  a  distance  comparatively 
great  they  finally  become  attractive  &  are  sensibly  proportional  to  the  inverse  squares  of 
the  distance. 

ic.  This  law  of  forces  differs  from  the  law  of  gravitation  enunciated  by  Newton  in  Difference  between 

.  J  -nii  r     i  i  •  i  i  •  •       this   'aw   °f   forces 

the  construction  &  development  or  the  curve  that  represents  it ;   thus,  the  curve  given  in  &  Newton's  law  of 
Fie.  2,  which  is  that  according  to  Newton,  is  DV,  a  hyperbola  of  the  third  degree,  lying  gravitation  ;     i  t  s 

ii  •  i         r     i  •          i  •   i     •       i  •  nil'6    use      ln     Physics ; 

altogether  on  one  side  of  the  axis,  which  it  does  not  cut  at  any  point ;  all  the  ordmates,  the  order  in  which 
such  as  vs,  op,  bt,  ag  lie  on  the  side  of  the  axis  representing  attractive  forces,  &  there-  ^ets^ects  are  to 
fore  there  is  no  change  from  positive  to  negative,  i.e.,  from  attraction  to  repulsion,  or 
vice  versa.  On  the  other  hand,  each  of  the  laws  is  represented  by  the  construction  of  a 
continuous  curve  possessing  two  infinite  asymptotic  branches  in  each  of  its  members,  if 
produced  to  infinity  on  both  sides.  Now,  from  a  law  of  forces  of  this  kind,  &  with  the 
help  of  well-known  mechanical  principles  only,  such  as  that  a  force  or  motion  can  be  com- 
pounded from  several  forces  or  motions  by  the  help  of  parallelograms  whose  sides  represent 
the  component  forces  or  motions,  or  that  the  forces  of  this  kind,  acting  on  single  points 
for  single  small  equal  intervals  of  time,  produce  in  them  velocities  that  are  proportional  to 
themselves ;  from  these  alone,  I  say,  there  have  burst  forth  on  me  in  a  regular  flood  all 
the  general  &  some  of  the  most  important  particular  properties  of  bodies,  as  I  intimated 
above.  Nor,  indeed,  for  the  purpose  of  deriving  special  properties,  do  I  assert  that  they 
ought  to  be  obtained  owing  to  some  special  combination  of  points ;  on  the  contrary  I 
consider  the  combinations  themselves,  &  prove  geometrically  what  phenomena,  or  what 
species  of  bodies,  ought  to  arise  from  this  or  that  combination.  Of  course,  before  I 
come  to  consider,  both  in  the  second  part  and  in  the  third,  all  the  matters  mentioned 
above,  I  will  show  in  this  first  part  in  what  way,  &  by  what  direct  reasoning,  I  have  arrived 
at  this  law  of  forces,  &  by  what  argument  I  have  made  out  the  simplicity  of  the  elements 
of  matter  ;  then  I  will  give  an  explanation  of  every  point  that  may  seem  to  present  any 
possible  difficulty. 

16.  In  the  year  1745,  I  was  putting  together  my  dissertation  De  Firibus  vivis,  &  had  The  occasion  that 
derived  everything  that  they  who  adhere  to  the  idea  of  Leibniz,  &  the  greater  number  of  o^my^L^Trom 
those  who  measure  '  living  forces '  by  means  of  velocity  only,  derive  from  these  '  living  the     consideration 
forces ' ;  as,  I  say  I  had  derived  everything  directly  &  solely  from  the  velocity  generated  by  of  imPulsive  action, 
the  forces  of  those  influences,  which,  according  to  the  generally  accepted  view  taken  by 

all  Mechanicians,  either  generate,  or  in  some  way  induce,  velocities  that  are  proportional 
to  themselves  &  the  intervals  of  time  during  which  they  act ;  take,  for  instance,  gravity, 
elasticity,  &  other  forces  of  the  same  kind.  I  then  began  to  investigate  somewhat  more 
carefully  that  production  of  velocity  which  is  thought  to  arise  through  impulsive  action, 
in  which  the  whole  of  the  velocity  is  credited  with  being  produced  in  an  instant  of  time  by 
those,  who  think,  because  of  that,  that  the  force  of  percussion  is  infinitely  greater  than  all 
forces  which  merely  exercise  pressure  for  single  instants.  It  immediately  forced  itself  upon 
me  that,  for  percussions  of  this  kind,  which  really  induce  a  finite  velocity  in  an  instant  of 
time,  laws  for  their  actions  must  be  obtained  different  from  the  rest. 

17.  However,  when  I  considered  the  matter  more  thoroughly,  it  struck  me  that,  if  The      cause      of 
we  employ  a  straightforward  method  of  argument,  such  a  mode  of  action  must  be  with-  w^s  the^pposftion 
drawn  from  Nature,  which  in  every  case  adheres  to  one  &  the  same  law  of  forces,  &  the  raised  to  the  Law 
same  mode  of  action.     I  came  to  the  conclusion  that  really  immediate  impulsive  action  of  °he  idea' 

one  body  on  another,  &  immediate  percussion,  could  not  be  obtained,  without  the  pro-  impulse, 
duction  of  a  finite  velocity  taking  place  in  an  indivisible  instant  of  time,  &  this  would  have 
to  be  accomplished  without  any  sudden  change  or  violation  of  what  is  called  the  Law  of 
Continuity  ;  this  law  indeed  I  considered  as  existing  in  Nature,  &  that  this  could  be  shown 
to  be  so  by  a  sufficiently  valid  argument.  The  following  is  the  line  of  argument  that  I 
employed  initially ;  afterwards  I  made  it  clearer  &  confirmed  it  by  further  arguments  & 
fresh  reflection. 

1 8.  Suppose  there  are  two  equal  bodies,  moving  in  the  same  straight  line  &  in  the  violation    of    the 
same  direction  ;  &  let  the  one  that  is  in  front  have  a  degree  of  velocity  represented  by  ^  tod^movrng1 
6,  &  the  one  behind  a  degree  represented  by  12.     If  the  latter,  i.e.,  the  body  that  was  be-  more  swiftly  comes 
hind,  should  ever  reach  with  its  velocity  undiminished,  &  come  into  absolute  contact  with,  J"*°  with^another 
the  former  body  which  was  in  front,  then  in  every  case  it  would  be  necessary  that,  at  the  body  moving  more 
very  instant  of  time  at  which  this  contact  happened,  the  hindermost  body  should  diminish  slowlv- 

its  velocity,  &  the  foremost  body  increase  its  velocity,  in  each  case  by  a  sudden  change  : 
one  of  them  would  pass  from  12  to  9,  the  other  from  6  to  9,  without  any  passage  through 
the  intermediate  degrees,  n  &  7,  10  &  8,  9$  &  8f,  &  so  on.  For  it  cannot  possibly  happen 


46  PHILOSOPHIC  NATURALIS  THEORIA 

temporis  particulam  ejusmodi  mutatio  fiat  per  intermedios  gradus,  durante  contactu.  Si 
enim  aliquando  alterum  corpus  jam  habuit  7  gradus  velocitatis,  &  alterum  adhuc  retinet 
1 1  ;  toto  illo  tempusculo,  quod  effluxit  ab  initio  contactus,  quando  velocitates  erant  12,  &  6, 
ad  id  tempus,  quo  sunt  n,  &  7,  corpus  secundum  debuit  moveri  cum  velocitate  majore, 
quam  primum,  adeoque  plus  percurrere  spatii,  quam  illud,  £  proinde  anterior  ejus  superficies 
debuit  transcurrere  ultra  illius  posteriorem  superficiem,  &  idcirco  pars  aliqua  corporis 
sequentis  cum  aliqua  antecedentis  corporis  parte  compenetrari  debuit,  quod  cum  ob 
impenetrabilitatem,  quam  in  materia  agnoscunt  passim  omnes  Physici,  &  quam  ipsi  tri- 
buendam  omnino  esse,  facile  evincitur,  fieri  omnino  non  possit ;  oportuit  sane,  in  ipso 
primo  initio  contactus,  in  ipso  indivisibili  momento  temporis,  quod  inter  tempus  continuum 
praecedens  contactum,  &  subsequens,  est  indivisibilis  limes,  ut  punctum  apud  Geometras 
est  limes  indivisibilis  inter  duo  continue  lineae  segmenta,  mutatio  velocitatum  facta  fuerit 
per  saltum  sine  transitu  per  intermedias,  laesa  penitus  ilia  continuitatis  lege,  quae  itum  ab 
una  magnitudine  ad  aliam  sine  transitu  per  intermedias  omnino  vetat.  Quod  autem  in 
corporibus  aequalibus  diximus  de  transitu  immediato  utriusque  ad  9  gradus  velocitatis, 
recurrit  utique  in  iisdem,  vel  in  utcunque  inaequalibus  de  quovis  alio  transitu  ad  numeros 
quosvis.  Nimirum  ille  posterioris  corporis  excessus  graduum  6  momento  temporis  auferri 
debet,  sive  imminuta  velocitate  in  ipso,  sive  aucta  in  priore,  vel  in  altero  imminuta  utcunque, 
&  aucta  in  altero,  quod  utique  sine  saltu,  qui  omissis  infinitis  intermediis  velocitatibus 
habeatur,  obtineri  omnino  non  poterit. 


Objectio    petita  a  ig.  Sunt,  qui  difficultatem  omnem  submoveri  posse  censeant,  dicendo,  id  quidem  ita  se 

cofporum.dl  '  habere  debere,  si  corpora  dura  habeantur,  quae  nimirum  nullam  compressionem  sentiant, 
nullam  mutationem  figurae  ;  &  quoniam  hsec  a  multis  excluduntur  penitus  a  Natura  ;  dum 
se  duo  globi  contingunt,  introcessione,  [10]  &  compressione  partium  fieri  posse,  ut  in  ipsis 
corporibus  velocitas  immutetur  per  omnes  intermedios  gradus  transitu  facto,  &  omnis 
argumenti  vis  eludatur. 

Ea  uti  non  posse,  2O    fa  mprjmis  ea  responsione  uti  non  possunt,  quicunque  cum  Newtono,  &  vero  etiam 

qui  admittunt  ele-  _,  \  .  .  r  .  j      •  o 

menta    soiida,    &  cum  plerisquc  veterum  Pnilosopnorum  pnma  elementa  matenae  omnino  dura  admittunt,  & 

dura-  soiida,  cum  adhaesione  infinita,  &  impossibilitate  absoluta  mutationis  figurae.     Nam  in  primis 

elementis  illis  solidis,  &  duris,  quae  in  anteriore  adsunt  sequentis  corporis  parte,  &  in  praece- 

dentis  posteriore,  quae  nimirum  se  mutuo  immediate  contingunt,  redit  omnis  argumenti  vis 

prorsus  illaesa. 

Extensionem   con-  2i.  Deinde  vero  illud  omnino  intelligi  sane  non  potest,  quo  pacto  corpora  omnia  partes 

primoT  pores,1™*!  aliquas  postremas  circa  superficiem  non  habeant  penitus  solidas,  quae  idcirco  comprimi 
parietes  soiidos,  ac  ornnino  non  possint.  In  materia  quidem,  si  continua  sit,  divisibilitas  in  infinitum  haberi 
potest,  &  vero  etiam  debet  ;  at  actualis  divisio  in  infinitum  difficultates  secum  trahit  sane 
inextricablies  ;  qua  tamen  divisione  in  infinitum  ii  indigent,  qui  nullam  in  corporibus 
admittunt  particulam  utcunque  exiguam  compressionis  omnis  expertem  penitus,  atque 
incapacem.  Ii  enim  debent  admittere,  particulam  quamcunque  actu  interpositis  poris 
distinctam,  divisamque  in  plures  pororum  ipsorum  velut  parietes,  poris  tamen  ipsis  iterum 
distinctos.  Illud  sane  intelligi  non  potest,  qui  fiat,  ut,  ubi  e  vacuo  spatio  transitur  ad  corpus, 
non  aliquis  continuus  haberi  debeat  alicujus  in  se  determinatae  crassitudinis  paries  usque  ad 
primum  porum,  poris  utique  carens  ;  vel  quomodo,  quod  eodem  recidit,  nullus  sit  extimus, 
&  superficiei  externae  omnium  proximus  porus,  qui  nimirum  si  sit  aliquis,  parietem  habeat 
utique  poris  expertem,  &  compressionis  incapacem,  in  quo  omnis  argumenti  superioris  vis 
redit  prorsus  illaesa. 


legis    Con-  22.  At  ea  etiam,  utcunque  penitus  inintelligibili,  sententia  admissa,  redit  omnis  eadem 

iprimis  su^r™  argument!  vis  in  ipsa  prima,  &  ultima  corporum  se  immediate  contingentium  superficie,  vel 
debus,  vel  punctis.  s{  nullae  continuae  superficies  congruant,  in  lineis,  vel  punctis.  Quidquid  enim  sit  id,  in  quo 
contactus  fiat,  debet  utique  esse  aliquid,  quod  nimirum  impenetrabilitati  occasionem 
praestet,  &  cogat  motum  in  sequente  corpore  minui,  in  prascedente  augeri  ;  id,  quidquid  est, 
in  quo  exeritur  impenetratibilitatis  vis,  quo  fit  immediatus  contactus,  id  sane  velocitatem 
mutare  debet  per  saltum,  sine  transitu  per  intermedia,  &  in  eo  continuitatis  lex  abrumpi 


A  THEORY  OF  NATURAL  PHILOSOPHY  47 

that  this  kind  of  change  is  made  by  intermediate  stages  in  some  finite  part,  however  small, 
of  continuous  time,  whilst  the  bodies  remain  in  contact.  For  if  at  any  time  the  one 
body  then  had  7  degrees  of  velocity,  the  other  would  still  retain  1 1  degrees ;  thus,  during 
the  whole  time  that  has  passed  since  the  beginning  of  contact,  when  the  velocities  were 
respectively  12  Si  6,  until  the  time  at  which  they  are  1 1  &  7,  the  second  body  must  be  moved 
with  a  greater  velocity  than  the  first ;  hence  it  must  traverse  a  greater  distance  in  space 
than  the  other.  It  follows  that  the  front  surface  of  the  second  body  must  have  passed 
beyond  the  back  surface  of  the  first  body  ;  &  therefore  some  part  of  the  body  that  follows 
behind  must  be  penetrated  by  some  part  of  the  body  that  goes  in  front.  Now,  on  account 
of  impenetrability,  which  all  Physicists  in  all  quarters  recognize  in  matter,  &  which  can  be 
easily  proved  to  be  rightly  attributed  to  it,  this  cannot  possibly  happen.  There  really 
must  be,  in  the  commencement  of  contact,  in  that  indivisible  instant  of  time  which  is  an 
indivisible  limit  between  the  continuous  time  that  preceded  the  contact  &  that  subsequent 
to  it  (just  in  the  same  way  as  a  point  in  geometry  is  an  indivisible  limit  between  two  seg- 
ments of  a  continuous  line),  a  change  of  velocity  taking  place  suddenly,  without  any  passage 
through  intermediate  stages  ;  &  this  violates  the  Law  of  Continuity,  which  absolutely 
denies  the  possibility  of  a  passage  from  one  magnitude  to  another  without  passing  through 
intermediate  stages.  Now  what  has  been  said  in  the  case  of  equal  bodies  concerning  the 
direct  passing  of  both  to  9  degrees  of  velocity,  in  every  case  holds  good  for  such  equal  bodies, 
or  for  bodies  that  are  unequal  in  any  way,  concerning  any  other  passage  to  any  numbers. 
In  fact,  the  excess  of  velocity  in  the  hindmost  body,  amounting  to  6  degrees,  has  to  be  got 
rid  of  in  an  instant  of  time,  whether  by  diminishing  the  velocity  of  this  body,  or  by  increasing 
the  velocity  of  the  other,  or  by  diminishing  somehow  the  velocity  of  the  one  &  increasing 
that  of  the  other  ;  &  this  cannot  possibly  be  done  in  any  case,  without  the  sudden  change 
that  is  obtained  by  omitting  the  infinite  number  of  intermediate  velocities. 

19.  There  are  some  people,  who  think  that  the  whole  difficulty  can  be  removed  by  An   objection   de- 
saying  that  this  is  just  as  it  should  be,  if  hard  bodies,  such  as  indeed  experience  no  com-  ^edexr^ncenyilo1 
pression  or  alteration  of  shape,  are  dealt  with  ;   whereas  by  many  philosophers  hard  bodies  hard  bodies. 

are  altogether  excluded  from  Nature  ;  &  therefore,  so  long  as  two  spheres  touch  one 
another,  it  is  possible,  by  introcession  &  compression  of  their  parts,  for  it  to  happen  that  in 
these  bodies  the  velocity  is  changed,  the  passage  being  made  through  all  intermediate  stages ; 
&  thus  the  whole  force  of  the  argument  will  be  evaded. 

20.  Now  in  the  first  place,  this  reply  can  not  be  used  by  anyone  who,  following  New-  This  reP'y  cannot 
ton,  &  indeed  many  of  the  ancient  philosophers  as  well,  admit  the  primary  elements  of  ^"admit^oiid0* 
matter  to  be  absolutely  hard  &  solid,  possessing  infinite  adhesion  &  a  definite  shape  that  it  hard  elements. 

is  perfectly  impossible  to  alter.  For  the  whole  force  of  my  argument  then  applies  quite 
unimpaired  to  those  solid  and  hard  primary  elements  that  are  in  the  anterior  part  of  the 
body  that  is  behind,  &  in  the  hindmost  part  of  the  body  that  is  in  front ;  &  certainly  these 
parts  touch  one  another  immediately. 

21.  Next  it  is  truly  impossible  to  understand  in  the  slightest  degree  how  all  bodies  do  Continuous  exten- 
not  have  some  of  their  last  parts  just  near  to  the  surface  perfectly  solid,  &  on  that  account  mary  ^resT*  walls 
altogether  incapable  of  being  compressed.     If  matter  is  continuous,  it  may  &  must  be  sub-  bounding      them, 
ject  to  infinite  divisibility ;    but  actual  division  carried  on  indefinitely  brings  in  its  train 

difficulties  that  are  truly  inextricable  ;  however,  this  infinite  division  is  required  by  those 
who  do  not  admit  that  there  are  any  particles,  no  matter  how  small,  in  bodies  that  are 
perfectly  free  from,  &  incapable  of,  compression.  For  they  must  admit  the  idea  that  every 
particle  is  marked  off  &  divided  up,  by  the  action  of  interspersed  pores,  into  many  boundary 
walls,  so  to  speak,  for  these  pores ;  &  these  walls  again  are  distinct  from  the  pores  them- 
selves. It  is  quite  impossible  to  understand  why  it  comes  about  that,  in  passing  from 
empty  vacuum  to  solid  matter,  we  are  not  then  bound  to  encounter  some  continuous  wall  of 
some  definite  inherent  thickness  from  the  surface  to  the  first  pore,  this  wall  being  everywhere 
devoid  of  pores ;  nor  why,  which  comes  to  the  same  thing  in  the  end,  there  does  not  exist 
a  pore  that  is  the  last  &  nearest  to  the  external  surface  ;  this  pore  at  least,  if  there  were  one, 
certainly  has  a  wall  that  is  free  from  pores  &  incapable  of  compression ;  &  here  then  the 
whole  force  of  the  argument  used  above  applies  perfectly  unimpaired. 

22.  Moreover,  even  if  this  idea  is  admitted,  although  it  may  be  quite  unintelligible,  Violation    of    the 
then  the  whole  force  of  the  same  argument  applies  to  the  first  or  last  surface  of  the  bodies  ta^s'piace^any 
that  are  in  immediate  contact  with  one  another  ;    or,  if  there  are  no  continuous  surfaces  rate,  in  prime  sur- 
congruent,  then  to  the  lines  or  points.     For,  whatever  the  manner  may  be  in  which  contact 

takes  place,  there  must  be  something  in  every  case  that  certainly  affords  occasion  for 
impenetrability,  &  causes  the  motion  of  the  body  that  follows  to  be  diminished,  &  that  of 
the  one  in  front  to  be  increased.  This,  whatever  it  may  be,  from  which  the  force  of  impene- 
trability is  derived,  at  the  instant  at  which  immediate  contact  is  obtained,  must  certainly 
change  the  velocity  suddenly,  &  without  any  passage  through  intermediate  stages ;  &  by 


48 


PHILOSOPHIC  NATURALIS  THEORIA 


debet,  atque  labefactari,  si  ad  ipsum  immediatum  contactum  illo  velocitatum  discrimine 
deveniatur.  Id  vero  est  sane  aliquid  in  quacunque  e  sententiis  omnibus  continuam 
extensionem  tribuentibus  materise.  Est  nimirum  realis  affectio  qusedam  corporis,  videlicet 
ejus  limes  ultimus  realis,  superficies,  realis  superficiei  limes  linea,  realis  lineae  limes  punctum, 
qua  affectiones  utcunque  in  iis  sententiis  sint  prorsus  inseparabiles  [n]  ab  ipso  corpore, 
sunt  tamen  non  utique  intellectu  confictae,  sed  reales,  quas  nimirum  reales  dimensiones 
aliquas  habent,  ut  superficies  binas,  linea  unam,  ac  realem  motum,  &  translationem  cum  ipso 
corpore,  cujus  idcirco  in  iis  sententiis  debent,  esse  affectiones  quaedam,  vel  modi. 


Objectio  petita   a  27.  Est,  qui  dicat,  nullum  in  iis  committi  saltum  idcirco,  quod  censendum  sit,  nullum 

vucemassa,    &,.J  r    .  ..  ,  ,,  i\/r    x 

motns.  quae  super-  habere  motum,  superficiem,  Imeam,  punctum,  quae  massam  habeant  nullam.    Motus,  mquit, 

ficiebus,  &  punctis  a  Mechanicis  habet  pro  mensura  massam  in  velocitatem  ductam  :   massa  autem  est  super- 
non  convemant.          _..  ,  .  •    •   «•  •          i  •      j  •  •  •  •         /-^ 

ficies  baseos  ducta  in  crassitudmem,  sive  altitudmem,  ex.  gr.  m  pnsmatis.  Quo  minor  est 
ejusmodi  crassitude,  eo  minor  est  massa,  &  motus,  ac  ipsa  crassitudine  evanescente,  evanescat 
oportet  &  massa,  &  motus. 

Kesponsionis    ini-  24.  Verum  qui  sic  ratiocinatur,  inprimis  ludit  in  ipsis  vocibus.     Massam  vulgo  appellant 

tacam.^punctmn!  quantitatem  materiae,  &  motum  corporum  metiuntur  per  massam  ejusmodi,  ac  velocitatem. 

posita    extensione  At  quemadmodum  in  ipsa  geometrica  quantitate  tria  genera  sunt  quantitatum,  corpus,  vel 

contmua,  e          -  ^11^^  qUO(J  trinam  dimensionem  habet,  superficies  quae  binas,  linae,  quae  unicam,  quibus 

accedit  linese  limes  punctum,  omni  dimensione,  &  extensione  carens  ;   sic  etiam  in  Physica 

habetur  in  communi  corpus  tribus  extensionis  speciebus  praeditum  ;  superficies  realis  extimus 

corporis  limes,  praedita  binis  ;    linea,  limes  realis  superficiei,  habens  unicam;    &  ejusdem 

lineae  indivisibilis  limes  punctum.     Utrobique  alterum  alterius  est  limes,  non  pars,  &  quatuor 

diversa  genera  constituunt.     Superficies  est  nihil  corporeum,  sed  non  &  nihil  superficial, 

quin  immo  partes  habet,  &  augeri  potest,  &  minui  ;  &  eodem  pacto  linea  in  ratione  quidem 

superficiei  est  nihil,  sed  aliquid  in  ratione  linese  ;  ac  ipsum  demum  punctum  est  aliquid  in 

suo  genere,  licet  in  ratione  lineae  sit  nihil. 


QUO  pacto  nomen  25.  Hinc  autem  in  iis  ipsis  massa  quaedam  considerari  potest  duarum  dimensionum,  vel 

motus  'debeat8  con-  unius,  vel  etiam  nullius  continuae  dimensionis,  sed  numeri  punctorum  tantummodo,  uti 

venire     superficie-  quantitas  ejus  genere  designetur  ;  quod  si  pro  iis  etiam  usurpetur  nomen  massae  generaliter, 

bus,  imeis,  punctis.  motus  quantitas  definiri  poterit  per  productum  ex  velocitate,  &  massa  ;  si  vero  massae  nomen 

tribuendum  sit  soli  corpori,  turn  motus  quidem  corporis  mensura  erit  massa  in  velocitatem 

ducta  ;   superficiei,  lineae,  punctorum  quotcunque  motus  pro  mensura  habebit  quantitatem 

superficiei,  vel  lineae,  vel  numerum  punctorum  in  velocitatem  ducta  ;  sed  motus  utique  iis 

omnibus  speciebus  tribuendus  erit,  eruntque  quatuor  motuum  genera,  ut  quatuor  sunt 

quantitatum,  solidi,  superficiei,  lineae,  punctorum  ;   ac  ut  altera  harum  erit  nihil  in  alterius 

ratione,  non  in  sua  ;  ita  alterius  motus  erit  nihil  in  ratione  alterius  sed  erit  sane  aliquid  in 

ratione  sui,  non  purum  nihil. 


Fore,  ut  ea  laedatur 
saltern  in  velocitate 
punctorum. 


Motum    passim  rI2i  2Q-    gt  quidem  jpSj  Mechanici  vulgo  motum  tribuunt  &  superficiebus  &  lineis,  & 

tnbui     punctis;  ,'•..*  ,  .  '.  -m       •    •  j 

fore,  lit  in  eo  ixda-  punctis,  ac  centri  gravitatis  motum  ubique  nommant  rhysici,  quod  centrum  utique  punctum 
i^r  Continuitatis  est  aliquod,  non  corpus  trina  praeditum  dimensione,  quam  iste  ad  motus  rationem,  & 
appellationem  requirit,  ludendo,  ut  ajebam,  in  verbis.  Porro  in  ejusmodi  motibus  exti- 
marum  saltern  superficierum,  vel  linearum,  vel  punctorum,  saltus  omnino  committi  debet, 
si  ea  ad  contactum  immediatum  deveniant  cum  illo  velocitatum  discrimine,  &  continuitatis 
lex  violari. 

27.  Verum  hac  omni  disquisitione  omissa  de  notione  motus,  &  massae,  si  factum  ex 
velocitate,  &  massa,  evanescente  una  e  tribus  dimensionibus,  evanescit ;  remanet  utique 
velocitas  reliquarum  dimensionum,  quae  remanet,  si  eae  reapse  remanent,  uti  quidem  omnino 
remanent  in  superficie,  &  ejus  velocitatis  mutatio  haberi  deberet  per  saltum,  ac  in  ea  violari 
continuitatis  lex  jam  toties  memorata. 

-,    ti°exin?P<!ne-  28.  Haec  quidem  ita  evidentia  sunt,  ut  omnino  dubitari  non  possit,  quin  continuitatis 

trabilitate  admissa    ,.,..,/  .-KT  •     »    •  •      i  «  i      •        •     j-       •      •  j         •• 

in    minimis  parti-  lex  infnngi  debeat,  &  saltus  m  Naturam  induci,  ubi  cum  velocitatis  discrimine  ad  se  invicem 

cuiis.  &  ejus  confu-  accedant  corpora,  &    ad    immediatum    contactum    deveniant,  si  modo  impenetrabilitas 

corporibus  tribuenda  sit,  uti  revera  est.     Earn  quidem  non  in  integris  tantummodo  corpori- 

bus,  sed  in  minimis  etiam  quibusque  corporum  particulis,  atque  elementis  agnoverunt 

Physici  universi.     Fuit  sane,  qui  post  meam  editam  Theoriam,  ut  ipsam  vim  mei  argument} 


A  THEORY  OF  NATURAL  PHILOSOPHY 


49 


that  the  Law  of  Continuity  must  be  broken  &  destroyed,  if  immediate  contact  is  arrived 
at  with  such  a  difference  of  velocity.  Moreover,  there  is  in  truth  always  something  of  this 
sort  in  every  one  of  the  ideas  that  attribute  continuous  extension  to  matter.  There  is  some 
real  condition  of  the  body,  namely,  its  last  real  boundary,  or  its  surface,  a  real  boundary  of 
a  surface,  a  line,  &  a  real  boundary  of  a  line,  a  point ;  &  these  conditions,  however  insepar- 
able they  may  be  in  these  theories  from  the  body  itself,  are  nevertheless  certainly  not 
fictions  of  the  brain,  but  real  things,  having  indeed  certain  real  dimensions  (for  instance,  a 
surface  has  two  dimensions,  &  a  line  one)  ;  they  also  have  real  motion  &  movement  of  trans- 
lation along  with  the  body  itself ;  hence  in  these  theories  they  must  be  certain  conditions 
or  modes  of  it. 

23.  Someone  may  say  that  there  is  no  sudden  change  made,  because  it  must  be  con-  Objection    derived 
sidered  that  a  surface,  a  line  or  a  point,  having  no  mass,  cannot  have  any  motion.     He  may  1™™  mo/io^w^idi 
say  that  motion  has,  according  to  Mechanicians,  as  its  measure,  the  mass  multiplied  by  the  do  not  accord  with 
velocity  ;    also  mass  is  the  surface  of  the  base  multiplied  by  the  thickness  or  the  altitude,  surfaces  &  P°mis- 
as  for  instance  in  prisms.     Hence  the  less  the  thickness,  the  less  the  mass  &  the  motion  ; 

thus,  if  the  thickness  vanishes,  then  both  the  mass  &  therefore  the  motion  must  vanish 
as  well. 

24.  Now  the  man  who  reasons  in  this  manner  is  first  of  all  merely  playing  with  words.  Commencement  of 
Mass  is  commonly  called  quantity  of  matter,  &  the  motion  of  bodies  is  measured  by  mass  the  answer  to  tl?ls : 

.      i    •       *   •        i  «  .         ,         •*•        —-  .  *  ,        .  c  •  -_*  cl  SUrlclCC,   OF  ii  11116, 

of  this  kind  &  the  velocity.     But,  just  as  in  a  geometrical  quantity  there  are  three  kinds  of  or  a  point,  is  some- 
quantities,  namely,  a  body  or  a  solid  having  three  dimensions,  a  surface  with  two,  &  a  line  \ 
with  one  :  to  which  is  added  the  boundary  of  a  line,  a  point,  lacking  dimensions  altogether,  is  supposed  to  ex- 
&  of  no  extension.     So  also  in  Physics,  a  body  is  considered  to  be  endowed  with  three  lst' 
species  of  extension  ;   a  surface,  the  last  real  boundary  of  a  body,  to  be  endowed  with  two ; 
a  line,  the  real  boundary  of  a  surface,  with  one  ;   &  the  indivisible  boundary  of  the  line,  to 
be  a  point.     In  both  subjects,  the  one  is  a  boundary  of  the  other,  &  not  a  part  of  it ;   & 
they  form  four  different  kinds.     There  is  nothing  solid  about  a  surface  ;   but  that  does  not 
mean  that  there  is  also  nothing  superficial  about  it ;    nay,  it  certainly  has  parts  &  can  be 
increased  or  diminished.     In  the  same  way  a  line  is  nothing  indeed  when  compared  with 
a  surface,  but  a  definite  something  when  compared  with  a  line  ;  &  lastly  a  point  is  a  definite 
something  in  its  own  class,  although  nothing  in  comparison  with  a  line. 

25.  Hence  also  in  these  matters,  a  mass  can  be  considered  to  be  of  two  dimensions,  or  The     manner    in 
of  one,  or  even  of  no  continuous  dimension,  but  only  numbers  of  points,  just  as  quantity  of  wn'^ma^and^the 
this  kind  is  indicated.     Now,  if  for  these  also,  the  term  mass  is  employed  in  a  generalized  term  motus  is  bound 
sense,  we  shall  be  able  to  define  the  quantity  of  motion  by  the  product  of  the  velocity  &  !:°;^p^'to?ujfaces' 

1  Ii  •  '  •  1  1  1      •  •  •     1  1  *    1      1  1  i  HOPS,    <X    pOintS. 

the  mass.  But  if  the  term  mass  is  only  to  be  used  in  connection  with  a  solid  body,  then 
indeed  the  motion  of  a  solid  body  will  be  measured  by  the  mass  multiplied  by  the  velocity ; 
but  the  motion  of  a  surface,  or  a  line,  or  any  number  of  points  will  have  as  their  measure 
the  quantity  of  the  surface,  or  line,  or  the  number- of  the  points,  multiplied  by  the  velocity. 
Motion  at  any  rate  will  be  ascribed  in  all  these  cases,  &  there  will  be  four  kinds  of  motion, 
as  there  are  four  kinds  of  quantity,  namely,  for  a  solid,  a  surface,  a  line,  or  for  points ;  and,  as 
each  class  of  the  latter  will  be  as  nothing  compared  with  the  class  before  it,  but  something 
in  its  own  class,  so  the  motion  of  the  one  will  be  as  nothing  compared  with  the  motion 
of  the  other,  but  yet  really  something,  &  not  entirely  nothing,  compared  with  those  of 
its  own  class. 

26.  Indeed,  Mechanicians  themselves  commonly  ascribe  motion  to  surfaces,  lines  &  Motion  is  ascribed 
points,  &  Physicists  universally  speak  of  the  motion  of  the  centre  of  gravity ;  this  centre  is  minateiy3  the'i^w 
undoubtedly  some  point,  &  not  a  body  endowed  with  three  dimensions,  which  the  objector  of  Continuity  is  vio- 
demands  for  the  idea  &  name  of  motion,  by  playing  with  words,  as  I  said  above.     On  the    ' 

other  hand,  in  this  kind  of  motions  of  ultimate  surfaces,  or  lines,  or  points,  a  sudden  change 
must  certainly  be  made,  if  they  arrive  at  immediate  contact  with  a  difference  of  velocity 
as  above,  &  the  Law  of  Continuity  must  be  violated. 

27.  But,  omitting  all  debate  about  the  notions  of  motion  &  mass,  if  the  product  of  it  is  at  least  a  fact 
the  velocity  &  the  mass  vanishes  when  one  of  the  three  dimensions  vanish,  there  will  still  fated^tf^the^idea 
remain  the  velocity  of  the  remaining  dimensions ;   &  this  will  persist  so  long  as  the  dimen-  of  the  velocity  of 
sions  persist,  as  they  do  persist  undoubtedly  in  the  case  of  a  surface.     Hence  the  change  P°mts- 

in  its  velocity  must  have  been  made  suddenly,  &  thereby  the  Law  of  Continuity,  which  I 
have  already  mentioned  so  many  times,  is  violated. 

28.  These  things  are  so  evident  that  it  is  absolutely  impossible  to  doubt  that  the  Law  objection    derived 

/./-!••••    r-  i     „      i  j  j  i  .     .  j          j  .  »T  iv        from  the  admission 

of  Continuity  is  infringed,  &  that  a  sudden  change  is  introduced  into  Nature,  when  bodies  Of   impenetrability 
approach  one  another  with  a  difference  of  velocity  &  come  into  immediate  contact,  if  only  in  verv  small  Par- 
we  are  to  ascribe  impenetrability  to  bodies,  as  we  really  should.     And  this  property  too,  tion. ' 
not  in  whole  bodies  only,  but  in  any  of  the  smallest  particles  of  bodies,  &  in  the  elements  as 
well,  is  recognized  by  Physicists  universally.     There  was  one,  I  must  confess,  who,  after  I 


50 


PHILOSOPHIC  NATURALIS   THEORIA 


infringeret,  affirmavit,  minimas  corporum  particulas  post  contactum  superficierum  com- 
penetrari  non  nihil,  &  post  ipsam  compenetrationem  mutari  velocitates  per  gradus.  At  id 
ipsum  facile  demonstrari  potest  contrarium  illi  inductioni,  &  analogiae,  quam  unam  habemus 
in  Physica  investigandis  generalibus  naturae  legibus  idoneam,  cujus  inductionis  vis  quae  sit, 
&  quibus  in  locis  usum  habeat,  quorum  locorum  unus  est  hie  ipse  impenetrabilitatis  ad 
minimas  quasque  particulas  extendendae,  inferius  exponam. 


Objectio   a  voce 

motus      assumpta 

pro  mutatione; 
confutatio  ex 
reahtate  motus 


2Q.  Fuit  itidem  e  Leibnitianorum  familia,  qui  post  evulgatam  Theoriam  meam  cen- 

.    '    ,./>-      •,  •  j-  •  j«  j  j          -i  • 

suerit,  dimcultatem  ejusmodi  amoveri  posse  dicendo,  duas  monades  sibi  etiam  mvicem 

occurrentes  cum  velocitatibus  quibuscunque  oppositis  aequalibus,  post  ipsum  contactum 
.....  .  i,  .  .   .'    r  ..... 

pergere  moven  sine  locali  progressione.  Ham  progressionem,  ajebat,  revera  omnmo  nihil 
esse,  si  a  spatio  percurso  sestimetur,  cum  spatium  sit  nihil  ;  motum  utique  perseverare,  & 
extingui  per  gradus,  quia  per  gradus  extinguatur  energia  ilia,  qua  in  se  mutuo  agunt,  sese 
premendo  invicem.  Is  itidem  ludit  in  voce  motus,  quam  adhibet  pro  mutatione  quacunque, 
&  actione,  vel  actionis  modo.  Motus  locaiis,  &  velocitas  motus  ipsius,  sunt  ea,  quse  ego 
quidem  adhibeo,  &  quae  ibi  abrumpuntur  per  saltum.  Ea,  ut  evidentissime  constat,  erant 
aliqua  ante  contactum,  &  post  contactum  mo-[i3]-mento  temporis  in  eo  casu  abrumpuntur  ; 
nee  vero  sunt  nihil  ;  licet  spatium  pure  imaginarium  sit  nihil.  Sunt  realis  affectio  rei 
mobilis  fundata  in  ipsis  modis  localiter  existendi,  qui  modi  etiam  relationes  inducunt  dis- 
tantiarum  reales  utique.  Quod  duo  corpora  magis  a  se  ipsis  invicem  distent,  vel  minus  ; 
quod  localiter  celerius  moveantur,  vel  lentius  ;  est  aliquid  non  imaginarie  tantummodo,  sed 
realiter  diversum  ;  in  eo  vero  per  immediatum  contactum  saltus  utique  induceretur  in  eo 
casu,  quo  ego  superius  sum  usus. 


Qui  Continuitatu,  30.  Et  sane  summus  nostri  aevi  Geometra,  &  Philosophus  Mac-Laurinus,  cum  etiam  ipse 

jegem  summover-  conisjonem  corporum  contemplatus  vidisset,  nihil  esse,  quod  continuitatis  legem  in  collisione 
corporum  facta  per  immediatum  contactum  conservare,  ac  tueri  posset,  ipsam  continuitatis 
legem  deferendam  censuit,  quam  in  eo  casu  omnino  violari  affirmavit  in  eo  opere,  quod  de 
Newtoni  Compertis  inscripsit,  lib.  I,  cap.  4.  Et  sane  sunt  alii  nonnulli,  qui  ipsam  con- 
tinuitatis legem  nequaquam  admiserint,  quos  inter  Maupertuisius,  vir  celeberrimus,  ac  de 
Republica  Litteraria  optime  meritus,  absurdam  etiam  censuit,  &  quodammodo  inexplica- 
bilem.  Eodem  nimirum  in  nostris  de  corporum  collisione  contemplationibus  devenimus 
Mac-Laurinus,  &  ego,  ut  viderimus  in  ipsa  immediatum  contactum,  atque  impulsionem  cum 
continuitatis  lege  conciliari  non  posse.  At  quoniam  de  impulsione,  &  immediate  corporum 
contactu  ille  ne  dubitari  quidem  posse  arbitrabatur,  (nee  vero  scio,  an  alius  quisquam  omnem 
omnium  corporum  immediatum  contactum  subducere  sit  ausus  antea,  utcunque  aliqui  aeris 
velum,  corporis  nimirum  alterius,  in  collisione  intermedium  retinuerint)  continuitatis 
legem  deseruit,  atque  infregit. 


Theorise  exortus, 
t^'t  Uf   fien 


31.  Ast  ego  cum  ipsam  continuitatis  legem  aliquanto  diligentius  considerarim,  & 
,  quibus  ea  innititur,  perpenderim,  arbitratus  sum,  ipsam  omnino  e  Natura 
submoveri  non  posse,  qua  proinde  retenta  contactum  ipsum  immediatum  submovendum 
censui  in  collisionibus  corporum,  ac  ea  consectaria  persecutus,  quae  ex  ipsa  continuitate 
servata  sponte  profluebant,  directa  ratiocinatione  delatus  sum  ad  earn,  quam  superius 
exposui,  virium  mutuarum  legem,  quae  consectaria  suo  quaeque  ordine  proferam,  ubi  ipsa, 
quae  ad  continuitatis  legem  retinendam  argumenta  me  movent,  attigero. 


Lex    Continuitatis  32.  Continuitatis  lex,  de  qua  hie  agimus,  in  eo  sita  est,  uti  superius  innui,  ut  quaevis 

quid     sit  :     discn-  •  j  i  •        i-  T     i-  •  •    r 

men  inter  status,  quantitas,  dum  ab  una  magmtudme  ad  aliam  migrat,  debeat  transire  per  omnes  intermedias 
&  incrementa.  ejusdem  generis  magnitudines.  Solet  etiam  idem  exprimi  nominandi  transitum  per  gradus 
intermedios,  quos  quidem  gradus  Maupertuisius  ita  accepit,  quasi  vero  quaedam  exiguae 
accessiones  fierent  momento  temporis,  in  quo  quidem  is  censuit  violari  jam  necessario  legem 
ipsam,  quae  utcunque  exiguo  saltu  utique  violatur  nihilo  minus,  quam  maximo  ;  cum 
nimi-[l4]-rum  magnum,  &  parvum  sint  tantummodo  respectiva  ;  &  jure  quidem  id  censuit  ; 
si  nomine  graduum  incrementa  magnitudinis  cujuscunque  momentanea  intelligerentur. 


A  THEORY  OF  NATURAL  PHILOSOPHY  51 

had  published  my  Theory,  endeavoured  to  overcome  the  force  of  the  argument  I  had  used 
by  asserting  that  the  minute  particles  of  the  bodies  after  contact  of  the  surfaces  were 
subject  to  compenetration  in  some  measure,  &  that  after  compenetration  the  velocities 
were  changed  gradually.  But  it  can  be  easily  proved  that  this  is  contrary  to  that  induction 
&  analogy,  such  as  we  have  in  Physics,  one  peculiarly  adapted  for  the  investigation  of  the 
general  laws  of  Nature.  What  the  power  of  this  induction  is,  &  where  it  can  be  used  (one 
of  the  cases  is  this  very  matter  of  extending  impenetrability  to  the  minute  particles  of  a 
body),  I  will  set  forth  later. 

29.  There  was  also  one  of  the  followers  of  Leibniz  who,  after  I  had  published  my  Objection   to    the 
Theory,  expressed  his  opinion  that  this  kind  of  difficulty  could  be  removed  by  saying  that  used  for°a"change^ 
two  monads  colliding  with  one  another  with  any  velocities  that  were  equal  &  opposite  refutation  from  the 

,,,.,  ..  .  .-I  ,        ,  •  TT      reality  of  local  mo- 

would,  alter  they  came  into  contact,  go  on  moving  without  any  local  progression,  rle  tion. 
added  that  that  progression  would  indeed  be  absolutely  nothing,  if  it  were  estimated  by  the 
space  passed  over,  since  the  space  was  nothing  ;  but  the  motion  would  go  on  &  be  destroyed 
by  degrees,  because  the  energy  with  which  they  act  upon  one  another,  by  mutual  pressure, 
would  be  gradually  destroyed.  He  also  is  playing  with  the  meaning  of  the  term  motus, 
which  he  uses  both  for  any  change,  &  for  action  &  mode  of  action.  Local  motion,  &  the 
velocity  of  that  motion  are  what  I  am  dealing  with,  &  these  are  here  broken  off  suddenly. 
These,  it  is  perfectly  evident,  were  something  definite  before  contact,  &  after  contact  in 
an  instant  of  time  in  this  case  they  are  broken  off.  Not  that  they  are  nothing ;  although 
purely  imaginary  space  is  nothing.  They  are  real  conditions  of  the  movable  thing 
depending  on  its  modes  of  extension  as  regards  position  ;  &  these  modes  induce  relations 
between  the  distances  that  are  certainly  real.  To  account  for  the  fact  that  two  bodies 
stand  at  a  greater  distance  from  one  another,  or  at  a  less ;  or  for  the  fact  that  they  are 
moved  in  position  more  quickly,  or  more  slowly ;  to  account  for  this  there  must  be  some- 
thing that  is  not  altogether  imaginary,  but  real  &  diverse.  In  this  something  there  would 
be  induced,  in  the  question  under  consideration,  a  sudden  change  through  immediate 
contact. 

30.  Indeed  the  finest  geometrician  &  philosopher  of  our  times,  Maclaurin,  after  he  too  There  are  some  who 
had  considered  the  collision  of  solid  bodies  &  observed  that  there  is  nothing  which  could  i^doi  continuity5 
maintain  &  preserve  the  Law  of  Continuity  in  the  collision  of  bodies  accomplished  by 

immediate  contact,  thought  that  the  Law  of  Continuity  ought  to  be  abandoned.  He 
asserted  that,  in  general  in  the  case  of  collision,  the  law  was  violated,  publishing  his  idea  in 
the  work  that  he  wrote  on  the  discoveries  of  Newton,  bk.  i,  chap.  4.  True,  there  are  some 
others  too,  who  would  not  admit  the  Law  of  Continuity  at  all ;  &  amongst  these,  Mauper- 
tuis,  a  man  of  great  reputation  &  the  highest  merit  in  the  world  of  letters,  thought  it  was 
senseless,  &  in  a  measure  inexplicable.  Thus,  Maclaurin  came  to  the  same  conclusion  as 
myself  with  regard  to  our  investigations  on  the  collision  of  bodies ;  for  we  both  saw  that,  in 
collision,  immediate  contact  &  impulsive  action  could  not  be  reconciled  with  the  Law  of 
Continuity.  But,  whereas  he  came  to  the  conclusion  that  there  could  be  no  doubt  about 
the  fact  of  impulsive  action  &  immediate  contact  between  the  bodies,  he  impeached  & 
abrogated  the  Law  of  Continuity.  Nor  indeed  do  I  know  of  anyone  else  before  me,  who 
has  had  the  courage  to  deny  the  existence  of  all  immediate  contact  for  any  bodies  whatever, 
although  there  are  some  who  would  retain  a  thin  layer  of  air,  (that  is  to  say,  of  another  body), 
in  between  the  two  in  collision. 

31.  But   I,  after  considering  the  Law  of    Continuity  somewhat  more  carefully,  &  The  origin  of  my 
pondering  over  the  fundamental  ideas  on  which  it  depends,  came  to  the  conclusion  that  this°Law,  as'shouid 
it  certainly  could  not  be  withdrawn  altogether  out  of  Nature.     Hence,  since  it  had  to  be  be  done, 
retained,  I  came  to  the  conclusion  that  immediate  contact  in  the  collision  of  solid  bodies 

must  be  got  rid  of ;  &,  investigating  the  deductions  that  naturally  sprang  from  the 
conservation  of  continuity,  I  was  led  by  straightforward  reasoning  to  the  law  that  I  have  set 
forth  above,  namely,  the  law  of  mutual  forces.  These  deductions,  each  set  out  in  order, 
I  will  bring  forward  when  I  come  to  touch  upon  those  arguments  that  persuade  me  to 
retain  the  Law  of  Continuity. 

32.  The  Law  of   Continuity,  as  we  here  deal  with  it,  consists  in  the  idea  that,  as  I  Jhe  nature  of  the 

.  j     ,  ..''...  .  ,  ,     Law  of  Continuity ; 

intimated  above,  any  quantity,  in  passing  from  one  magnitude  to  another,  must  pass  through  distinction  between 
all  intermediate  magnitudes  of  the  same  class.     The  same  notion  is  also  commonly  expressed  stat<~s     &     incre- 

,  ,     °  .,,.  ,.  -11    ments. 

by  saying  that  the  passage  is  made  by  intermediate  stages  or  steps ;  these  steps  indeed 
Maupertuis  accepted,  but  considered  that  they  were  very  small  additions  made  in  an 
instant  of  time.  In  this  he  thought  that  the  Law  of  Continuity  was  already  of  necessity 
violated,  the  law  being  indeed  violated  by  any  sudden  change,  no  matter  how  small,  in  no 
less  a  degree  than  by  a  very  great  one.  For,  of  a  truth,  large  &  small  are  only  relative  terms ; 
&  he  rightly  thought  as  he  did,  if  by  the  name  of  steps  we  are  to  understand  momentaneous 


PHILOSOPHIC  NATURALIS  THEORIA 


Geometriae  usus  ad 
earn  exponendam  : 
momenta  punctis, 
tempera  continua 
lineis  expressa. 


Fluxus  ordinatae 
transeuntis  per 
m  agnit  u  d  i  nes 
omnes  intermedias. 


Idem  in  quantitate 
variabili  expressa  : 
aequivocatio  in 
voce  gradus. 


FKMH     K'  M'  D' 


FIG.    3. 


Verum  id  ita  intelligendum  est ;    ut  singulis  momentis  singuli  status  respondeant ;   incre- 
menta,  vel  decrementa  non  nisi  continuis  tempusculis. 

33.  Id  sane  admodum  facile  concipitur  ope  Geometriae.     Sit  recta  quaedam  AB  in 
fig.  3,  ad  quam  referatur  quaedam  alia  linea  CDE.     Exprimat  prior  ex  iis  tempus,  uti  solet 
utique  in  ipsis  horologiis  circularis  peripheria 

ab  indicis  cuspide  denotata  tempus  definire. 
Quemadmodum  in  Geometria  in  lineis 
puncta  sunt  indivisibiles  limites  continuarum 
lineas  partium,  non  vero  partes  linese  ipsius ; 
ita  in  tempore  distinguenda;  erunt  partes 
continui  temporis  respondentes  ipsis  lines 
partibus,  continue  itidem  &  ipsas,  a  mo- 
mentis, quae  sunt  indivisibiles  earum  partium 
limites,  &  punctis  respondent ;  nee  inpos- 
terum  alio  sensu  agens  de  tempore  momenti 
nomen  adhibebo,  quam  eo  indivisibilis 
limitis ;  particulam  vero  temporis  utcunque 
exiguam,  &  habitam  etiam  pro  infinitesima, 
tempusculum  appellabo. 

34.  Si  jam  a  quovis  puncto  rectae  AB,  ut  F,  H,  erigatur  ordinata  perpendicularis  FG, 
HI,  usque    ad    lineam    CD  ;    ea  poterit  repraesentare  quantitatem  quampiam  continuo 
variabilem.  Cuicunque  momento  temporis  F,  H,  respondebit  sua  ejus  quantitatis  magnitudo 
FG,  HI ;  momentis  autem  intermediis  aliis  K,  M,  aliae  magnitudines,  KL,  MN,  respondebunt ; 
ac  si  a  puncto  G  ad  I  continua,  &  finita  abeat  pars  linese  CDE,  facile  patet  &  accurate  de- 
monstrari  potest,  utcunque  eadem  contorqueatur,  nullum  fore  punctum  K  intermedium, 
cui  aliqua  ordinata  KL  non  respondeat ;    &  e   converse  nullam  fore  ordinatam  magnitu- 
dinis  intermediae  inter  FG,  HI,  quae  alicui  puncto  inter  F,  H  intermedio  non  respondeat. 

35.  Quantitas  ilia  variabilis  per  hanc  variabilem  ordinatam  expressa  mutatur  juxta 
continuitatis  legem,  quia  a  magnitudine  FG,  quam  habet  momento  temporis  F,  ad  magni- 
tudinem  HI,  quae  respondet  momento  temporis  H,  transit  per  omnes  intermedias  magnitu- 
dines KL,  MN,  respondentes  intermediis  momentis  K,  M,  &  momento  cuivis  respondet 
determinata    magnitudo.     Quod   si   assumatur   tempusculum   quoddam    continuum    KM 
utcunque  exiguum  ita,  ut  inter  puncta  L,  N  arcus  ipse  LN  non  mutet  recessum  a  recta  AB 
in  accessum  ;   ducta  LO  ipsi  parallela,  habebitur  quantitas  NO,  quas  in  schemate  exhibito 
est   incrementum   magnitudinis   ejus   quantitatis   continuo  variatae.     Quo   minor   est  ibi 
temporis  particula  KM,  eo  minus  est  id  incrementum  NO,  &  ilia  evanescente,  ubi  congruant 
momenta  K,  M,  hoc  etiam  evanescit.     Potest  quaevis  magnitudo  KL,  MN  appellari  status 
quidam  variabilis   illius  quantitatis,  &  gradus  nomine  deberet  potius  in-[i5]-telligi  illud 
incrementum  NO,  quanquam  aliquando  etiam  ille  status,  ilia  magnitudo  KL  nomine  gradus 
intelligi  solet,  ubi  illud  dicitur,  quod  ab  una  magnitudine  ad  aliam  per  omnes  intermedios 
gradus  transeatur ;  quod  quidem  aequivocationibus  omnibus  occasionem  exhibuit. 


status   singuios  36.  Sed  omissis  aequivocationibus  ipsis,  illud,  quod  ad  rem  facit,  est  accessio  incremen- 

menta^vero'utcun"  torum  facta  non  momento  temporis,  sed  tempusculo  continuo,  quod  est  particula  continui 

que  parva  tem-  temporis.     Utcunque  exiguum  sit  incrementum  ON,  ipsi  semper  respondet  tempusculum 

respondereC°ntinuis  q.u°ddam  KM  continuum.     Nullum  est  in  linea  punctum  M  ita  proximum  puncto  K,  ut  sit 

primum  post  ipsum  ;   sed  vel  congruunt,  vel  intercipiunt  lineolam  continua  bisectione  per 

alia  intermedia  puncta  perpetuo  divisibilem  in  infinitum.     Eodem  pacto  nullum  est  in 

tempore  momentum  ita  proximum  alteri  praecedenti  momento,  ut  sit  primum  post  ipsum, 

sed  vel  idem  momentum  sunt,  vel  inter jacet  inter  ipsa  tempusculum  continuum  per  alia 

intermedia  momenta  divisibile  in  infinitum  ;    ac  nullus  itidem  est  quantitatis  continuo 

variabilis  status  ita  proximus  praecedenti  statui,  ut  sit  primus  post  ipsum  accessu  aliquo 

momentaneo  facto  :    sed  differentia,  quae  inter  ejusmodi  status  est,  debetur  intermedio 

continuo   tempusculo ;    ac  data  lege  variationis,  sive  natura  lineae  ipsam    exprimentis,  & 

quacunque  utcunque  exigua  accessione,  inveniri  potest  tempusculum  continuum,  quo  ea 

accessio  advenerit. 

Transitus  sine  sal-  37-  Atque  sic  quidem  intelligitur,  quo  pacto  fieri  possit  transitus  per  intermedias 

tu, etiamapositivis  magnitudines  omnes,  per  intermedios  status,  per  gradus  intermedios,  quin  ullus  habeatur 

ad  negativa  perm-        ,°  .     r  -,          .     ...     ,   '   " 

hiium,  quod  tamen  saltus  utcunque  exiguus  momento  temporis  factus.  Notari  mud  potest  tantummodo, 
m°"  eSstedVereu'ida1m  mutati°nem  neri  alicubi  per  incrementa,  ut  ubi  KL  abit,  in  MN  per  NO  ;  alicubi  per 
reaiis  status,1"0  '  decrementa,  ut  ubi  K'L'  abeat  in  N'M'  per  O'N' ;  quin  immo  si  linea  CDE,  quse  legem 


A  THEORY  OF  NATURAL  PHILOSOPHY  53 

increments  of  any  magnitude  whatever.  But  the  idea  should  be  interpreted  as  follows  : 
single  states  correspond  to  single  instants  of  time,  but  increments  or  decrements  only  to 
small  intervals  of  continuous  time. 

33.  The  idea  can  be  very  easily  assimilated  by  the  help  of  geometry.  Explanation  by  the 
Let  AB  be  any  straight  line  (Fig.  3),  to  which  as  axis  let  any  other  line  CDE  be  referred.  "nsseta°tfs  ^eTes^ 

Let  the  first  of  them  represent  the  time,  in  the  same  manner  as  it  is  customary  to  specify  ted  by  points,  con- 
the  time  in  the  case  of  circular  clocks  by  marking  off  the  periphery  with  the  end  of  a  pointer.  1™°^  "^s***  °f 
Now,  just  as  in  geometry,  points  are  the  indivisible  boundaries  of  the  continuous  parts  of 
a  line,  so,  in  time,  distinction  must  be  made  between  parts  of  continuous  time,  which  cor- 
respond to  these  parts  of  a  line,  themselves  also  continuous,  &  instants  of  time,  which  are 
the  indivisible  boundaries  of  those  parts  of  time,  &  correspond  to  points.     In  future  I  shall 
not  use  the  term  instant  in  any  other  sense,  when  dealing  with  time,  than  that  of  the 
indivisible  boundary ;   &  a  small  part  of  time,  no  matter  how  small,  even  though  it  is 
considered  to  be  infinitesimal,  I  shall  term  a  tempuscule,  or  small  interval  of  time. 

34.  If  now  from  any  points  F,H  on  the  straight  line  AB  there  are  erected  at  right  angles  T.he  flux  °.f  the  or~ 
to  it  ordinates  FG,  HI,  to  meet  the  line  CD  ;  any  of  these  ordinates  can  be  taken  to  repre-  through^  ail  *interS 
sent  a  quantity  that  is  continuously  varying.     To  any  instant  of  time  F,  or  H,  there  will  mediate  values, 
correspond  its  own  magnitude  of  the  quantity  FG,  or  HI  ;  &  to  other  intermediate  instants 

K,  M,  other  magnitudes  KL,  MN  will  correspond.  Now,  if  from  the  point  G,  there  pro- 
ceeds a  continuous  &  finite  part  of  the  line  CDE,  it  is  very  evident,  &  it  can  be  rigorously 
proved,  that,  no  matter  how  the  curve  twists  &  turns,  there  is  no  intermediate  point  K, 
to  which  some  ordinate  KL  does  not  correspond  ;  &,  conversely,  there  is  no  ordinate  of 
magnitude  intermediate  between  FG  &  HI,  to  which  there  does  not  correspond  a  point 
intermediate  between  F  &  H. 

35.  The  variable  quantity  that  is  represented  by  this  variable  ordinate  is  altered  in  The    same     holds 
accordance  with  the  Law  of  Continuity  ;    for,  from  the  magnitude  FG,  which  it  has  at  able1  quantity    w 
the  instant  of  time  F,  to  the  magnitude  HI,  which  corresponds  to  the  instant  H,  it  passes  represented  ;  equi- 
through  all  intermediate  magnitudes  KL,  MN,  which  correspond  to  the  intermediate  oUhe1(term Itep^ 
instants  K,  M  ;  &  to  every  instant  there  corresponds  a  definite  magnitude.     But  if  we  take 

a  definite  small  interval  of  continuous  time  KM,  no  matter  how  small,  so  that  between  the 
points  L  &  N  the  arc  LN  does  not  alter  from  recession  from  the  line  AB  to  approach,  & 
draw  LO  parallel  to  AB,  we  shall  obtain  the  quantity  NO  that  in  the  figure  as  drawn  is  the 
increment  of  the  magnitude  of  the  continuously  varying  quantity.  Now  the  smaller  the 
interval  of  time  KM,  the  smaller  is  this  increment  NO  ;  &  as  that  vanishes  when  the 
instants  of  time  K,  M  coincide,  the  increment  NO  also  vanishes.  Any  magnitude  KL,  MN 
can  be  called  a  state  of  the  variable  quantity,  &  by  the  name  step  we  ought  rather  to  under- 
stand the  increment  NO  ;  although  sometimes  also  the  state,  or  the  magnitude  KL  is 
accustomed  to  be  called  by  the  name  step.  For  instance,  when  it  is  said  that  from  one 
magnitude  to  another  there  is  a  passage  through  all  intermediate  stages  or  steps ;  but  this 
indeed  affords  opportunity  for  equivocations  of  all  sorts. 

36.  But,  omitting  all  equivocation  of  this  kind,  the  point  is  this  :    that  addition  of  single   states  cor- 

.'  1-11  •  •  <•      •  i          •  11    .  respond  to  instants, 

increments  is    accomplished,  not  m  an  instant  01  time,  but  in  a    small  interval  of  con-  but      increments 
tinuous  time,  which  is  a  part  of  continuous  time.     However  small  the  increment  ON  may  however  sma11   to 

i  i  i    Tru  if        mi  •»•>    intervals     of     con- 

DC,  there  always  corresponds  to  it  some  continuous  interval  KM.      1  here  is  no  point  M  tinuous  time. 

in  the  straight  line  AB  so  very  close  to  the  point  K,  that  it  is  the  next  after  it ;  but  either 
the  points  coincide,  or  they  intercept  between  them  a  short  length  of  line  that  is  divisible 
again  &  again  indefinitely  by  repeated  bisection  at  other  points  that  are  in  between  M  & 
K.  In  the  same  way,  there  is  no  instant  of  time  that  is  so  near  to  another  instant  that  has 
gone  before  it,  that  it  is  the  next  after  it ;  but  either  they  are  the  same  instant,  or  there 
lies  between  them  a  continuous  interval  that  can  be  divided  indefinitely  at  other  inter- 
mediate instants.  Similarly,  there  is  no  state  of  a  continuously  varying  quantity  so  very 
near  to  a  preceding  state  that  it  is  the  next  state  to  it,  some  momentary  addition  having 
been  made  ;  any  difference  that  exists  between  two  states  of  the  same  kind  is  due  to  a 
continuous  interval  of  time  that  has  passed  in  the  meanwhile.  Hence,  being  given  the 
law  of  variation,  or  the  nature  of  the  line  that  represents  it,  &  any  increment,  no  matter 
how  small,  it  is  possible  to  find  a  small  interval  of  continuous  time  in  which  the  increment 
took  place. 

37.  In  this  manner  we  can  understand  how  it  is  possible  for  a  passage  to  take  place  Passages     without 
through  all  intermediate  magnitudes,  through  intermediate  states,  or  through  intermediate  from^positive1 8  to 
stages,  without  any  sudden  change  being  made,  no  matter  how  small,  in  an  instant  of  time,  negative     through 

T'  11  1111  •  i  111-  /i  zero  :      zero     how- 

It  can  merely  be  remarked  that  change  in  some  places  takes  place  by  increments  (as  when  ever  ;s  not  realiy 

KL  becomes  MN  by  the  addition  of  NO),  in  other  places  by  decrements  (as  when  K'L'  nothing,  but  acer- 

'  tain  real  state. 


54  PHILOSOPHIC  NATURALIS  THEORIA 

variationis  exhibit,  alicubi  secet  rectam,  temporis  AB,  potest  ibidem  evanescere  magnitude, 
ut  ordinata  M'N',  puncto  M'  allapso  ad  D  evanesceret,  &  deinde  mutari  in  negativam  PQ, 
RS,  habentem  videlicet  directionem  contrariam,  quae,  quo  magis  ex  oppositae  parte  crescit, 
eo  minor  censetur  in  ratione  priore,  quemadmodum  in  ratione  possessionis,  vel  divitiarum, 
pergit  perpetuo  se  habere  pejus,  qui  iis  omnibus,  quae  habebat,  absumptis,  aes  alienum 
contrahit  perpetuo  majus.  Et  in  Geometria  quidem  habetur  a  positivo  ad  negativa 
transitus,  uti  etiam  in  Algebraicis  formulis,  tarn  transeundo  per  nihilum,  quam  per  innnitum, 
quos  ego  transitus  persecutus  sum  partim  in  dissertatione  adjecta  meis  Sectionibus  Conicis, 
partim  in  Algebra  §  14,  &  utrumque  simul  in  dissertatione  De  Lege  Continuitatis ;  sed  in 
Physica,  ubi  nulla  quantitas  in  innnitum  excrescit,  is  casus  locum  non  habet,  &  non,  nisi 
transeundo  per  nihilum,  transitus  fit  a  positi-[i6]-vis  ad  negativa,  ac  vice  versa  ;  quanquam, 
uti  inferius  innuam,  id  ipsum  sit  non  nihilum  revera  in  se  ipso,  sed  realis  quidem  status,  & 
habeatur  pro  nihilo  in  consideration  quadam  tantummodo,  in  qua  negativa  etiam,  qui  sunt 
veri  status,  in  se  positivi,  ut  ut  ad  priorem  seriem  pertinentes  negative  quodam  modo, 
negativa  appellentur. 


Proponitur  pro-  ,§_  Exposita  hoc  pacto,  &  vindicata  continuitatis  lege,  earn  in  Natura  existere  plerique 

banda       existentia   _,  .,  J       .  .    r  .  .  ....  ...  P  .          .        ,-,  r. 

legis  Continuitat.s.  Philosophi  arbitrantur,  contradicentibus  nonnullis,  uti  supra  mnui.  Ego,  cum  in  earn 
primo  inquirerem,  censui,  eandem  omitti  omnino  non  posse  ;  si  earn,  quam  habemus  unicam, 
Naturae  analogiam,  &  inductionis  vim  consulamus,  ope  cujus  inductionis  earn  demonstrare 
conatus  sum  in  pluribus  e  memoratis  dissertationibus,  ac  eandem  probationem  adhibet 
Benvenutus  in  sua  Synopsi  Num.  119;  in  quibus  etiam  locis,  prout  diversis  occasionibus 
conscripta  sunt,  repetuntur  non  nulla. 

Ejus   probatio  ab  ,g    Longum  hie  esset  singula  inde  excerpere  in  ordinem  redacta  :   satis  erit  exscribere 

mductione     satis     ,.        Jy    .          °_      ,  ~        .       P      .  r      ,-,        -n          •     i         •  •  j 

ampia.  dissertatioms  De  lege  Continuitatis  numerum  138.     Post  mductionem  petitam  praecedente 

numero  a  Geometria,  quae  nullum  uspiam  habet  saltum,  atque  a  motu  locali,  in  quo  nunquam 
ab  uno  loco  ad  alium  devenitur,  nisi  ductu  continue  aliquo,  unde  consequitur  illud,  dis- 
tantiam  a  dato  loco  nunquam  mutari  in  aliam,  neque  densitatem,  quae  utique  a  distantiis 
pendet  particularum  in  aliam,  nisi  transeundo  per  intermedias ;  fit  gradus  in  eo  numero  ad 
motuum  velocitates,  &  ductus,  quas  magis  hie  ad  rem  faciunt,  nimirum  ubi  de  velocitate 
agimus  non  mutanda  per  saltum  in  corporum  collisionibus.  Sic  autem  habetur  :  "  Quin 
immo  in  motibus  ipsis  continuitas  servatur  etiam  in  eo,  quod  motus  omnes  in  lineis  continuis 
fiunt  nusquam  abruptis.  Plurimos  ejusmodi  motus  videmus.  Planetae,  &  cometse  in  lineis 
continuis  cursum  peragunt  suum,  &  omnes  retrogradationes  fiunt  paullatim,  ac  in  stationibus 
semper  exiguus  quidem  motus,  sed  tamen  habetur  semper,  atque  hinc  etiam  dies  paullatim 
per  auroram  venit,  per  vespertinum  crepusculum  abit,  Solis  diameter  non  per  saltum,  sed 
continuo  motu  supra  horizontem  ascendit,  vel  descendit.  Gravia  itidem  oblique  projecta 
in  lineis  itidem  pariter  continuis  motus  exercent  suos,  nimirum  in  parabolis,  seclusa  ^aeris 
resistentia,  vel,  ea  considerata,  in  orbibus  ad  hyperbolas  potius  accedentibus,  &  quidem 
semper  cum  aliqua  exigua  obliquitate  projiciuntur,  cum  infinities  infinitam  improbabilitatem 
habeat  motus  accurate  verticalis  inter  infinities  infinitas  inclinationes,  licet  exiguas,  &  sub 
sensum  non  cadentes,  fortuito  obvenienfe,  qui  quidem  motus  in  hypothesi  Telluris^motae  a 
parabolicis  plurimum  distant,  &  curvam  continuam  exhibent  etiam  pro  casu  projectionis 
accurate  verticalis,  quo,  quiescente  penitus  Tellure,  &  nulla  ventorum  vi  deflectente  motum, 
haberetur  [17]  ascensus  rectilineus,  vel  descensus.  Immo  omnes  alii  motus  a  gravitate 
pendentes,  omnes  ab  elasticitate,  a  vi  magnetica,  continuitatem  itidem  servant ;  cum  earn 
servent  vires  illse  ipsae,  quibus  gignuntur.  Nam  gravitas,  cum  decrescat  in  ratione  reciproca 
duplicata  distantiarum,  &  distantise  per  saltum  mutari  non  possint,  mutatur  per  omnes 
intermedias  magnitudines.  Videmus  pariter,  vim  magneticam  a  distantiis  pendere  lege 
continua  ;  vim  elasticam  ab  inflexione,  uti  in  laminis,  vel  a  distantia,  ut  in  particulis  aeris 
compressi.  In  iis,  &  omnibus  ejusmodi  viribus,  &  motibus,  quos  gignunt,  continuitas  habetur 
semper,  tarn  in  lineis  quae  describuntur,  quam  in  velocitatibus,  quae  pariter  per  omnes 
intermedias  magnitudines  mutantur,  ut  videre  est  in  pendulis,  in  ascensu  corporum  gravium, 


A  THEORY  OF  NATURAL  PHILOSOPHY  55 

becomes  N'M'  by  the  subtraction  of  O'N')  ;  moreover,  if  the  line  CDE,  which  represents 
the  law  of  variation,  cuts  the  straight  AB,  which  is  the  axis  of  time,  in  any  point,  then  the 
magnitude  can  vanish  at  that  point  (just  as  the  ordinate  M'N'  would  vanish  when  the 
point  M'  coincided  with  D),  &  be  changed  into  a  negative  magnitude  PQ,  or  RS,  that  is 
to  say  one  having  an  opposite  direction  ;  &  this,  the  more  it  increases  in  the  opposite  sense, 
the  less  it  is  to  be  considered  in  the  former  sense  (just  as  in  the  idea  of  property  or  riches, 
a  man  goes  on  continuously  getting  worse  off,  when,  after  everything  he  had  has  been 
taken  away  from  him,  he  continues  to  get  deeper  &  deeper  into  debt).  In  Geometry  too 
we  have  this  passage  from  positive  to  negative,  &  also  in  algebraical  formulae,  the  passage 
being  made  not  only  through  nothing,  but  also  through  infinity ;  such  I  have  discussed, 
the  one  in  a  dissertation  added  to  my  Conic  Sections,  the  other  in  my  Algebra  (§  14),  &  both 
of  them  together  in  my  essay  De  Lege  Continuitatis ;  but  in  Physics,  where  no  quantity 
ever  increases  to  an  infinite  extent,  the  second  case  has  no  place  ;  hence,  unless  the  passage 
is  made  through  the  value  nothing,  there  is  no  passage  from  positive  to  negative,  or  vice 
versa.  Although,  as  I  point  out  below,  this  nothing  is  not  really  nothing  in  itself,  but  a 
certain  real  state  ;  &  it  may  be  considered  as  nothing  only  in  a  certain  sense.  In  the  same 
sense,  too,  negatives,  which  are  true  states,  are  positive  in  themselves,  although,  as  they 
belong  to  the  first  set  in  a  certain  negative  way,  they  are  called  negative. 

38.  Thus  explained  &  defended,  the  Law  of  Continuity  is  considered  by  most  philoso-  I  propose  to  prove 
phers  to  exist  in  Nature,  though  there  are  some  who  deny  it,  as  I  mentioned  above.     I,  LaVof^Continuity6 
when  first  I  investigated  the  matter,  considered  that  it  was  absolutely  impossible  that  it 
should  be  left  out  of  account,  if  we  have  regard  to  the  unparalleled  analogy  that  there  is 
with  Nature  &  to  the  power  of  induction  ;  &  by  the  help  of  this  induction  I  endeavoured 
to  prove  the  law  in  several  of  the  dissertations  that  I  have  mentioned,  &  Benvenutus  also 
used  the  same  form  of  proof  in  his  Synopsis  (Art.  119).     In  these  too,  as  they  were  written 
on  several  different  occasions,  there  are  some  repetitions. 

39.  It  would  take  too  long  to  extract  &  arrange  in  order  here  each  of  the  passages  in  Proof  by  induction 
these  essays ;  it  will  be  sufficient  if  I  give  Art.  138  of  the  dissertation  De  Lege  Continuitatis.  s~^^  for  the 
After  induction  derived  in  the  preceding  article  from  geometry,  in  which  there  is  no  sudden 
change  anywhere,  &  from  local  motion,  in  which  passage  from  one  position  to  another 
never  takes  place  unless  by  some  continuous  progress  (the  consequence  of  which  is  that  a 
distance  from  any  given  position  can  never  be  changed  into  another  distance,  nor  the 
density,  which  depends  altogether  on  the  distances  between  the  particles, into  another  density, 
except  by  passing  through  intermediate  stages),  the  step  is  made  in  that  article  to  the 
velocities  of  motions,  &  deductions,  which  have  more  to  do  with  the  matter  now  in  hand, 
namely,  where  we  are  dealing  with  the  idea  that  the  velocity  is  not  changed  suddenly  in  the 
collision  of  solid  bodies.  These  are  the  words  :  "  Moreover  in  motions  themselves 
continuity  is  preserved  also  in  the  fact  that  all  motions  take  place  in  continuous  lines  that 
are  not  broken  anywhere.  We  see  a  great  number  of  motions  of  this  kind.  The  planets  & 
the  comets  pursue  their  courses,  each  in  its  own  continuous  line,  &  all  retrogradations  are 
gradual ;  &  in  stationary  positions  the  motion  is  always  slight  indeed,  but  yet  there  is 
always  some  ;  hence  also  daylight  comes  gradually  through  the  dawn,  &  goes  through  the 
evening  twilight,  as  the  diameter  of  the  sun  ascends  above  the  horizon,  not  suddenly,  but 
by  a  continuous  motion,  &  in  the  same  manner  descends.  Again  heavy  bodies  projected 
obliquely  follow  their  courses  in  lines  also  that  are  just  as  continuous ;  namely,  in  para- 
bolae,  if  we  neglect  the  resistance  of  the  air,  but  if  that  is  taken  into  account,  then  in  orbits 
that  are  more  nearly  hyperbolae.  Now,  they  are  always  projected  with  some  slight  obli- 
quity, since  there  is  an  infinitely  infinite  probability  against  accurate  vertical  motion,  from 
out  of  the  infinitely  infinite  number  of  inclinations  (although  slight  &  not  capable  of  being 
observed),  happening  fortuitously.  These  motions  are  indeed  very  far  from  being  para- 
bolae,  if  the  hypothesis  that  the  Earth  is  in  motion  is  adopted.  They  give  a  continuous 
curve  also  for  the  case  of  accurate  vertical  projection,  in  which,  if  the  Earth  were  at  rest, 
&  no  wind-force  deflected  the  motion,  rectilinear  ascent  &  descent  would  be  obtained. 
All  other  motions  that  depend  on  gravity,  all  that  depend  upon  elasticity,  or  magnetic 
force,  also  preserve  continuity ;  for  the  forces  themselves,  from  which  the  motions  arise, 
preserve  it.  For  gravity,  since  it  diminishes  in  the  inverse  ratio  of  the  squares  of  the  dis- 
tances, &  the  distances  cannot  be  changed  suddenly,  is  itself  changed  through  every  inter- 
mediate stage.  Similarly  we  see  that  magnetic  force  depends  on  the  distances  according 
to  a  continuous  law ;  that  elastic  force  depends  on  the  amount  of  bending  as  in  plates,  or 
according  to  distance  as  in  particles  of  compressed  air.  In  these,  &  all  other  forces  of  the 
sort,  &  in  the  motions  that  arise  from  them,  we  always  get  continuity,  both  as  regards  the 
lines  which  they  describe  &  also  in  the  velocities  which  are  changed  in  similar  manner 
through  all  intermediate  magnitudes ;  as  is  seen  in  pendulums,  in  the  ascent  of  heavy 


56  PHILOSOPHISE  NATURALIS  THEORIA 

&  in  aliis  mille  ejusmodi,  in  quibus  mutationes  velocitatis  fiunt  gradatim,  nee  retro  cursus 
reflectitur,  nisi  imminuta  velocitate  per  omnes  gradus.  Ea  diligentissime  continuitatem 
servat  omnia.  Hinc  nee  ulli  in  naturalibus  motibus  habentur  anguli,  sed  semper  mutatio 
directionis  fit  paullatim,  nee  vero  anguli  exacti  habentur  in  corporibus  ipsis,  in  quibus 
utcunque  videatur  tennis  acies,  vel  cuspis,  microscopii  saltern  ope  videri  solet  curvatura, 
quam  etiam  habent  alvei  fluviorum  semper,  habent  arborum  folia,  &  frondes,  ac  rami,  habent 
lapides  quicunque,  nisi  forte  alicubi  cuspides  continuae  occurrant,  vel  primi  generis,  quas 
Natura  videtur  affectare  in  spinis,  vel  secundi  generis,  quas  videtur  affectare  in  avium 
unguibus,  &  rostro,  in  quibus  tamen  manente  in  ipsa  cuspide  unica  tangente  continuitatem 
servari  videbimus  infra.  Infinitum  esset  singula  persequi,  in  quibus  continuitas  in  Natura 
observatur.  Satius  est  generaliter  provocare  ad  exhibendum  casum  in  Natura,  in  quo 
eontinuitas  non  servetur,  qui  omnino  exhiberi  non  poterit." 


Duplex  inductionis  40.  Inductio  amplissima  turn  ex  hisce  motibus,  ac  velocitatibus,  turn  ex  aliis  pluribus 

vimhabeatittductio  exemPn's>  <lU3e  habemus  in  Natura,  in  quibus  ea  ubique,  quantum  observando  licet  depre- 
incompieta.  hendere,  continuitatem  vel  observat  accurate,  vel  affcctat,  debet  omnino  id  efficere,  ut  ab 

ea  ne  in  ipsa  quidem  corporum  collisione  recedamus.  Sed  de  inductionis  natura,  &  vi,  ac 
ejusdem  usu  in  Physica,  libet  itidem  hie  inserere  partem  numeri  134,  &  totum  135,  disserta- 
tionis  De  Lege  Continuitatis.  Sic  autem  habent  ibidem  :  "  Inprimis  ubi  generales  Naturae 
leges  investigantur,  inductio  vim  habet  maximam,  &  ad  earum  inventionem  vix  alia  ulla 
superest  via.  Ejus  ope  extensionem,  figurabilitem,  mobilitatem,  impenetrabilitatem 
corporibus  omnibus  tribuerunt  semper  Philosophi  etiam  veteres,  quibus  eodem  argumento 
inertiam,  &  generalem  gravitatem  plerique  e  recentioribus  addunt.  Inductio,  ut  demon- 
strationis  vim  habeat,  debet  omnes  singulares  casus,  quicunque  haberi  possunt  percurrere. 
Ea  in  Natu-[i8]-rae  legibus  stabiliendis  locum  habere  non  potest.  Habet  locum  laxior 
qusedam  inductio,  quae,  ut  adhiberi  possit,  debet  esse  ejusmodi,  ut  inprimis  in  omnibus  iis 
casibus,  qui  ad  trutinam  ita  revocari  possunt,  ut  deprehendi  debeat,  an  ea  lex  observetur, 
eadem  in  iis  omnibus  inveniatur,  &  ii  non  exiguo  numero  sint ;  in  reliquis  vero,  si  quse  prima 
fronte  contraria  videantur,  re  accuratius  perspecta,  cum  ilia  lege  possint  omnia  conciliari ; 
licet,  an  eo  potissimum  pacto  concilientur,  immediate  innotescere,  nequaquam  possit.  Si 
eae  conditiones  habeantur  ;  inductio  ad  legem  stabiliendam  censeri  debet  idonea.  Sic  quia 
videmus  corpora  tarn  multa,  quae  habemus  prae  manibus,  aliis  corporibus  resistere,  ne  in 
eorum  locum  adveniant,  &  loco  cedere,  si  resistendo  sint  imparia,  potius,  quam  eodem 
perstare  simul ;  impenetrabilitatem  corporum  admittimus  ;  nee  obest,  quod  qusedam 
corpora  videamus  intra  alia,  licet  durissima,  insinuari,  ut  oleum  in  marmora,  lumen  in 
crystalla,  &  gemmas.  Videmus  enim  hoc  phsenomenum  facile  conciliari  cum  ipsa  impene- 
trabilitate,  dicendo,  per  vacuos  corporum  poros  ea  corpora  permeare.  (Num.  135). 
Praeterea,  qusecunque  proprietates  absolutae,  nimirum  quae  relationem  non  habent  ad 
nostros  sensus,  deteguntur  generaliter  in  massis  sensibilibus  corporum,  easdem  ad  quascunque 
utcunque  exiguas  particulas  debemus  transferre  ;  nisi  positiva  aliqua  ratio  obstet,  &  nisi  sint 
ejusmodi,  quae  pendeant  a  ratione  totius,  seu  multitudinis,  contradistincta  a  ratione  partis. 
Primum  evincitur  ex  eo,  quod  magna,  &  parva  sunt  respectiva,  ac  insensibilia  dicuntur  ea, 
quse  respectu  nostrae  molis,  &  nostrorum  sensuum  sunt  exigua.  Quare  ubi  agitur  de 
proprietatibus  absolutis  non  respectivis,  quaecunque  communia  videmus  in  iis,  quse  intra 
limites  continentur  nobis  sensibiles,  ea  debemus  censere  communia  etiam  infra  eos  limites  : 
nam  ii  limites  respectu  rerum,  ut  sunt  in  se,  sunt  accidentales,  adeoque  siqua  fuisset  analogise 
Isesio,  poterat  ilia  multo  facilius  cadere  intra  limites  nobis  sensibiles,  qui  tanto  laxiores  sunt, 
quam  infra  eos,  adeo  nimirum  propinquos  nihilo.  Quod  nulla  ceciderit,  indicio  est,  nullam 
esse.  Id  indicium  non  est  evidens,  sed  ad  investigationis  principia  pertinet,  quae  si  juxta 


A  THEORY  OF  NATURAL  PHILOSOPHY  57 

bodies,  &  in  a  thousand  other  things  of  the  same  kind,  where  the  changes  of  velocity  occur 
gradually,  &  the  path  is  not  retraced  before  the  velocity  has  been  diminished  through  all 
degrees.  All  these  things  most  strictly  preserve  continuity.  Hence  it  follows  that  no 
sharp  angles  are  met  with  in  natural  motions,  but  in  every  case  a  change  of  direction  occurs 
gradually ;  neither  do  perfect  angles  occur  in  bodies  themselves,  for,  however  fine  an  edge 
or  point  in  them  may  seem,  one  can  usually  detect  curvature  by  the  help  of  the  microscope 
if  nothing  else.  We  have  this  gradual  change  of  direction  also  in  the  beds  of  rivers,  in  the 
leaves,  boughs  &  branches  of  trees,  &  stones  of  all  kinds  ;  unless,  in  some  cases  perchance, 
there  may  be  continuous  pointed  ends,  either  of  the  first  kind,  which  Nature  is  seen  to 
affect  in  thorns,  or  of  the  second  kind,  which  she  is  seen  to  do  in  the  claws  &  the  beak  of 
birds ;  in  these,  however,  we  shall  see  below  that  continuity  is  still  preserved,  since  we  are 
left  with  a  single  tangent  at  the  extreme  end.  It  would  take  far  too  long  to  mention  every 
single  thing  in  which  Nature  preserves  the  Law  of  Continuity  ;  it  is  more  than  sufficient 
to  make  a  general  statement  challenging  the  production  of  a  single  case  in  Nature,  in  which 
continuity  is  not  preserved  ;  for  it  is  absolutely  impossible  for  any  such  case  to  be  brought 
forward." 

40.  The  effect  of  the  very  complete  induction  from  such  motions  as  these  &  velocities,  induction  of  a  two- 
as  well  as  from  a  large  number  of  other  examples,  such  as  we  have  in  Nature,  where  Nature  *old ,  kil\d  '•    when 

e   c  ,...-..&  why   incomplete 

in  every  case,  as  far  as  can  be  gathered  from  direct  observation,  maintains  continuity  or  induction  has  vaii- 

tries  to  do  so,  should  certainly  be  that  of  keeping  us  from  neglecting  it  even  in  the  case 

of  collision  of  bodies.     As  regards  the  nature  &  validity  of  induction,  &  its  use  in  Physics, 

I  may  here  quote  part  of  Art.  134  &  the  wjiole  of  Art.  135  from  my  dissertation  De  Lege 

Continuitatis,     The  passage  runs  thus  :  "  Especially  when  we  investigate  the  general  laws 

of  Nature,  induction  has  very  great  power  ;   &  there  is  scarcely  any  other  method  beside 

it  for  the  discovery  of  these  laws.     By  its  assistance,  even  the  ancient  philosophers  attributed 

to  all  bodies  extension,  figurability,  mobility,  &  impenetrability ;    &  to  these  properties, 

by  the  use  of  the  same  method  of  reasoning,  most  of  the  later  philosophers  add  inertia  & 

universal  gravitation.     Now,  induction  should  take  account  of  every  single  case  that  can 

possibly  happen,  before  it  can  have  the  force  of  demonstration  ;  such  induction  as  this  has  no 

place  in  establishing  the  laws  of  Nature.     But  use  is  made  of  an  induction  of  a  less  rigorous 

type  ;   in  order  that  this  kind  of  induction  may  be  employed,  it  must  be  of  such  a  nature 

that  in  all  those  cases  particularly,  which  can  be  examined  in  a  manner  that  is  bound  to 

lead  to  a  definite  conclusion  as  to  whether  or  no  the  law  in  question  is  followed,  in  all  of 

them  the  same  result  is  arrived  at ;    &  that  these  cases  are  not  merely  a  few.     Moreover, 

in  the  other  cases,  if  those  which  at  first  sight  appeared  to  be  contradictory,  on  further  & 

more  accurate  investigation,  can  all  of  them  be  made  to  agree  with  the  law ;    although, 

whether  they  can  be  made  to  agree  in  this  way  better  than  in  any  other  whatever,  it  is 

impossible  to  know  directly  anyhow.     If  such  conditions  obtain,  then  it  must  be  considered 

that  the  induction  is  adapted  to  establishing  the  law.     Thus,  as  we  see  that  so  many  of 

the  bodies  around  us  try  to  prevent  other  bodies  from  occupying  the  position  which  they 

themselves  occupy,  or  give  way  to  them  if  they  are  not  capable  of  resisting  them,  rather 

than  that  both  should  occupy  the  same  place  at  the  same  time,  therefore  we  admit  the 

impenetrability  of  bodies.     Nor  is  there  anything  against  the  idea  in  the  fact  that  we  see 

certain  bodies  penetrating  into  the  innermost  parts  of  others,  although  the  latter  are  very 

hard  bodies ;   such  as  oil  into  marble,  &  light  into  crystals  &  gems.     For  we  see  that  this 

phenomenon  can  very  easily  be  reconciled  with  the  idea  of  impenetrability,  by  supposing 

that   the   former  bodies  enter  and  pass  through  empty  pores  in  the   latter    bodies  (Art. 

135).     In  addition,  whatever  absolute  properties,  for  instance  those  that  bear  no  relation 

to  our  senses,  are  generally  found  to  exist  in  sensible  masses  of  bodies,  we  are  bound  to 

attribute  these  same  properties  also  to  all  small  parts  whatsoever,  no  matter  how  small 

they  may  be.     That  is  to  say,  unless  some  positive  reason  prevents  this ;   such  as  that  they 

are  of  such  a  nature  that  they  depend  on  argument  having  to  do  with  a  body  as  a  whole, 

or  with  a  group  of  particles,  in  contradistinction  to  an  argument  dealing  with  a  part  only. 

The  proof  comes  in  the  first  place  from  the  fact  that  great  &  small  are  relative  terms,  & 

those  things  are  called  insensible  which  are  very  small  with  respect  to  our  own  size  &  with 

regard  to  our  senses.     Therefore,  when  we  consider  absolute,  &  not  relative,  properties, 

whatever  we  perceive  to  be  common  to  those  contained  within  the  limits  that  are  sensible 

to  us,  we  should  consider  these  things  to  be  still  common  to  those  beyond  those  limits. 

For  these  limits,  with  regard  to  such  matters  as  are  self-contained,  are  accidental ;  &  thus, 

if  there  should  be  any  violation  of  the  analogy,  this  would  be  far  more  likely  to  happen 

between  the  limits  sensible  to  us,  which  are  more  open,  than  beyond  them,  where  indeed 

they  are  so  nearly  nothing.     Because  then  none  did  happen  thus,  it  is  a  sign  that  there  is 

none.     This  sign  is  not  evident,  but  belongs  to  the  principles  of   investigation,  which 

generally  proves  successful  if  it  is  carried  out  in  accordance  with  certain  definite  wisely 


5  8  PHILOSOPHIC  NATURALIS  THEORIA 

quasdam  prudentes  regulas  fiat,  successum  habere  solet.  Cum  id  indicium  fallere  possit ; 
fieri  potest,  ut  committatur  error,  sed  contra  ipsum  errorem  habebitur  praesumptio,  ut 
etiam  in  jure  appellant,  donee  positiva  ratione  evincatur  oppositum.  Hinc  addendum  fuit, 
nisi  ratio  positiva  obstet.  Sic  contra  hasce  regulas  peccaret,  qui  diceret,  corpora  quidem 
magna  compenetrari,  ac  replicari,  &  inertia  carere  non  posse,  compenetrari  tamen  posse,  vel 
replicari,  vel  sine  inertia  esse  exiguas  eorum  partes.  At  si  proprietas  sit  respectiva,  respectu 
nostrorum  sensuum,  ex  [19]  eo,  quod  habeatur  in  majoribus  massis,  non  debemus  inferre, 
earn  haberi  in  particulis  minoribus,  ut  est  hoc  ipsum,  esse  sensibile,  ut  est,  esse  coloratas, 
quod  ipsis  majoribus  massis  competit,  minoribus  non  competit  ;  cum  ejusmodi  magnitudinis 
discrimen,  accidentale  respectu  materiae,  non  sit  accidentale  respectu  ejus  denominationis 
sensibile,  coloratum.  Sic  etiam  siqua  proprietas  ita  pendet  a  ratione  aggregati,  vel  totius,  ut 
ab  ea  separari  non  possit ;  nee  ea,  ob  rationem  nimirum  eandem,  a  toto,  vel  aggregate  debet 
transferri  ad  partes.  Est  de  ratione  totius,  ut  partes  habeat,  nee  totum  sine  partibus  haberi 
potest.  Est  de  ratione  figurabilis,  &  extensi,  ut  habeat  aliquid,  quod  ab  alio  distet,  adeoque, 
ut  habeat  partes ;  hinc  eae  proprietates,  licet  in  quovis  aggregate  particularum  materiae, 
sive  in  quavis  sensibili  massa,  inveniantur,  non  debent  inductionis  vi  transferri  ad  particulas 
quascunque." 


Et    impenetrabili-  41.  Ex  his  patet,  &  impenetrabilitatem,  &  continuitatis  legem  per  ejusmodi  inductionis 

ultatem  tvtad""pCT  genus  abunde  probari,  atque  evinci,  &  illam  quidem  ad  quascunque  utcunque   exiguas 

inductionem  :  "*  ad  particulas  corporum,  hanc  ad  gradus  utcunque  exiguos  momento  temporis  adjectos  debere 

ipsam  quid  requu-a-  exten(jj<     Requiritur  autem  ad  hujusmodi  inductionem  primo,  ut  ilia  proprietas,  ad  quam 

probandam  ea  adhibetur,  in  plurimis  casibus  observetur,  aliter  enim  probabilitas  esset  exigua  ; 

&  ut  nullus  sit  casus  observatus,  in  quo  evinci  possit,  earn  violari.     Non  est  necessarium  illud, 

ut  in  iis  casibus,  in  quibus  primo  aspectu  timeri  possit  defectus  proprietatis  ipsius,  positive 

demonstretur,  earn  non    deficere  ;    satis  est,  si  pro  iis    casibus    haberi  possit  ratio  aliqua 

conciliandi  observationem  cum  ipsa  proprietate,  &  id  multo  magis,  si  in  aliis  casibus  habeatur 

ejus   conciliationis   exemplum,  &  positive   ostendi   possit,   eo  ipso   modo  fieri   aliquando 

conciliationem. 


Ejus  appiicatio  ad  42.  Id  ipsum  fit,  ubi  per  inductionem  impenetrabilitas  corporum  accipitur  pro  generali 

impenetrab;htatem.  jege  ]sjaturaEi  Nam  impenetrabilitatem  ipsam  magnorum  corporum  observamus  in  exemplis 
sane  innumeris  tot  corporum,  quae  pertractamus.  Habentur  quidem  &  casus,  in  quibus  earn 
violari  quis  credent,  ut  ubi  oleum  per  ligna,  &  marmora  penetrat,  atque  insinuatur,  &  ubi 
lux  per  vitra,  &  gemmas  traducitur.  At  praesto  est  conciliatio  phasnomeni  cum  impenetra- 
bilitate,  petita  ab  eo,  quod  ilia  corpora,  in  quse  se  ejusmodi  substantiae  insinuant,  poros 
habeant,  quos  633  permeent.  Et  quidem  haec  conciliatio  exemplum  habet  manifestissimum 
in  spongia,  quae  per  poros  ingentes  aqua  immissa  imbuitur.  Poros  marmorum  illorum,  & 
multo  magis  vitrorum,  non  videmus,  ac  multo  minus  videre  possumus  illud,  non  insinuari 
eas  substantias  nisi  per  poros.  Hoc  satis  est  reliquae  inductionis  vi,  ut  dicere  debeamus,  eo 
potissimum  pacto  se  rem  habere,  &  ne  ibi  quidem  violari  generalem  utique  impenetrabilitatis 
legem. 

simiiisad  continu-       [2O]  43-  Eodem  igitur  pacto  in  lege  ipsa  continuitatis  agendum  est.     Ilia  tarn  ampla 

itatem  :    duo  cas-  inductio,  quam  habemus,  debet  nos  movere  ad  illam  generaliter  admittendam  etiam  pro  iis 

quibus  ean<videatur  casibus,  in  quibus  determinare  immediate  per  observations  non  possumus,   an  eadem 

lacdi-  habeatur,  uti  est  collisio  corporum  ;   ac  si  sunt  casus  nonnulli,  in  quibus  eadem  prima  fronte 

violari  videatur  ;  ineunda  est  ratio  aliqua,  qua  ipsum  phsenomenum  cum  ea  lege  conciliari 

possit,  uti  revera  potest.     Nonnullos  ejusmodi  casus  protuli  in  memoratis  dissertationibus, 

quorum  alii  ad  geometricam  continuitatem  pertinent,  alii  ad  physicam.     In  illis  prioribus 

non  immorabor  ;    neque  enim  geometrica  continuitas  necessaria  est  ad  hanc  physicam 

propugnandam,  sed  earn  ut  exemplum  quoddam  ad  confirmationem  quandam  inductionis 

majoris  adhibui.     Posterior,  ut  saepe  &  ilia  prior,  ad  duas  classes  reducitur  ;  altera  est  eorum 

casuum,  in  quibus  saltus  videtur  committi  idcirco,  quia  nos  per  saltum  omittimus  intermedias 

quantitates  :   rem  exemplo  geometrico  illustro,  cui  physicum  adjicio. 


A  THEORY  OF  NATURAL  PHILOSOPHY  59 

chosen  rules.  Now,  since  the  indication  may  possibly  be  fallacious,  it  may  happen  that  an 
error  may  be  made  ;  but  there  is  presumption  against  such  an  error,  as  they  call  it  in  law, 
until  direct  evidence  to  the  contrary  can  be  brought  forward.  Hence  we  should  add  : 
unless  some  positive  argument  is  against  it.  Thus,  it  would  be  offending  against  these  rules 
to  say  that  large  bodies  indeed  could  not  suffer  compenetration,  or  enfolding,  or  be  deficient 
in  inertia,  but  yet  very  small  parts  of  them  could  suffer  penetration,  or  enfolding,  or  be 
without  inertia.  On  the  other  hand,  if  a  property  is  relative  with  respect  to  our  senses, 
then,  from  a  result  obtained  for  the  larger  masses  we  cannot  infer  that  the  same  is  to  be 
obtained  in  its  smaller  particles  ;  for  instance,  that  it  is  the  same  thing  to  be  sensible,  as 
it  is  to  be  coloured,  which  is  true  in  the  case  of  large  masses,  but  not  in  the  case  of  small 
particles  ;  since  a  distinction  of  this  kind,  accidental  with  respect  to  matter,  is  not  accidental 
with  respect  to  the  term  sensible  or  coloured.  So  also  if  any  property  depends  on  an  argu- 
ment referring  to  an  aggregate,  or  a  whole,  in  such  a  way  that  it  cannot  be  considered 
apart  from  the  whole,  or  the  aggregate  ;  then,  neither  must  it  (that  is  to  say,  by  that  same 
argument),  be  transferred  from  the  whole,  or  the  aggregate,  to  parts  of  it.  It  is  on  account 
of  its  being  a  whole  that  it  has  parts  ;  nor  can  there  be  a  whole  without  parts.  It  is  on 
account  of  its  being  figurable  &  extended  that  it  has  some  thing  that  is  apart  from  some 
other  thing,  &  therefore  that  it  has  parts.  Hence  those  properties,  although  they  are 
found  in  any  aggregate  of  particles  of  matter,  or  in  any  sensible  mass,  must  not  however  be 
transferred  by  the  power  of  induction  to  each  &  every  particle." 

41.  From  what  has  been  said  it  is  quite  evident  that  both  impenetrability  &  the  Law  Both     impenetra- 
of  Continuity  can  be  proved  by  a  kind  of  induction  of  this  type  ;   &  the  former  must  be  c^bf  dTm""^ 
extended  to  all  particles  of  bodies,  no  matter  how  small,  &  the  latter  to  all  additional  steps,  strated  by   indue- 
however  small,  made  in  an  instant  of  time.     Now,  in  the  first  place,  to  use  this  kind  of  quired  for  thisSpur- 
induction,  it  is  required  that  the  property,  for  the  proof  of  which  it  is  to  be  used,  must  be  pose. 
observed  in  a  very  large  number  of  cases  ;    for  otherwise  the  probability  would  be  very 

small.  Also  it  is  required  that  no  case  should  be  observed,  in  which  it  can  be  proved  that 
it  is  violated.  It  is  not  necessary  that,  in  those  cases  in  which  at  first  sight  it  is  feared  that 
there  may  be  a  failure  of  the  property,  that  it  should  be  directly  proved  that  there  is  no 
failure.  It  is  sufficient  if  in  those  cases  some  reason  can  be  obtained  which  will  make  the 
observation  agree  with  the  property  ;  &  all  the  more  so,  if  in  other  cases  an  example  of 
reconciliation  can  be  obtained,  &  it  can  be  positively  proved  that  sometimes  reconciliation 
can  be  obtained  in  that  way. 

42.  This  is  just  what  does  happen,  when  the  impenetrability  of  solid  bodies  is  accepted  Application  of  in- 
as  a  law  of  Nature  through  inductive  reasoning.     For  we  observe  this  impenetrability  of  tr'abiuty.*0  lmpene" 
large  bodies  in  innumerable  examples  of  the  many  bodies  that  we  consider.     There  are 

indeed  also  cases,  in  which  one  would  think  that  it  was  violated,  such  as  when  oil  penetrates 
wood  and  marble,  &  works  its  way  through  them,  or  when  light  passes  through  glasses  & 
gems.  But  we  have  ready  a  means  of  making  these  phenomena  agree  with  impenetrability, 
derived  from  the  fact  that  those  bodies,  into  which  substances  of  this  kind  work  their  way, 
possess  pores  which  they  can  permeate.  There  is  a  very  evident  example  of  this  recon- 
ciliation in  a  sponge,  which  is  saturated  with  water  introduced  into  it  by  means  of  huge 
pores.  We  do  not  see  the  pores  of  the  marble,  still  less  those  of  glass  ;  &  far  less  can  we  see 
that  these  substances  do  not  penetrate  except  by  pores.  It  satisfies  the  general  force  of 
induction  if  we  can  say  that  the  matter  can  be  explained  in  this  way  better  than  in  any 
other,  &  that  in  this  case  there  is  absolutely  no  contradiction  of  the  general  law  of  impene- 
trability. 

43.  In  the  same  way,  then,  we  must  deal  with  the  Law  of  Continuity.     The  full  Similar  application 

" 


induction  that  we  possess  should  lead  us  to  admit  in  general  this  law  even  in  those  cases  in  ^sisses  "oT  cases 

which  it  is  impossible  for  us  to  determine  directly  by  observation  whether  the  same  law  which  there  seems 

holds  good,  as  for  instance  in  the  collision  of  bodies.     Also,  if  there  are  some  cases  in  which  * 

the  law  at  first  sight  seems  to  be  violated,  some  method  must  be  followed,  through  which 

each  phenomenon  can  be  reconciled  with  the  law,  as  is  in  every  case  possible.     I  brought 

forward  several  cases  of  this  kind  in  the  dissertations  I  have  mentioned,  some  of  which 

pertained  to  geometrical  continuity,  &  others  to  physical  continuity.     I  will  not  delay  over 

the  first  of  these  :  for  geometrical  continuity  is  not  necessary  for  the  defence  of  the  physical 

variety  ;   I  used  it  as  an  example  in  confirmation  of  a  wider  induction.     The  latter,  as  well 

as  very  frequently  the  former,  reduces  to  two  classes  ;  &  the  first  of  these  classes  is  that  class 

in  which  a  sudden  change  seems  to  have  been  made  on  account  of  our  having  omitted  the 

intermediate  quantities  with  a  jump.     I  give  a  geometrical  illustration,  and  then  add  one 

in  physics. 


6o 


PHILOSOPHISE  NATURALIS  THEORIA 


Exemplum  geome- 
tricum  primi  gene- 
ris, ubi  nos  inter- 
mcdias  magnitu- 
dines  omittimus. 


Quando  id  accidat 
exempla  physica 
dierum,  &  oscilla- 
tionum  consequen- 
tium. 


44.  In  axe  curvae  cujusdam  in  fig.  4.    sumantur  segmenta  AC,  CE,  EG  aequalia,  & 
erigantur  ordinatae  AB,  CD,  EF,  GH.     Area;  BACD,  DCEF,  FEGH  videntur  continue 
cujusdam  seriei  termini  ita,  ut  ab  ilia  BACD  acl  DCEF,  &   inde    ad   FEGH  immediate 
transeatur,    &   tamen  secunda  a    prima,  ut 

&  tertia  a  secunda,  differunt  per  quanti- 
tates  finitas  :  si  enim  capiantur  CI,  EK 
sequales  BA,  DC,  &  arcus  BD  transferatur 
in  IK  ;  area  DIKF  erit  incrementum  se- 
cundae  supra  primam,  quod  videtur  imme- 
diate advenire  totum  absque  eo,  quod 
unquam  habitum  sit  ejus  dimidium,  vel 
quaevis  alia  pars  incrementi  ipsius  ;  ut  idcirco 
a  prima  ad  secundam  magnitudinem  areae 
itum  sit  sine  transitu  per  intermedias.  At 
ibi  omittuntur  a  nobis  termini  intermedii, 

qui  continuitatem  servant ;  si  enim  ac  aequalis  FIG.  4. 

AC  motu  continue  feratur  ita,  ut  incipiendo 

ab  AC  desinat  in  CE  ;  magnitude  areae  BACD  per  omnes  intermedias  bacd  abit  in  magnitu- 
dinem DCEF  sine  ullo  saltu,  &  sine  ulla  violatione  continuitatis. 

45.  Id  sane  ubique  accidit,  ubi  initium  secundae  magnitudinis  aliquo  intervallo  distal 
ab  initio  primas ;  sive  statim  veniat  post  ejus  finem,  sive  qua  vis  alia  lege  ab  ea  disjungatur. 
Sic  in  pliysicis,  si  diem  concipiamus  intervallum  temporis  ab  occasu  ad  occasum,  vel  etiam 
ab  ortu  ad  occasum,  dies  praecedens  a  sequent!  quibusdam  anni  temporibus  differt  per  plura 
secunda,  ubi  videtur  fieri  saltus  sine  ullo  intermedio  die,  qui  minus  differat.     At  seriem 
quidem  continuam  ii  dies  nequaquam  constituunt.     Concipiatur  parallelus  integer  Telluris, 
in  quo  sunt  continuo  ductu  disposita  loca  omnia,  quae  eandem  latitudinem  geographicam 
habent ;    ea  singula  loca  suam  habent  durationem  diei,  &  omnium  ejusmodi  dierum  initia, 
ac  fines  continenter  fluunt ;   donee  ad  eundem  redeatur  locum,  cujus  pr£e-[2i]-cedens  dies 
est  in  continua  ilia  serie  primus,  &  sequens  postremus.     Illorum  omnium  dierum  magni- 
tudines  continenter  fluunt  sine  ullo  saltu  :    nos,  intermediis  omissis,  saltum  committimus 
non  Natura.     Atque  huic  similis  responsio  est  ad  omnes  reliquos  casus  ejusmodi,  in  quibus 
initia,  &  fines  continenter  non  fluunt,  sed  a  nobis  per  saltum  accipiuntur.     Sic  ubi  pendulum 
oscillat  in  acre  ;  sequens  oscillatio  per  finitam  magnitudinem  distat  a  praecedente  ;   sed  & 
initium  &  finis  ejus  finite  intervallo  temporis  distat  a  prascedentis  initio,  &  fine,  ac  intermedii 
termini  continua  serie  fluente  a  prima  oscillatione  ad  secundam  essent  ii,  qui  haberentur,  si 
primae,  &  secundae  oscillationis  arcu  in  aequalem  partium  numerum  diviso,  assumeretur  via 
confecta,  vel  tempus  in  ea  impensum,  inter jacens  inter  fines  partium  omnium  proportion- 
alium,  ut  inter  trientem,  vel  quadrantem  prioris  arcus,  &  trientem,vel  quadrantem  posterioris, 
quod  ad  omnes  ejus  generis  casus  facile  transferri  potest,  in  quibus  semper  immediate  etiam 
demonstrari  potest  illud,  continuitatem  nequaquam  violari. 


Exempla     secundi  46.  Secunda  classis  casuum  est  ea,  in  qua  videtur  aliquid  momento  temporis  peragi, 

atne«iOTimeUtSsed  &  tamen  peragitur  tempore  successive,  sed  perbrevi.  Sunt,  qui  objiciant  pro  violatione 
non  momento' tem-  continuitatis  casum,  quo  quisquam  manu  lapidem  tenens,  ipsi  statim  det  velocitatem 
quandam  finitam  :  alius  objicit  aquae  e  vase  effluentis,  foramine  constitute  aliquanto  infra 
superficiem  ipsius  aquae,  velocitatem  oriri  momento  temporis  finitam.  At  in  priore  casu 
admodum  evidens  est,  momento  temporis  velocitatem  finitam  nequaquam  produci.  Tempore 
opus  est,  utcunque  brevissimo,  ad  excursum  spirituum  per  nervos,  &  musculos,  ad  fibrarum 
tensionem,  &  alia  ejusmodi  :  ac  idcirco  ut  velocitatem  aliquam  sensibilem  demus  lapidi, 
manum  retrahimus,  &  ipsum  aliquandiu,  perpetuo  accelerantes,  retinemus.  Sic  etiam,  ubi 
tormentum  bellicum  exploditur,  videtur  momento  temporis  emitti  globus,  ac  totam 
celeritatem  acquirere  ;  at  id  successive  fieri,  patet  vel  inde,  quod  debeat  inflammari  tota 
massa  pulveris  pyrii,  &  dilatari  aer,  ut  elasticitate  sua  globum  acceleret,  quod  quidem  fit 
omnino  per  omnes  gradus.  Successionem  multo  etiam  melius  videmus  in  globe,  qui  ab 
elastro  sibi  relicto  propellatur  :  quo  elasticitas  est  major,  eo  citius,  sed  nunquam  momento 
temporis  velocitas  in  globum  inducitur. 


AppUcatio  ipsorum  47.  Hsec    exempla   illud   praestant,  quod   aqua    per   pores  spongiae  ingressa  respectu 

ad  emuxum1IaquK  impenetrabilitatis,  ut  ea  responsione  uti  possimus  in  aliis  casibus  omnibus,  in  quibus  accessio 
e  vase.  aliqua  magnitudinis  videtur  fieri  tota  momento  temporis ;  ut  nimirum  dicamus  fieri  tempore 


A  THEORY  OF  NATURAL  PHILOSOPHY  61 

44.  In  the  axis  of  any  curve  (Fig.  4)  let  there  be  taken  the  segments  AC,  CE,  EG  equal  Geometrical    ex- 
to  one  another  ;  &  let  the  ordinates  AB,  CD,  EF,  GH  be  erected.     The  areas  BACD,  DCEF,  kind    where6  Ivl 
FEGH  seem  to  be  terms  of  some  continuous  series  such  that  we  can  pass  directly  from  BACD  omit .  intermediate 
to  DCEF  and  then  on  to  FEGH,  &  yet  the  second  differs  from  the  first,  &  also  the  third  from 

the  second,  by  a  finite  quantity.  For  if  CI,  EK  are  taken  equal  to  BA,  DC,  &  the  arc  BD 
is  transferred  to  the  position  IK  ;  then  the  area  DIKF  will  be  the  increment  of  the  second 
area  beyond  the  first ;  &  this  seems  to  be  directly  arrived  at  as  a  whole  without  that  which 
at  any  one  time  is  considered  to  be  the  half  of  it,  or  indeed  any  other  part  of  the  increment 
itself  :  so  that,  in  consequence,  we  go  from  the  first  to  the  second  magnitude  of  area  without 
passing  through  intermediate  magnitudes.  But  in  this  case  we  omit  intermediate  terms 
which  maintain  the  continuity ;  for  if  ac  is  equal  to  AC,  &  this  is  carried  by  a  continuous 
motion  in  such  a  way  that,  starting  from  the  position  AC  it  ends  up  at  the  position  CE, 
then  the  magnitude  of  the  area  BACD  will  pass  through  all  intermediate  values  such  as 
bacd  until  it  reaches  the  magnitude  of  the  area  DCEF  without  any  sudden  change,  &  hence 
without  any  breach  of  continuity. 

45.  Indeed  this  always  happens  when  the  beginning  of  the  second  magnitude  is  distant  when    this  will 
by  a  definite  interval  from  the  beginning  of  the  first ;   whether  it  comes  immediately  after  happen  =     physical 
the  end  of  the  first  or  is  disconnected  from  it  by  some  other  law.     Thus  in  physics,  if  we  casTof Consecutive 
look  upon  the  day  as  the  interval  of  time  between  sunset  &  sunset,  or  even  between  sunrise  day^  OI.  consecutive 
&  sunset,  the  preceding  day  differs  from  that  which  follows  it  at  certain  times  of  the  year 

by  several  seconds ;  in  which  case  we  see  that  there  is  a  sudden  change  made,  without  there 
being  any  intermediate  day  for  which  the  change  is  less.  But  the  fact  is  that  these  days  do 
not  constitute  a  continuous  series.  Let  us  consider  a  complete  parallel  of  latitude  on  the 
Earth,  along  which  in  a  continuous  sequence  are  situated  all  those  places  that  have  the  same 
geographical  latitude.  Each  of  these  places  has  its  own  duration  of  the  day,  &  the  begin- 
nings &  ends  of  days  of  this  kind  change  uninterruptedly  ;  until  we  get  back  again  to  the 
same  place,  where  the  preceding  day  is  the  first  of  that  continuous  series,  &  the  day  that  fol- 
lows is  the  last  of  the  series.  The  magnitudes  of  all  these  days  continuously  alter  without  there 
being  any  sudden  change  :  it  was  we  who,  by  omitting  the  intermediates,  made  the  sudden 
change,  &  not  Nature.  Similar  to  this  is  the  answer  to  all  the  rest  of  the  cases  of  the  same 
kind,  in  which  the  beginnings  &  the  ends  do  not  change  uninterruptedly,  but  are  observed  by 
us  discontinuously.  Similarly,  when  a  pendulum  oscillates  in  air,  the  oscillation  that  follows 
differs  from  the  oscillation  that  has  gone  before  by  a  finite  magnitude.  But  both  the  begin- 
ning &  the  end  of  the  second  differs  from  the  beginning  &  the  end  of  the  first  by  a  finite  inter- 
val of  time  ;  &  the  intermediate  terms  in  a  continuously  varying  series  from  the  first  oscillation 
to  the  second  would  be  those  that  would  be  obtained,  if  the  arcs  of  the  first  &  second  oscilla- 
tions were  each  divided  into  the  same  number  of  equal  parts,  &  the  path  traversed  (or  the 
time  spent  in  traversing  the  path)  is  taken  between  the  ends  of  all  these  proportional  paths ; 
such  as  that  between  the  third  or  fourth  part  of  the  first  arc  &  the  third  or  fourth  part 
of  the  second  arc.  This  argument  can  be  easily  transferred  so  as  to  apply  to  all  cases  of  this 
kind  ;  &  in  such  cases  it  can  always  be  directly  proved  that  there  is  no  breach  of  continuity. 

46.  The  second  class  of  cases  is  that  in  which  something  seems  to  have  been  done  in  an  Examples   of   the 
instant  of  time,  but  still  it  is  really  done  in  a  continuous,  but  very  short,  interval  of  time.  ^iS?  the^chan'e 
There  are  some  who  bring  forward,  as  an  objection  in  favour  of  a  breach  of  continuity,  the  is  very  rapid,  but 
case  in  which  a  man,  holding  a  stone  in  his  hand,  gives  to  it  a  definite  velocity  all  at  once  ;  f^an^nstant^of 
another  raises  an  objection  that  favours  a  breach  of  continuity,  in  the  case  of  water  flowing  time. 

from  a  vessel,  where,  if  an  opening  is  made  below  the  level  of  the  surface  of  the  water,  a 
finite  velocity  is  produced  in  an  instant  of  time.  But  in  the  first  case  it  is  perfectly  clear 
that  a  finite  velocity  is  in  no  wise  produced  in  an  instant  of  time.  For  there  is  need  of 
time,  although  this  is  exceedingly  short,  for  the  passage  of  cerebral  impulses  through 
the  nerves  and  muscles,  for  the  tension  of  the  fibres,  and  other  things  of  that  sort ;  and 
therefore,  in  order  to  give  a  definite  sensible  velocity  to  the  stone,  we  draw  back  the  hand, 
and  then  retain  the  stone  in  it  for  some  time  as  we  continually  increase  its  velocity  forwards. 
So  too  when  an  engine  of  war  is  exploded,  the  ball  seems  to  be  driven  forth  and  to  acquire 
the  whole  of  its  speed  in  an  instant  of  time.  But  that  it  is  done  continuously  is  clear,  if 
only  from  the  fact  that  the  whole  mass  of  the  gunpowder  has  to  be  inflamed  and  the  gas 
has  to  be  expanded  in  order  that  it  may  accelerate  the  ball  by  its  elasticity  ;  and  this  latter 
certainly  takes  place  by  degrees.  The  continuous  nature  of  this  is  far  better  seen  in  the 
case  of  a  ball  propelled  by  releasing  a  spring  ;  here  the  stronger  the  elasticity,  the  greater 
the  speed  ;  but  in  no  case  is  the  speed  imparted  to  the  ball  in  an  instant  of  time. 

47.  These  examples  are  superior  to  that  of  water  entering  through  the  pores  of  a  sponge,  Application  of 
which  we  employed  in  the  matter  of  impenetrability  ;  so  that  we  can  make  use  of  this  reply  *£.gss.e particularly 
in  all  other  cases  in  which  some  addition  to  a  magnitude  seems  to  have  taken  place  entirely  in  to  the  flow  of  water 
an  instant  of  time.     Thus,  without  doubt  we  may  say  that  it  takes  place  in  an  exceedingly  from  a  vesse1' 


62 


PHILOSOPHIC  NATURALIS  THEORIA 


brevissimo,  utique  per  omnes  intermedias  magnitudines,  ac  illsesa  penitus  lege  continuitatis. 
Hinc  &  in  aquae  effluentis  exemplo  res  eodem  redit,  ut  non  unico  momento,  sed  successive 
aliquo  tempore,  &  per  [22]  omnes  intermedias  magnitudines  progignatur  velocitas,  quod 
quidem  ita  se  habere  optimi  quique  Physici  affirmant.  Et  ibi  quidem,  qui  momento 
temporis  omnem  illam  velocitatem  progigni,  contra  me  affirmet,  principium  utique,  ut 
ajunt,  petat,  necesse  est.  Neque  enim  aqua,  nisi  foramen  aperiatur,  operculo  dimoto, 
effluet ;  remotio  vero  operculi,  sive  manu  fiat,  sive  percussione  aliqua,  non  potest  fieri 
momento  temporis,  sed  debet  velocitatem  suam  acquirere  per  omnes  gradus ;  nisi  illud 
ipsum,  quod  quaerimus,  supponatur  jam  definitum,  nimirum  an  in  collisione  corporum 
communicatio  motus  fiat  momento  temporis,  an  per  omnes  intermedios  gradus,  &  magni- 
tudines. Verum  eo  omisso,  si  etiam  concipiamus  momento  temporis  impedimentum 
auferri,  non  idcirco  momento  itidem  temporis  omnis  ilia  velocitas  produceretur  ;  ilia  enim 
non  a  percussione  aliqua,  sed  a  pressione  superincumbentis  aquae  orta,  oriri  utique  non 
potest,  nisi  per  accessiones  continuas  tempusculo  admodum  parvo,  sed  non  omnino  nullo  : 
nam  pressio  tempore  indiget,  ut  velocitatem  progignat,  in  communi  omnium  sententia. 


Transitus  ad  meta- 


continuis 

ut  in  Geometria. 


48.  Illaesa  igitur  esse  debet  continuitatis  lex,  nee  ad  earn  evertendam  contra  inductionem, 
tam  uberem  quidquam  poterunt  casus  allati  hucusque,  vel  iis  similes.  At  ejusdem  con- 
umcus,  tinuitatis  aliam  metaphysicam  rationem  adinveni,  &  proposui  in  dissertatione  De  Lege 
Continuitatis,  petitam  ab  ipsa  continuitatis  natura,  in  qua  quod  Aristoteles  ipse  olim 
notaverat,  communis  esse  debet  limes,  qui  praecedentia  cum  consequentibus  conjungit,  qui 
idcirco  etiam  indivisibilis  est  in  ea  ratione,  in  qua  est  limes.  Sic  superficies  duo  solida 
dirimens  &  crassitudine  caret,  &  est  unica,  in  qua  immediatus  ab  una  parte  fit  transitus  ad 
aliam  ;  linea  dirimens  binas  superficiei  continuae  partes  latitudine  caret ;  punctum  continuae 
lineae  segmenta  discriminans,  dimensione  omni  :  nee  duo  sunt  puncta  contigua,  quorum 
alterum  sit  finis  prioris  segmenti,  alterum  initium  sequentis,  cum  duo  contigua  indivisibilia, 
&  inextensa  haberi  non  possint  sine  compenetratione,  &  coalescentia  quadam  in  unum. 


idem   in   tempore  49.  Eodem  autem  pacto  idem  debet  accidere  etiam  in  tempore,  ut  nimirum  inter  tempus 

«>  ti  ua^'evide"6  contmuum  praecedens,  &  continuo  subsequens  unicum  habeatur  momentum,  quod  sit 
tius  in  quibusdam.  indivisibilis  terminus  utriusque  ;  nee  duo  momenta,  uti  supra  innuimus,  contigua  esse 
possint,  sed  inter  quodvis  momentum,  &  aliud  momentum  debeat  intercedere  semper 
continuum  aliquod  tempus  divisibile  in  infinitum.  Et  eodem  pacto  in  quavis  quantitate, 
quae  continuo  tempore  duret,  haberi  debet  series  quasdam  magnitudinum  ejusmodi,  ut 
momento  temporis  cuivis  respondeat  sua,  quae  praecedentem  cum  consequente  conjungat, 
&  ab  ilia  per  aliquam  determinatam  magnitudinem  differat.  Quin  immo  in  illo  quantitatum 
genere,  in  quo  [23]  binae  magnitudines  simul  haberi  non  possunt,  id  ipsum  multo  evidentius 
conficitur,  nempe  nullum  haberi  posse  saltum  immediatum  ab  una  ad  alteram.  Nam  illo 
momento  temporis,  quo  deberet  saltus  fieri,  &  abrumpi  series  accessu  aliquo  momentaneo, 
deberent  haberi  duae  magnitudines,  postrema  seriei  praecedentis,  &  prima  seriei  sequentis. 
Id  ipsum  vero  adhuc  multo  evidentius  habetur  in  illis  rerum  statibus,  in  quibus  ex  una 
parte  quovis  momento  haberi  debet  aliquis  status  ita,  ut  nunquam  sine  aliquo  ejus  generis 
statu  res  esse  possit ;  &  ex  alia  duos  simul  ejusmodi  status  habere  non  potest. 


inde  cur  motus  ip-  ro>  \&  quidem  satis  patebit  in  ipso  locali  motu,  in  quo  habetur  phsenomenum  omnibus 

calls   non  fiat,    nisi  •>       .     .    *  ,        .  r  r.       ......  \  ,.  .     ,  .     . 

per  Hneam  contin-  sane  notissimum,  sed  cujus  ratio  non  ita  facile  ahunde  redditur,  inde  autem  patentissima  est, 
Corpus  a  quovis  loco  ad  alium  quemvis  devenire  utique  potest  motu  continuo  per  lineas 
quascunque  utcunque  contortas,  &  in  immensum  productas  quaquaversum,  quae  numero 
infinities  infinitae  sunt  :  sed  omnino  debet  per  continuam  aliquam  abire,  &  nullibi  inter- 
ruptam.  En  inde  rationem  ejus  rei  admodum  manifestam.  Si  alicubi  linea  motus  abrum- 
peretur  ;  vel  momentum  temporis,  quo  esset  in  primo  puncto  posterioris  lineae,  esset 
posterius  eo  momento,  quo  esset  in  puncto  postremo  anterioris,  vel  esset  idem,  vel  anterius  ? 
In  primo,  &  tertio  casu  inter  ea  momenta  intercederet  tempus  aliquod  continuum  divisibile 
in  infinitum  per  alia  momenta  intermedia,  cum  bina  momenta  temporis,  in  eo  sensu  accepta, 
in  quo  ego  hie  ea  accipio,  contigua  esse  non  possint,  uti  superiusexposui.  Quamobrem  in 


A  THEORY  OF  NATURAL  PHILOSOPHY  63 

short  interval  of  time,  and  certainly  passes  through  every  intermediate  magnitude,  and  that 
the  Law  of  Continuity  is  not  violated.  Hence  also  in  the  case  of  water  flowing  from  a 
vessel  it  reduces  to  the  same  example  :  so  that  the  velocity  is  generated,  not  in  a  single 
instant,  but  in  some  continuous  interval  of  time,  and  passes  through  all  intermediate  magni- 
tudes ;  and  indeed  all  the  most  noted  physicists  assert  that  this  is  what  really  happens. 
Also  in  this  matter,  should  anyone  assert  in  opposition  to  me  that  the  whole  of  the  speed 
is  produced  in  an  instant  of  time,  then  he  must  use  a  •petitio  principii,  as  they  call  it.  For 
the  water  can-not  flow  out,  unless  the  hole  is  opened,  &  the  lid  removed  ;  &  the  removal  of 
the  lid,  whether  done  by  hand  or  by  a  blow,  cannot  be  effected  in  an  instant  of  time,  but 
must  acquire  its  own  velocity  by  degrees ;  unless  we  suppose  that  the  matter  under  investi- 
gation is  already  decided,  that  is  to  say,  whether  in  collision  of  bodies  communication  of 
motion  takes  place  in  an  instant  of  time  or  through  all  intermediate  degrees  and  magnitudes. 
But  even  if  that  is  left  out  of  account,  &  if  also  we  assume  that  the  barrier  is  removed 
in  an  instant  of  time,  none  the  more  on  that  account  would  the  whole  of  the  velocity 
also  be  produced  in  an  instant  of  time  ;  for  it  is  impossible  that  such  velocity  can  arise, 
not  from  some  blow,  but  from  a  pressure  arising  from  the  superincumbent  water,  except  by 
continuous  additions  in  a  very  short  interval  of  time,  which  is  however  not  absolutely 
nothing  ;  for  pressure  requires  time  to  produce  velocity,  according  to  the  general  opinion 
of  everybody. 

48.  The  Law  of  Continuity  ought  then  to  be  subject  to  no  breach,  nor  will  the  cases  Passing  to  a  meta- 
hitherto  brought  forward,  nor  others  like  them,  have  any  power  at  all  to  controvert  this  haveT'smrie'iinUt 
law  in  opposition  to  induction  so  copious.     Moreover  I  discovered  another  argument,  a  in  the  case  of  con- 
metaphysical  one,  in  favour  of  this  continuity,  &  published  it  in  my  dissertation  De  Lege  g'^n^iy1"11^' &S  "* 
Continuitatis,  having  derived  it  from  the  very  nature  of  continuity  ;  as  Aristotle  himself  long 

ago  remarked,  there  must  be  a  common  boundary  which  joins  the  things  that  precede  to 
those  that  follow ;  &  this  must  therefore  be  indivisible  for  the  very  reason  that  it  is  a 
boundary.  In  the  same  way,  a  surface  of  separation  of  two  solids  is  also  without  thickness 
&  is  single,  &  in  it  there  is  immediate  passage  from  one  side  to  the  other  ;  the  line  of 
separation  of  two  parts  of  a  continuous  surface  lacks  any  breadth  ;  a  point  determining 
segments  of  a  continuous  line  has  no  dimension  at  all ;  nor  are  there  two  contiguous  points, 
one  of  which  is  the  end  of  the  first  segment,  &  the  other  the  beginning  of  the  next ;  for 
two  contiguous  indivisibles,  of  no  extent,  cannot  possibly  be  considered  to  exist,  unless 
there  is  compenetration  &  a  coalescence  into  one. 

49.  In  the  same  way,  this  should  also  happen  with  regard  to  time,  namely,  that  between  similarly  for  time 
a  preceding  continuous  time  &  the  next  following  there  should  be  a  single  instant,  which  ^£y.    mor^evi- 
is  the  indivisible  boundary  of  either.     There  cannot  be  two  instants,  as  we  intimated  above,  dent  in  some  than 
contiguous  to  one  another  ;  but  between  one  instant  &  another  there  must  always  intervene  m  others- 

some  interval  of  continuous  time  divisible  indefinitely.  In  the  same  way,  in  any  quantity 
which  lasts  for  a  continuous  interval  of  time,  there  must  be  obtained  a  series  of  magnitudes 
of  such  a  kind  that  to  each  instant  of  time  there  is  its  corresponding  magnitude  ;  &  this 
magnitude  connects  the  one  that  precedes  with  the  one  that  follows  it,  &  differs  from  the 
former  by  some  definite  magnitude.  Nay  even  in  that  class  of  quantities,  in  which  we 
cannot  have  two  magnitudes  at  the  same  time,  this  very  point  can  be  deduced  far  more 
clearly,  namely,  that  there  cannot  be  any  sudden  change  from  one  to  another.  For  at  that 
instant,  when  the  sudden  change  should  take  place,  &  the  series  be  broken  by  some  momen- 
tary definite  addition,  two  magnitudes  would  necessarily  be  obtained,  namely,  the  last  of 
the  first  series  &  the  first  of  the  next.  Now  this  very  point  is  still  more  clearly  seen  in  those 
states  of  things,  in  which  on  the  one  hand  there  must  be  at  any  instant  some  state  so  that 
at  no  time  can  the  thing  be  without  some  state  of  the  kind,  whilst  on  the  other  hand  it  can 
never  have  two  states  of  the  kind  simultaneously. 

50.  The  above  will  be  sufficiently  clear  in  the  case  of  local  motion,  in  regard  to  which  Hence  the  reason 
the  phenomenon  is  perfectly  well  known  to  all ;   the  reason  for  it,  however,  is  not  so  easily  ^Jj^  Recurs™;:!10" 
derived  from  any  other  source,  whilst  it  follows  most  clearly  from  this  idea.     A  body  can  continuous  line, 
get  from  any  one  position  to  any  other  position  in  any  case  by  a  continuous  motion  along 

any  line  whatever,  no  matter  how  contorted,  or  produced  ever  so  far  in  any  direction  ; 
these  lines  being  infinitely  infinite  in  number.  But  it  is  bound  to  travel  by  some  continuous 
line,  with  no  break  in  it  at  any  point.  Here  then  is  the  reason  of  this  phenomenon  quite 
clearly  explained.  If  the  motion  in  the  line  should  be  broken  at  any  point,  either  the 
instant  of  time,  at  which  it  was  at  the  first  point  of  the  second  part  of  the  line,  would  be 
after  the  instant,  at  which  it  was  at  the  last  point  of  the  first  part  of  the  line,  or  it  would 
be  the  same  instant,  or  before  it.  In  the  first  &  third  cases,  there  would  intervene  between 
the  two  instants  some  definite  interval  of  continuous  time  divisible  indefinitely  at  other 
intermediate  instants ;  for  two  instants  of  time,  considered  in  the  sense  in  which  I  have 


PHILOSOPHIC   NATURALIS  THEORIA 


primo  casu  in  omnibus  iis  infinitis  intermediis  momentis  nullibi  esset  id  corpus,  in  secundo 
casu  idem  esset  eodem  illo  memento  in  binis  locis,  adeoque  replicaretur  ;  in  terio  haberetur 
replicatio  non  tantum  respectu  eorum  binorum  momentorum,  sed  omnium  etiam  inter- 
mediorum,  in  quibus  nimirum  omnibus  id  corpus  esset  in  binis  locis.  Cum  igitur  corpus 
existens  nee  nullibi  esse  possit,  nee  simul  in  locis  pluribus ;  ilia  vias  mutatio,  &  ille  saltus 
haberi  omnino  non  possunt. 

51.  Idem  ope  Geometric  magis  adhuc  oculis  ipsis  subjicitur.     Exponantur  per  rectam 
AB  tempora,  ac  per  ordinatas  ad  lineas  CD,  EF,  abruptas  alicubi,  diversi  status  rei  cujuspiam. 
e  metaphysica,  Ductis  ordinatis  DG,  EH,  vel  punctum  H  iaceret  post  G,  ut  in  Fie.  c  :    vel    cum    ipso 

ibus  exemphs  .     •       /•  i    •  ij  .     r  T  .  o     J  •  r 

congrueret,  ut  in  6  ;  vel  ipsum  prsccederet,  ut  in  7.  In  pnmo  casu  nulla  responderet 
ordinata  omnibus  punctis  rectae  GH  ;  in  secundo  binae  responderent  GD,  &  HE  eidem  puncto 
G ;  in  tertio  vero  binae  HI,  &  HE  puncto  H,  binas  GD,  GK  puncto  G,  &  binae  LM,  LN 


Illustratio  ejus 
i  ex  Geo- 
ratiocina- 

tione 

pluribus  exempl 


D    E. 


D 


G  H 

FIG.  5. 


B  A 


GH 


FIG.  6. 


H    L    G 


FIG.  7. 


puncto  cuivis  intermedio  L  ;  nam  ordinata  est  relatio  quaedam  distantly,  quam  habet 
punctum  curvae  cum  puncto  axis  sibi  respondente,  adeoque  ubi  jacent  in  recta  eadem 
perpendiculari  axi  bina  curvarum  puncta,  habentur  binae  ordinatae  respondentes  eidem 
puncto  axis.  Quamobrem  si  nee  o-[24]-mni  statu  carere  res  possit,  nee  haberi  possint 
status  simul  bini ;  necessario  consequitur,  saltum  ilium  committi  non  posse.  Saltus  ipse,  si 
deberet  accidere,  uti  vulgo  fieri  concipitur,  accideret  binis  momentis  G,  &  H,  quae  sibi  in 
fig.  6  immediate  succederent  sine  ullo  immediato  hiatu,  quod  utique  fieri  non  potest  ex 
ipsa  limitis  ratione,  qui  in  continuis  debet  esse  idem,  &  antecedentibus,  &  consequentibus 
communis,  uti  diximus.  Atque  idem  in  quavis  reali  serie  accidit ;  ut  hie  linea  finita  sine 
puncto  primo,  &  postremo,  quod  sit  ejus  limes,  &  superficies  sine  linea  esse  non  potest ;  unde 
fit,  ut  in  casu  figurae  6  binae  ordinatae  necessario  respondere  debeant  eidem  puncto  :  ita  in 
quavis  finita  reali  serie  statuum  primus  terminus,  &  postremus  haberi  necessario  debent ; 
adeoque  si  saltus  fit,  uti  supra  de  loco  diximus ;  debet  eo  momento,  quo  saltus  confici 
dicitur,  haberi  simul  status  duplex  ;  qui  cum  haberi  non  possit  :  saltus  itidem  ille  haberi 
omnino  non  potest.  Sic,  ut  aliis  utamur  exemplis,  distantia  unius  corporis  ab  alio  mutari 
per  saltum  non  potest,  nee  densitas,  quia  dux  simul  haberentur  distantiae,  vel  duae  densitates, 
quod  utique  sine  replicatione  haberi  non  potest ;  caloris  itidem,  &  frigoris  mutatio  in 
thermometris,  ponderis  atmosphaerae  mutatio  in  barometris,  non  fit  per  saltum,  quia  binae 
simul  altitudines  mercurii  in  instrumento  haberi  deberent  eodem  momento  temporis,  quod 
fieri  utique  non  potest ;  cum  quovis  momento  determinate  unica  altitude  haberi  debeat, 
ac  unicus  determinatus  caloris  gradus,  vel  frigoris  ;  quae  quidem  theoria  innumeris  casibus 
pariter  aptari  potest. 

52.  Contra  hoc  argumentum  videtur  primo  aspectu  adesse  aliquid,  quod  ipsum  pforsus 
non   esse    conjun-  evertat,  &   tamen    ipsi    illustrando    idoneum  est  maxime.     Videtur  nimirum  inde  erui, 

gend  s  in  creatione     •  •«  M  •  •          •         •  o    •  •  £»•          •  •  J 

&  annihiiatione,  ac  impossibilem  esse  &  creationem  rei  cujuspiam,  Scintentum.     01  enim  conjungendus  est 
ejus  soiutio.  postremus  terminus  praecedentis  seriei  cum  primo  sequentis ;"  in  ipso  transitu  a  non  esse  ad 

esse,  vel  vice  versa,  debebit  utrumque  conjungi,  ac  idem  simul  erit,  &  non  erit,  quod  est 
absurdum.  Responsio  in  promptu  est.  Seriei  finita;  realis,  &  existentis,  reales  itidem,  & 
existentes  termini  esse  debent ;  non  vero  nihili,  quod  nullas  proprietates  habet,  quas  exigat, 
Hinc  si  realium  statuum  seriei  altera  series  realium  itidem  statuum  succedat,  quae  non 
sit  communi  termino  conjuncta  ;  bini  eodem  momento  debebuntur  status,  qui  nimirum 
sint  bini  limites  earundem.  At  quoniam  non  esse  est  merum  nihilum ;  ejusmodi  series 
limitem  nullum  extremum  requirit,  sed  per  ipsum  esse  immediate,  &  directe  excluditur. 
Quamobrem  primo,  &  postremo  momento  temporis  ejus  continui,  quo  res  est,  erit  utique, 
nee  cum  hoc  esse  suum  non  esse  conjunget  simul ;  at  si  densitas  certa  per  horam  duret,  turn 
momento  temporis  in  aliam  mutetur  duplam,  duraturam  itidem  per  alteram  sequentem 
horam  ;  momento  temporis,  [25]  quod  horas  dirimit,  binae  debebunt  esse  densitates  simul, 
nimirum  &  simplex,  &  dupla,  quae  sunt  reales  binarum  realium  serierum  termini. 


Objectio  ab  esse,  & 


A  THEORY  OF  NATURAL  PHILOSOPHY  65 

considered  them,  cannot  be  contiguous,  as  I  explained  above.  Wherefore  in  the  first  case, 
at  all  those  infinite  intermediate  instants  the  body  would  be  nowhere  at  all  ;  in  the  second 
case,  it  would  be  at  the  same  instant  in  two  different  places  &  so  there  would  be  replication. 
In  the  third  case,  there  would  not  only  occur  replication  in  respect  of  these  two  instants 
but  for  all  those  intermediate  to  them  as  well,  in  all  of  which  the  body  would  forsooth  be 
in  two  places  at  the  same  time.  Since  then  a  body  that  exists  can  never  be  nowhere,  nor 
in  several  places  at  one  &  the  same  time,  there  can  certainly  be  no  alteration  of  path  &  no 
sudden  change. 

51.  The  same  thing  can  be  visualized  better  with  the  aid  of  Geometry.  illustration  of  this 

Let  times  be  represented  by  the  straight  line  AB,  &  diverse  states  of  any  thing  by  SSyT^STS 
ordinates  drawn  to  meet  the  lines  CD,  EF,  which  are  discontinuous  at  some  point.  If  the  reasoning  being 
ordmates  DG,  EH  are  drawn,  either  the  point  H  will  fall  after  the  point  G,  as  in  Fig.  5  ; 
or  it  will  coincide  with  it,  as  in  Fig.  6  ;  or  it  will  fall  before  it,  as  in  Fig.  7.  In  the  first 
case,  no  ordinate  will  correspond  to  any  one  of  the  points  of  the  straight  line  GH  ;  in  the 
second  case,  GD  and  HE  would  correspond  to  the  same  point  G  ;  in  the  third  case,  two 
ordinates,  HI,  HE,  would  correspond  to  the  same  point  H,  two,  GD,  GK,  to  the  same 
point  G,  and  two,  LM,  LN,  to  any  intermediate  point  L.  Now  the  ordinate  is  some  relation 
as  regards  distance,  which  a  point  on  the  curve  bears  to  the  point  on  the  axis  that  corresponds 
with  it  ;  &  thus,  when  two  points  of  the  curve  lie  in  the  same  straight  line  perpendicular 
to  the  axis,  we  have  two  ordinates  corresponding  to  the  same  point  of  the  axis.  Wherefore, 
if  the  thing  in  question  can  neither  be  without  some  state  at  each  instant,  nor  is  it  possible 
that  there  should  be  two  states  at  the  same  time,  then  it  necessarily  follows  that  the  sudden 
change  cannot  be  made.  For  this  sudden  change,  if  it  is  bound  to  happen,  would  take  place 
at  the  two  instants  G  &  H,  which  immediately  succeed  the  one  the  other  without  any  direct 
gap  between  them  ;  this  is  quite  impossible,  from  the  very  nature  of  a  limit,  which  should 
be  the  same  for,&  common  to,  both  the  antecedents  &  the  consequents  in  a  continuous  set, 
as  has  been  said.  The  same  thing  happens  in  any  series  of  real  things  ;  as  in  this  case  there 
cannot  be  a  finite  line  without  a  first  &  last  point,  each  to  be  a  boundary  to  it,  neither  can 
there  be  a  surface  without  a  line.  Hence  it  comes  about  that  in  the  case  of  Fig.  6  two 
ordinates  must  necessarily  correspond  to  the  same  point.  Thus,  in  any  finite  real  series  of 
states,  there  must  of  necessity  be  a  first  term  &  a  last  ;  &  so  if  a  sudden  change  is  made,  as 
we  said  above  with  regard  to  position,  there  must  be  at  the  instant,  at  which  the  sudden 
change  is  said  to  be  accomplished,  a  twofold  state  at  one  &  the  same  time.  Now  since  this 
can  never  happen,  it  follows  that  this  sudden  change  is  also  quite  impossible.  Similarly,  to 
make  use  of  other  illustrations,  the  distance  of  one  body  from  another  can  never  be  altered 
suddenly,  no  more  can  its  density  ;  for  there  would  be  at  one  &  the  same  time  two  distances, 
or  two  densities,  a  thing  which  is  quite  impossible  without  replication.  Again,  the  change 
of  heat,  or  cold,  in  thermometers,  the  change  in  the  weight  of  the  air  in  barometers,  does 
not  happen  suddenly  ;  for  then  there  would  necessarily  be  at  one  &  the  same  time  two 
different  heights  for  the  mercury  in  the  instrument  ;  &  this  could  not  possibly  be  the  case. 
For  at  any  given  instant  there  must  be  but  one  height,  &  but  one  definite  degree  of  heat, 
&  but  one  definite  degree  of  cold  ;  &  this  argument  can  be  applied  just  as  well  to  innu- 
merable other  cases. 


52.  Against  this  argument  it  would  seem  at  first  sight  that  there  is  something  ready  to 
hand  which  overthrows  it  altogether  ;    whilst  as  a  matter  of  fact  it  is  peculiarly  fitted  to  together  of  existence 
exemplify  it.     It  seems  that  from  this  argument  it  follows  that  both  the  creation  of  any  *  non-existence  a.t 

•>  •         „    •        i  •  •  -11          rf       >r    T      i  <•  -i  i       •  the  time  of  creation 

thing,  &  its  destruction,  are  impossible,     r  or,  it  the  last  term  of  a  series  that  precedes  is  to  Or  annihilation  ;  & 

be  connected  with  the  first  term  of  the  series  that  follows,  then  in  the  passage  from  a  state  its  solution. 

of  existence  to  one  of  non-existence,  or  vice  versa,  it  will  be  necessary  that  the  two  are 

connected  together  ;   &  then  at  one  &  the  same  time  the  same  thing  will  both  exist  &  not 

exist,  which  is  absurd.     The  answer  to  this  is  immediate.     For  the  ends  of  a  finite  series 

that  is  real  &  existent  must  themselves  be  real  &  existent,  not  such  as  end  up  in  absolute 

nothing,  which  has  no  properties.     Hence,  if  to  one  series  of  real  states  there  succeeds 

another  series  of  real  states  also,  which  is  not  connected  with  it  by  a  common  term,  then 

indeed  there  must  be  two  states  at  the  same  instant,  namely  those  which  are  their  two 

limits.     But  since  non-existence  is  mere  nothing,  a  series  of  this  kind  requires  no  last  limiting 

term,  but  is  immediately  &  directly  cut  off  by  fact  of  existence.     Wherefore,  at  the  first  & 

at  the  last  instant  of  that  continuous  interval  of  time,  during  which  the  matter  exists,  it  will 

certainly  exist  ;  &  its  non-existence  will  not  be  connected  with  its  existence  simultaneously. 

On  the  other  hand  if  a  given  density  persists  for  an  hour,  &  then  is  changed  in  an  instant 

of  time  into  another  twice  as  great,  which  will  last  for  another  hour  ;   then  in  that  instant 

of  time  which  separates  the  two  hours,  there  would  have  to  be  two  densities  at  one  &  the 

same  time,  the  simple  &  the  double,  &  these  are  real  terms  of  two  real  series. 


66 


PHILOSOPHIC  NATURALIS  THEORIA 


Unde  hue  transfer- 
enda  solutio  ipsa. 


Solutio  petita  ex 
geometrico  exem- 
plo. 


Solutio 
physica 
atione. 


ex    meta- 
consider- 


Illustratio    ulterior 
geometrica. 


Applicatio  ad  crea- 
tionem,  &  annihi- 
lationem. 


D 

F 

i 

\ 

F 

D 

f 

m   m* 

\ 

G 

G' 

P 

L 

5 

\ 

MJVI, 

' 

A 

B 

C  E  H         H'E'C7 

FIG.  8. 

53.  Id  ipsum  in  dissertatione  De  lege  virium  in  Natura  existentium  satis,  ni  fallor, 
luculenter  exposui,  ac  geometricis  figuris  illustravi,  adjectis  nonnullis,  quae  eodem  recidunt, 
&  quae  in  applicatione  ad  rem,  de  qua  agimus,  &  in  cujus  gratiam  haec  omnia  ad  legem  con- 
tinuitatis  pertinentia  allata  sunt,  proderunt  infra  ;    libet  autem  novem  ejus  dissertationis 
numeros  hue  transferre  integros,  incipiendo  ab  octavo,  sed  numeros  ipsos,  ut  &  schematum 
numeros  mutabo  hie,  ut  cum  superioribus  consentiant. 

54.  "  Sit  in  fig.  8   circulus  GMM'wz,  qui  referatur  ad  datam  rectam  AB  per  ordinatas 
HM  ipsi  rectae  perpendiculares ;    uti  itidem  perpendiculares  sint  binae  tangentes  EGF, 
E'G'F'.     Concipiantur  igitur  recta  quaedam  indefinita  ipsi  rectse  AB  perpendicularis,  motu 
quodam  continuo  delata  ab  A  ad  B.     Ubi  ea  habuerit,  positionem  quamcumque  GD,  quae 
praecedat  tangentem  EF,  vel  C'D',  quae  consequatur  tangentem  E'F'  ;  ordinata  ad  circulum 
nulla    erit,    sive  erit  impossibilis,  &  ut  Geometrae 

loquuntur,    imaginaria.      Ubicunque  autem  ea   sit 

inter  binas  tangentes  EGF,   E'G'F',  in  HI,    HT, 

occurret  circulo  in  binis  punctis  M,  m,  vel  M',  m', 

&  habebitur    valor  ordinate  HM,  HOT,  vel  H'M', 

H'm'.     Ordinata  quidem  ipsa  respondet   soli  inter- 

vallo    EE'  :   &  si  ipsa  linea   AB   referat   tempus  ; 

momentum  E  est  limes  inter  tempus    praecedens 

continuum  AE,  quo  ordinata   non  est,  &  tempus 

continuum  EE'  subsequens,  quo  ordinata  est  ;  punc- 

tum  E'  est  limes  inter  tempus  praecedens  EE',  quo 

ordinata  est,  &  subsequens  E'B,  quo  non  est.     Vita 

igitur  quaedam    ordinatae  est    tempus    EE' ;  ortus 

habetur  in    E,   interitus   in  E'.      Quid   autem  in 

ipso  ortu,  &  interitu  ?     Habetur-ne  quoddam  esse 

ordinatas,  an  non  esse  ?     Habetur  utique  esse,  nimi- 

rum   EG,  vel   E'G',  non  autem  non  esse.     Oritur 

tota  finitae  magnitudinis  ordinata  EG,  interit  tota  finite    magnitudinis  E'G',  nee  tamen 

ibi  conjungit  esse,  &  non   esse,  nee  ullum  absurdum  secum  trahit.  Habetur  momento    E 

primus  terminus  seriei  sequentis  sine  ultimo  seriei  praecedentis,  &  habetur  momento  E' 

ultimus  terminus  seriei  praecedentis  sine  primo  termino  seriei  sequentis." 

55.  "  Quare  autem  id  ipsum  accidat,  si  metaphysica  consideratione  rem  perpendimus, 
statim  patebit.     Nimirum  veri  nihili  nullae  sunt  verae  proprietates  :    entis  realis  verae,  & 
reales  proprietates  sunt.     Quaevis  realis  series  initium  reale  debet,  &  finem,  sive  primum,  & 
ultimum  terminum.     Id,  quod  non  est,  nullam  habet  veram  proprietatem,  nee  proinde  sui 
generis  ultimum  terminum,  aut  primum  exigit.     Series  praecedens  ordinatae  nullius,  ultimum 
terminum  non  [26]  habet,  series  consequens  non  habet  primum  :    series  realis  contenta 
intervallo  EE',  &  primum  habere  debet,  &  ultimum.     Hujus  reales  termini  terminum  ilium 
nihili  per  se  se  excludunt,  cum  ipsum  esse  per  se  excludat  non  esse." 

56.  "  Atque    id    quidem    manifestum    fit    magis  :  si    consideremus  seriem  aliquam 
praecedentem  realem,  quam  exprimant  ordinatae  ad  lineam  continuam  PLg,  quae  respondeat 
toti  tempori  AE  ita,  ut  cuivis  momento  C  ejus  temporis  respondeat  ordinata  CL.     Turn 
vero  si  momento  E  debeat  fieri  saltus  ab  ordinata  Eg  ad  ordinatam  EG  :    necessario    ipsi 
momento  E  debent  respondere  binae  ordinatae  EG,  Eg.     Nam  in  tota  linea  PLg  non  potest 
deesse  solum  ultimum  punctum  g  ;    cum  ipso  sublato  debeat  adhuc  ilia  linea  terminum 
habere  suum,  qui  terminus  esset  itidem  punctum  :    id  vero  punctum  idcirco  fuisset  ante 
contiguum  puncto  g,  quod  est  absurdum,  ut  in  eadem  dissertatione  De  Lege  Continuitatis 
demonstravimus.     Nam  inter  quodvis  punctum,  &  aliud  punctum  linea  aliqua  interjacere 
debet ;  quae  si  non  inter jaceat ;   jam  ilia  puncta  in  unicum  coalescunt.     Quare  non  potest 
deesse  nisi  lineola  aliqua  gL  ita,  ut  terminus  seriei  praecedentis  sit  in  aliquo  momento  C 
praecedente  momentum  E,  &  disjuncto  ab  eo  per  tempus  quoddam  continuum,  in  cujus 
temporis  momentis  omnibus  ordi'nata  sit  nulla." 

57.  "  Patet  igitur  discrimen  inter  transitum  a  vero  nihilo,  nimirum  a  quantitate 
imaginaria,  ad  esse,  &  transitum  ab  una  magnitudine  ad  aliam.     In  primo  casu  terminus 
nihili  non  habetur  ;   habetur  terminus  uterque  seriei  veram  habentis  existentiam,  &  potest 
quantitas,  cujus  ea  est  series,  oriri,  vel  occidere  quantitate  finita,  ac  per  se  excludere  non  esse. 
In  secundo  casu  necessario  haberi  debet  utriusque  seriei  terminus,  alterius  nimirum  postre- 
mus,  alterius  primus.     Quamobrem  etiam  in  creatione,  &  in  annihilatione  potest  quantitas 
oriri,  vel  interire  magnitudine  finita,  &  primum,  ac  ultimum  esse  erit  quoddam  esse,  quod 
secum  non  conjunget  una  non  esse.     Contra  vero  ubi  magnitude  realis  ab  una  quantitate  ad 


A  THEORY  OF  NATURAL  PHILOSOPHY  67 

c*.  I  explained  this  very  point  clearly  enough,  if  I  mistake  not,  in  my  dissertation  The    s0"166    from 

n     i  •    •          •      JIT-    .  •  '       .  •  a    T  -11  j   v  i  ...  •      i   A  'IT  ^      which  the  solution 

D,?  lege  vmum  in  Natura  existentium,  &  1  illustrated  it  by  geometrical  figures ;   also  I  made  u  to  be  borrowed. 

some  additions  that  reduced  to  the  same  thing.     These  will  appear  below,  as  an  application 

to  the  matter  in  question  ;    for  the  sake  of  which  all  these  things  relating  to  the  Law  of 

Continuity  have  been  adduced.     It  is  allowable  for  me  to  quote  in  this  connection  the 

whole  of  nine  articles  from  that   dissertation,   beginning  with    Art.  8  ;    but   I  will  here 

change  the  numbering  of  the  articles,  &  of  the  diagrams  as  well,  so  that  they  may  agree 

with  those  already  given. 

54.  "  In  Fig.  8,  let  GMM'm  be  a  circle,  referred  to  a  given  straight  line  AB  as  axis,  by  Sotoion     derived 
means  of  ordinates  HM  drawn  perpendicular  to  that  straight  line  ;    also  let  the  two  tan-  exampief" 
gents  EGF,  E'G'F'  be  perpendiculars  to  the  axis.     Now  suppose  that  an  unlimited  straight 

line  perpendicular  to  the  axis  AB  is  carried  with  a  continuous  motion  from  A  to  B.  When 
it  reaches  some  such  position  as  CD  preceding  the  tangent  EF,  or  as  C'D'  subsequent  to 
the  tangent  E'F',  there  will  be  no  ordinate  to  the  circle,  or  it  will  be  impossible  &,  as  the 
geometricians  call  it,  imaginary.  Also,  wherever  it  falls  between  the  two  tangents  EGF, 
E'G'F',  as  at  HI  or  HT,  it  will  meet  the  circle  in  two  points,  M,  m  or  M',  m' ;  &  for  the 
value  of  the  ordinate  there  will  be  obtained  HM  &  Hm,  or  H'M'  &  H'm'.  Such  an  ordinate 
will  correspond  to  the  interval  EE'  only ;  &  if  the  line  AB  represents  time,  the  instant  E 
is  the  boundary  between  the  preceding  continuous  time  AE,  in  which  the  ordinate  does 
not  exist,  £  the  subsequent  continuous  time  EE',  in  which  the  ordinate  does  exist.  The 
point  E'  is  the  boundary  between  the  preceding  time  EE',  in  which  the  ordinate  does  exist, 
&  the  subsequent  time  E'B,  in  which  it  does  not ;  the  lifetime,  as  it  were,  of  the  ordinate, 
is  EE'  ;  its  production  is  at  E  &  its  destruction  at  E'.  But  what  happens  at  this  production 
&  destruction  ?  Is  it  an  existence  of  the  ordinate,  or  a  non-existence  I  Of  a  truth  there 
is  an  existence,  represented  by  EG  &  E'G',  &  not  a  non-existence.  The  whole  ordinate  EG 
of  finite  magnitude  is  produced,  &  the  whole  ordinate  E'G'  of  finite  magnitude  is  destroyed; 
&  yet  there  is  no  connecting  together  of  the  states  of  existence  &  non-existence,  nor  does  it 
bring  in  anything  absurd  in  its  train.  At  the  instant  E  we  get  the  first  term  of  the  sub- 
sequent series  without  the  last  term  of  the  preceding  series ;  &  at  the  instant  E'  we  have 
the  last  term  of  the  preceding  series  without  the  first  term  of  the  subsequent  series." 

55.  "  The  reason  why  this  should  happen  is  immediately  evident,  if  we  consider  the  Sol«tion     from   a 
matter  metaphysically.     Thus,  to  absolute  nothing  there  belong  no  real  properties ;  but  Sderatwn!* 

the  properties  of  a  real  absolute  entity  are  also  real.  Any  real  series  must  have  a  real 
beginning  &  end,  or  a  first  term  &  a  last.  That  which  does  not  exist  can  have  no  true 
property ;  &  on  that  account  does  not  require  a  last  term  of  its  kind,  or  a  first.  The 
preceding  series,  in  which  there  is  no  ordinate,  does  not  have  a  last  term  ;  &  the  subsequent 
series  has  likewise  no  first  term  ;  whilst  the  real  series  contained  within  the  interval  EE' 
must  have  both  a  first  term  &  a  last  term.  The  real  terms  of  this  series  of  themselves 
exclude  the  term  of  no  value,  since  the  fact  of  existence  of  itself  excludes  non-existence" 

56.  "  This  indeed  will  be  still  more  evident,  if  we  consider  some  preceding  series  of  Further  illustration 

i  •   •  11  i  •  i  i     i  •          T.T        r  „      i  i  •  by  geometry. 

real  quantities,  expressed  by  the  ordinates  to  the  curved  line  PLg  ;  &  let  this  curve 
correspond  to  the  whole  time  AE  in  such  a  way  that  to  every  instant  C  of  the  time  there 
corresponds  an  ordinate  CL.  Then,  if  at  the  instant  E  there  is  bound  to  be  a  sudden 
change  from  the  ordinate  Eg  to  the  ordinate  EG,  to  that  instant  E  there  must  of  necessity 
correspond  both  the  ordinates  EG,  Eg.  For  it  is  impossible  that  in  the  whole  line  PLg 
the  last  point  alone  should  be  missing ;  because,  if  that  point  is  taken  away,  yet  the  line 
is  Bound  to  have  an  end  to  it,  &  that  end  must  also  be  a  point ;  hence  that  point  would  be 
before  &  contiguous  to  the  point  g  ;  &  this  is  absurd,  as  we  have  shown  in  the  same 
dissertation  De  Lege  Continuitatis.  For  between  any  one  point  &  any  other  point  there 
must  lie  some  line  ;  &  if  such  a  line  does  not  intervene,  then  those  points  must  coalesce 
into  one.  Hence  nothing  can  be  absent,  except  it  be  a  short  length  of  line  gL,  so  that 
the  end  of  the  series  that  precedes  occurs  at  some  instant,  C,  preceding  the  instant  E,  & 
separated  from  it  by  an  interval  of  continuous  time,  at  all  instants  of  which  there  is  no 
ordinate." 

157.  "Evidently,  then,  there  is  a  distinction  between  passing  from  absolute  nothing,  Application  to  crea- 

•f'  . ''  .  ...  °.         ,  .      Y     tion&  annihilation. 

i.e.,  from  an  imaginary  quantity,  to  a  state  of  existence,  &  passing  from  one  magnitude 
to  another.  In  the  first  case  the  term  which  is  naught  is  not  reckoned  in  ;  the  term  at 
either  end  of  a  series  which  has  real  existence  is  given,  &  the  quantity,  of  which  it  is  the 
series,  can  be  produced  or  destroyed,  finite  in  amount ;  &  of  itself  it  will  exclude  non- 
existence.  In  the  second  case,  there  must  of  necessity  be  an  end  to  either  series,  namely 
the  last  of  the  one  series  &  the  first  of  the  other.  Hence,  in  creation  &  annihilation, 
a  quantity  can  be  produced  or  destroyed,  finite  in  magnitude ;  &  the  first  &  last 
state  of  existence  will  be  a  state  of  existence  of  some  kind  ;  &  this  will  not  associate  with 
itself  a  state  of  non-existence.  But,  on  the  other  hand,  where  a  real  magnitude  is  bound 


68 


PHILOSOPHIC  NATURALIS  THEORIA 


Aliquando  videri 
nihtium  id,  quod 
est  aliquid. 


Ordinatam  nullam, 
ut  &  distantiam 
nullam  existentium 
esse  compenetra- 
tionem. 


Ad  idem  pertinere 
seriei  realis  genus 
earn  distan  t  i  a  m 
nullam,  &  aliquam. 


Alia,  quje  videntur 
nihil,  &  sunt  ali- 
quid :  discrimen 
inter  radicem  ima- 
ginariam,  &  zero. 


aliam  transire  debet  per  saltum  ;  momento  temporis,  quo  saltus  committitur,  uterque 
terminus  haberi  deberet.  Manet  igitur  illaesum  argumentum  nostrum  metaphysicum  pro 
exclusione  saltus  a  creatione  &  annihilatione,  sive  ortu,  &  interitu." 

58.  "At  hie  illud  etiam  notandum  est ;  quoniam  ad  ortum,  &  interitum  considerandum 
geometricas  contemplationes  assumpsimus,  videri  quidem  prima  fronte,  aliquando  etiam 
realis  seriei  terminum  postremum  esse  nihilum  ;  sed  re  altius  considerata,  non  erit  vere 
nihilum  ;  sed  status  quidam  itidem  realis,  &  ejusdem  generis  cum  prsecedentibus,  licet  alio 
nomine  insignitus." 

[27]  59.  "  Sit  in  Fig.  9.  Linea  AB,  ut  prius,  ad  quam  linea  qusedam  PL  deveniat  in  G 
(pertinet  punctum  G  ad  lineam  PL,  E  ad  AB  continuatas,  &  sibi  occurrentes  ibidem),  &  sive 
pergat  ultra  ipsam  in  GM',  sive  retro  resiliat  per  GM'.  Recta  CD  habebit  ordinatam  CL, 
quas  evanescet,  ubi  puncto  C  abeunte  in  E,  ipsa  CD  abibit  in  EF,  turn  in  positione  ulteriori 
rectse  perpendicularis  HI,  vel  abibit  in  nega- 
tivam  HM,  vel  retro  positiva  regredietur 
in  HM'.  Ubi  linea  altera  cum  altera  coit, 
&  punctum  E  alterius  cum  alterius  puncto 
G  congreditur,  ordinata  CL  videtur  abire  in 
nihilum  ita,  ut  nihilum,  quemadmodum  & 
supra  innuimus,  sit  limes  quidam  inter  seriem 
ordinatarum  positivarum  CL,  &  negativarum 
HM  ;  vel  positivarum  CL,  &  iterum  posi- 
tivarum HM'.  Sed,  si  res  altius  considere- 
tur  ad  metaphysicum  conceptum  reducta, 
in  situ  EF  non  habetur  verum  nihilum. 
In  situ  CD,  HI  habetur  distantia  quaedam 
punctorum  C,  L  ;  H,  M  :  in  situ  EF 
habetur  eorundem  punctorum  compene- 

tratio.        Distantia     est     relatio     quaedam  FJG 

binorum    modorum,    quibus    bina     puncta 

existunt ;  compenetratio  itidem  est  relatio  binorum  modorum,  quibus  ea  existunt, 
quae  compenetratio  est  aliquid  reale  ejusdem  prorsus  generis,  cujus  est  distantia,  constituta 
nimirum  per  binos  reales  existendi  modos." 

60.  "  Totum  discrimen  est  in  vocabulis,  quae  nos  imposuimus.  Bini  locales  existendi 
modi  infinitas  numero  relationes  possunt  constituere,  alii  alias.  Hae  omnes  inter  se  & 
differunt,  &  tamen  simul  etiam  plurimum  conveniunt ;  nam  reales  sunt,  &  in  quodam  genere 
congruunt,  quod  nimirum  sint  relationes  ortae  a  binis  localibus  existendi  modis.  Diversa 
vero  habent  nomina  ad  arbitrarium  instituta,  cum  alise  ex  ejusmodi  relationibus,  ut  CL, 
dicantur  distantiae  positivae,  relatio  EG  dicatur  compenetratio,  relationes  HM  dicantur 
distantiae  negativse.  Sed  quoniam,  ut  a  decem  palmis  distantiae  demptis  5,  relinquuntur  5, 
ita  demptis  aliis  5,  habetur  nihil  (non  quidem  verum  nihil,  sed  nihil  in  ratione  distantiae  a 
nobis  ita  appellatae,  cum  remaneat  compenetratio)  ;  ablatis  autem  aliis  quinque,  remanent 
quinque  palmi  distantiae  negativae  ;  ista  omnia  realia  sunt,  &  ad  idem  genus  pertinent ;  cum 
eodem  prorsus  modo  inter  se  differant  distantia  palmorum  10  a  distantia  palmorum  5,  haec 
a  distantia  nulla,  sed  reali,  quas  compenetrationem  importat,  &  haec  a  distantia  negativa 
palmorum  5.  Nam  ex  prima  ilia  quantitate  eodem  modo  devenitur  ad  hasce  posteriores  per 
continuam  ablationem  palmorum  5.  Eodem  autem  pacto  infinitas  ellipses,  ab  infinitis 
hyperbolis  unica  interjecta  parabola  discriminat,  quae  quidem  unica  nomen  peculiare  sortita 
est,  cum  illas  numero  infinitas,  &  a  se  invicem  admodum  discrepantes  unico  vocabulo  com- 
plectamur  ;  licet  altera  magis  oblonga  ab  altera  minus  oblonga  plurimum  itidem  diversa  sit." 

[28]  61.  "  Et  quidem  eodem  pacto  status  quidam  realis  est  quies,  sive  perseverantia  in 
eodem  modo  locali  existendi ;  status  quidam  realis  est  velocitas  nulla  puncti  existentis. 
nimirum  determinatio  perseverandi  in  eodem  loco ;  status  quidam  realis  puncti  existentis 
est  vis  nulla,  nimirum  determinatio  retinendi  praecedentem  velocitatem,  &  ita  porro ; 
plurimum  haec  discrepant  a  vero  non  esse.  Casus  ordinatae  respondentis  lineae  EF  in  fig.  9, 
differt  plurimum  a  casu  ordinatae  circuli  respondentis  lineae  CD  figurae  8  :  in  prima  existunt 
puncta,  sed  compenetrata,  in  secunda  alterum  punctum  impossible  est.  Ubi  in  solutione 
problematum  devenitur  ad  quantitatem  primi  generis,  problema  determinationem  peculiarem 
accipit ;  ubi  devenitur  ad  quantitatem  secundi  generis,  problema  evadit  impossible ;  usque 
adeo  in  hoc  secundo  casu  habetur  verum  nihilum,  omni  reali  proprietate  carens ;  in  illo 
primo  habetur  aliquid  realibus  proprietatibus  praeditum,  quod  ipsis  etiam  solutionibus 
problematum,  &  constructionibus  veras  sufficit,  &  reales  determinationes ;  cum  realis,  non 
imaginaria  sit  radix  equationis  cujuspiam,  quae  sit  =  o,  sive  nihilo  aequalis." 


A  THEORY  OF  NATURAL  PHILOSOPHY  69 

to  pass  suddenly  from  one  quantity  to  another,  then  at  the  instant  in  which  the  sudden 
change  is  accomplished,  both  terms  must  be  obtained.  Hence,  our  argument  on 
metaphysical  grounds  in  favour  of  the  exclusion  of  a  sudden  change  from  creation  or 
annihilation,  or  production  &  destruction,  remains  quite  unimpaired." 

58.  "  In  this  connection  the  following  point  must  be  noted.  As  we  have  used  geometrical  Sometimes  what  is 
ideas  for  the  consideration  of   production   &   destruction,  it  seems  also  that    sometimes  reallysomethingap- 
the  last  term  of  a  real  series  is  nothing.     But  if  we  go  deeper  into    the  matter,  we  find 

that  it  is  not  in  reality  nothing,  but  some  state  that  is  also  real  and  of  the  same  kind  as 
those  that  precede  it,  though  designated  by  another  name." 

59.  "  In  Fig.  9,  let  AB  be  a  line,  as  before,  which  some  line  PL  reaches  at  G  (where  the  When  the  ordinate 
point  G  belongs  to  the  line  PL,  &  E  to  the  line  AB,  both  being  produced  to  meet  one  whe^thT'dlst^n'13 
another  at  this   point)  ;   &  suppose  that  PL  either  goes  on  beyond  the  point  as  GM,  or  between  two  exis- 
recoils  along  GM'.     Then  the  straight  line  CD  will  contain  the  ordinate  CL,  which  will  ^  tJ1™gs  .u  no" 

_  &       „,  .  .   .  '  .  thing,  there  is  com- 

vanish  when,  as  the  point  L,  gets  to  H,  L-D  attains  the  position  r,r  ;   &  after  that,  in  the  penetration. 

further  position  of  the  perpendicular  straight  line  HI,  will  either  pass  on  to  the  negative 

ordinate  HM  or  return,  once  more  positive,  to  HM'.     Now  when  the  one  line  meets  the 

other,  &  the  point  E  of  the  one  coincides  with  the  point  G  of  the  other,  the  ordinate 

CL  seems  to  run  off  into  nothing  in  such  a  manner  that  nothing,  as  we  remarked  above, 

is  a  certain  boundary  between  the  series  of  positive  ordinates  CL  &  the  negative  ordinates 

HM,  or  between  the  positive  ordinates  CL  &  the  ordinates  HM'  which  are  also  positive. 

But  if  the  matter  is  more  deeply  considered  &  reduced  to  a  metaphysical  concept,  there 

is  not  an  absolute  nothing  in  the  position  EF.     In  the  position  CD,  or  HI,  we  have  given 

a   certain   distance   between   the  points   C,L,   or   H,M ;    in  the   position   EF,   there  is 

compenetration  of  these  points.     Now  distance  is  a  relation  between  the  modes  of  existence 

of  two   points ;    also   compenetration  is  a  relation  between  two  modes  of   existence  ;    & 

this  compenetration  is  something  real  of  the  very  same  nature  as  distance,  founded  as  it  is 

on  two  real  modes  of  existence." 

60.  "  The  whole  difference  lies  in  the  words  that  we  have  given  to  the  things  in  question.  ™s ' no  '  distance 
Two  local  modes  of  existence  can  constitute  an  infinite  number  of  relations,  some  of  one  kmdT^f  °series  "of 
sort  &  some  of  another.     All  of  these   differ  from  one  another,  &  yet  agree  with  one  real  quantities  as 

•i         •          i  •    i     j  r  ia  •  •  j        •     i      •  •    j      j    '  some  '  distance. 

another  in  a  high  degree ;  ior  they  are  real  &  to  a  certain  extent  identical,  since  indeed 
they  are  all  relations  arising  from  a  pair  of  local  modes  of  existence.  But  they  have  different 
names  assigned  to  them  arbitrarily,  so  that  some  of  the  relations  of  this  kind,  as  CL,  are 
called  positive  distances,  the  relation  EG  is  called  compenetration,  &  relations  like  HM 
are  called  negative  distances.  But,  just  as  when  five  palms  of  distance  are  taken  away 
from  ten  palms,  there  are  left  five  palms,  so  when  five  more  are  taken  away,  there  is  nothing 
left  (&  yet  not  really  nothing,  but  nothing  in  comparison  with  what  we  usually  call 
distance ;  for  compenetration  is  left).  Again,  if  we  take  away  another  five,  there  remain 
five  palms  of  negative  distance.  All  of  these  are  real  &  belong  to  the  same  class ;  for 
they  differ  amongst  themselves  in  exactly  the  same  way,  namely,  the  distance  of  ten  palms 
from  the  distance  of  five  palms,  the  latter  from  '  no  '  distance  (which  however  is  something 
real  that  denotes  compenetration),  &  this  again  from  a  negative  distance  of  five  palms. 
For  starting  with  the  first  quantity,  the  others  that  follow  are  obtained  in  the  same  manner, 
by  a  continual  subtraction  of  five  palms.  In  a  similar  manner  a  single  intermediate 
parabola  discriminates  between  an  infinite  number  of  ellipses  &  an  infinite  number  of 
hyperbolas  ;  &  this  single  curve  receives  a  special  name,  whilst  under  the  one  term  we  include 
an  infinite  number  of  them  that  to  a  certain  extent  are  all  different  from  one  another, 
although  one  that  is  considerably  elongated  may  be  very  different  from  another  that  is 
less  elongated." 

61.  "In  the  same  way,  rest,  i.e.,  a  perseverance  in  the  same  mode  of  local  existence,  other  things  that 
is  some  real  state  ;  so  is '  no  '  velocity  a  real  state  of  an  existent  point,  namely,  a  propensity  ^ndVet^re^eaJi^ 
to  remain  in  the  same  place  ;  so  also  is  '  no  '  force  a  real  state  of  an  existent  point,  namely,  something  ;    d  i  s- 
a  propensity  to  retain  the  velocity  that  it  has  already;    &  so  on.     All  these  differ  from  a'~" 

a  state  of  non-existence  in  the  highest  degree.  The  case  of  the  ordinate  corresponding  &  zero/ 
to  the  line  EF  in  Fig.  9  differs  altogether  from  the  case  of  the  ordinate  of  the  circle 
corresponding  to  the  line  CD  in  Fig.  8.  In  the  first  there  exist  two  points,  but  there  is 
compenetration  of  these  points ;  in  the  other  case,  the  second  point  cannot  possibly  exist. 
When,  in  the  solution  of  problems,  we  arrive  at  a  quantity  of  the  first  kind,  the  problem 
receives  a  special  sort  of  solution  ;  but  when  the  result  is  a  quantity  of  the  second  kind, 
the  problem  turns  out  to  be  incapable  of  solution.  So  much  indeed  that,  in  this  second  case, 
there  is  obtained  a  true  nothing  that  lacks  every  real  property ;  in  the  first  case,  we  get 
something  endowed  with  real  properties,  which  also  supplies  true  &  real  values  to  the 
solutions  &  constructions  of  the  problems.  For  the  root  of  any  equation  that  =  o,  or  is 
equal  to  nothing,  is  something  that  is  real,  &  is  not  an  imaginary  thing." 


70  PHILOSOPHIC  NATURALIS  THEORIA 

Conciusip  prosolu-  fa.  "  Firmum  igitur  manebit  semper.  &  stabile,  seriem  realem  quamcunque.  quas 

tione     ejus    objec-  .  ~  °.          ,  ,    ,  v    »  „  ...  a        i   •  r 

contmuo  tempore  finito  duret,  debere  habere  £  primum  prmcipium,  &  ultimum  nnem 
realem,  sine  ullo  absurdo,  &  sine  conjunctione  sui  esse  cum  non  esse,  si  forte  duret  eo  solo 
tempore  :  dum  si  prascedenti  etiam  exstitit  tempore,  habere  debet  &  ultimum  terminum 
seriei  praecedentis,  &  primum  sequentis,  qui  debent  esse  unicus  indivisibilis  communis  limes, 
ut  momentum  est  unicus  indivisibilis  limes  inter  tempus  continuum  praecedens,  &  subsequens. 
Sed  haec  de  ortu,  &  interitu  jam  satis." 


Apphcatio     leg  is  ft-    ijt  igitur  contrahamus  iam  vela,  continuitatis  lex  &  inductione,  &  metaphysico 

contmuitatis       ad  J  ,  °      ,        .   .  •  i    •  •          •  •      •        .  ..  .         .  r     '   . 

coiiisionem    corpo-  argumento  abunde  nititur,  quas  idcirco  etiam  in  velocitatis  commumcatione  retmeri  omnmo 
rum-  debet,  ut  nimirum  ab  una  velocitate  ad  aliam  numquam  transeatur,  nisi  per  intermedias 

velocitates  omnes  sine  saltu.  Et  quidem  in  ipsis  motibus,  &  velocitatibus  inductionem 
habuimus  num.  39,  ac  difficultates  solvimus  num.  46,  &  47  pertinentes  ad  velocitates,  quae 
videri  possent  mutatse  per  saltum.  Quod  autem  pertinet  ad  metaphysicum  argumentum,  si 
toto  tempore  ante  contactum  subsequentis  corporis  superficies  antecedens  habuit  12  gradus 
velocitatis,  &  sequenti  9,  saltu  facto  momentaneo  ipso  initio  contactus  ;  in  ipso  momento  ea 
tempora  dirimente  debuisset  habere  &  12,  &  9  simul,  quod  est  absurdum.  Duas  enim 
velocitates  simul  habere  corpus  non  potest,  quod  ipsum  aliquanto  diligentius  demonstrabo. 

DUO  velocitatum  g,    Velocitatis  nomen,  uti  passim  usurpatur  a  Mechanicis,  asquivocum  est;    potest 

genera,  potentials,  T  r  r      .  T.  .  r 

&  actuaiis.  enim  sigmncare  velocitatem  actuaiem,  quas  nimirum  est  relatio  quaedam  in  motu  asquabm 

spatii  percursi  divisi  per  tempus,  quo  percurritur  ;  &  potest  significare  [29]  quandam,  quam 
apto  Scholiasticorum  vocabulo  potentialem  appello,  quae  nimirum  est  determinatio,  ad 
actuaiem,  sive  determinatio,  quam  habet  mobile,  si  nulla  vis  mutationem  inducat,  percur- 
rendi  motu  asquabili  determinatum  quoddam  spatium  quovis  determinato  tempore,  quas 
quidem  duo  &  in  dissertatione  De  Viribus  Fivis,  &  in  Stayanis  Supplements  distinxi, 
distinctione  utique  .necessaria  ad  aequivocationes  evitandas.  Prima  haberi  non  potest 
momento  temporis,  sed  requirit  tempus  continuum,  quo  motus  fiat,  &  quidem  etiam  motum 
aequabilem  requirit  ad  accuratam  sui  mensuram  ;  secunda  habetur  etiam  momento  quovis 
determinata  ;  &  hanc  alteram  intelligunt  utique  Mechanici,  cum  scalas  geometricas  effor- 
mant  pro  motibus  quibuscunque  difformibus,  sive  abscissa  exprimente  tempus,  &  ordinata 
velocitatem,  utcunque  etiam  variatam,  area  exprimat  spatium  :  sive  abscissa  exprimente 
itidem  tempus,  &  ordinata  vim,  area  exprimat  velocitatem  jam  genitam,  quod  itidem  in  aliis 
ejusmodi  scalis,  &  formulis  algebraicis  fit  passim,  hac  potentiali  velocitate  usurpata,  quas  sit 
tantummodo  determinatio  ad  actuaiem,  quam  quidem  ipsam  intelligo,  ubi  in  collisione 
corporum  earn  nego  mutari  posse  per  saltum  ex  hoc  posteriore  argumento. 


^5'  Jam  vero  velocitates  actuales  non  posse  simul  esse  duas  in  eodem  mobili,  satis  patet  ; 
potentials  'simul  quia  oporteret,  id  mobile,  quod  initio  dati  cujusdam  temporis  fuerit  in  dato  spatii  puncto, 
ne^eturn<vei  exf<*a-  ^n  omnibus  sequentibus  occupare  duo  puncta  ejusdem  spatii,  ut  nimirum  spatium  percursum 
tur  compenetratfo.  sit  duplex,  alterum  pro  altera  velocitate  determinanda,  adeoque  requireretur  actuaiis 
replicatio,  quam  non  haberi  uspiam,  ex  principio  inductionis  colligere  sane  possumus 
admodum  facile.  Cum  nimirum  nunquam  videamus  idem  mobile  simul  ex  eodem  loco 
discedere  in  partes  duas,  &  esse  simul  in  duobis  locis  ita,  ut  constet  nobis,  utrobique  esse  illud 
idem.  At  nee  potentiales  velocitates  duas  simul  esse  posse,  facile  demonstratur.  Nam 
velocitas  potentialis  est  determinatio  ad  existendum  post  datum  tempus  continuum  quodvis 
in  dato  quodam  puncto  spatii  habente  datam  distantiam  a  puncto  spatii,  in  quo  mobile  est 
eo  temporis  momento,  quo  dicitur  habere  illam  potentialem  velocitatem  determinatam. 
Quamobrem  habere  simul  illas  duas  potentiales  velocitates  est  esse  determinatum  ad  occu- 
panda  eodem  momento  temporis  duo  puncta  spatii,  quorum  singula  habeant  suam  diversam 
distantiam  ab  eo  puncto  spatii,  in  quo  turn  est  mobile,  quod  est  esse  determinatum  ad 
replicationem  habendam  momentis  omnibus  sequentis  temporis.  Dicitur  utique  idem 
mobile  a  diversis  causis  acquirere  simul  diversas  velocitates,  sed  eae  componuntur  in  unicam 
ita,  ut  singulas  constituant  statum  mobilis,  qui  status  respectu  dispositionum,  quas  eo 
momento,  in  quo  turn  est,  habet  ipsum  mobile,  complectentium  omnes  circumstantias 
praeteritas,  &  praesentes,  est  tantummodo  conditionatus,  non  absolutus  ;  nimirum  ut  con- 
tineant  determi-[3o]-nationem,  quam  ex  omnibus  praeteritis,  &  praesentibus  circumstantiis 
haberet  ad  occupandum  illud  determinatum  spatii  punctum  determinato  illo  momento 


A  THEORY  OF  NATURAL  PHILOSOPHY  71 

62.  "Hence  in  all  cases  it  must  remain  a  firm  &stable  conclusion  that  any  real  series,  Conclusion  in 

,.,,  ,  c.  .  .  .1  i,  c         i        •       •  r-       i    favour  of  a  solution 

which  lasts  for  some  finite  continuous  time,  is  bound  to  have  a  first  beginning  &  a  final  Of  this  difficulty. 

end,  without  any  absurdity  coming  in,  &  without  any  linking  up  of  its  existence  with 

a  state  of  non-existence,  if  perchance  it  lasts  for  that  interval  of  time  only.     But  if  it  existed 

at  a  previous  time  as  well,  it  must  have  both  a  last  term  of  the  preceding  series  &  a  first 

term  of  the  subsequent  series ;   just  as  an  instant  is  a  single  indivisible  boundary  between 

the  continuous  time  that   precedes   &  that  which  follows.     But  what  I  have  said  about 

production  &  destruction  is  already  quite  enough." 

63.  But,  to  come  back  at  last  to  our  point,  the  Law  of  Continuity  is  solidly  founded  Application  of  the 
both  on  induction  &  on  metaphysical  reasoning  ;  &  on  that  account  it  should  be  retained  ^The*  co5ision"af 
in  every  case  of  communication  of  velocity.     So  that  indeed  there  can  never  be  any  passing  solid  bodies. 
from  one  velocity  to  another  except  through  all  intermediate  velocities,  &  then  without 

any  sudden  change.  We  have  employed  induction  for  actual  motions  &  velocities  in 
Art.  39  &  solved  difficulties  with  regard  to  velocities  in  Art.  46,  47,  in  cases  in  which  they 
might  seem  to  be  subject  to  sudden  changes.  As  regards  metaphysical  argument,  if  in  the 
whole  time  before  contact  the  anterior  surface  of  the  body  that  follows  had  12  degrees  of 
velocity  &  in  the  subsequent  time  had  9,  a  sudden  change  being  made  at  the  instant  of  first 
contact ;  then  at  the  instant  that  separates  the  two  times,  the  body  would  be  bound  to  have 
12  degrees  of  velocity,  &  9,  at  one  &  the  same  time.  This  is  absurd  ;  for  a  body  cannot  at 
the  same  time  have  two  velocities,  as  I  will  now  demonstrate  somewhat  more  carefully. 

64.  The  term  velocity,  as  it  is  used  in  general  by  Mechanicians  is  equivocal.     For  it  Two  kinds  of  veio- 
may  mean  actual  velocity,  that  is  to  say,  a  certain  relation  in  uniform  motion  given  by  Clty<  P°tentlal    & 
the  space  passed  over  divided  by  the  time  taken  to  traverse  it.     It  may  mean  also  something 

which,  adopting  a  term  used  by  the  Scholastics,  I  call  potential  velocity.  The  latter  is 
a  propensity  for  actual  velocity,  or  a  propensity  possessed  by  the  movable  body  (should 
no  force  cause  an  alteration)  for  traversing  with  uniform  motion  some  definite  space  in 
any  definite  time.  I  made  the  distinction  between  these  two  meanings,  both  in  the 
dissertation  De  Firibus  Fivis  &  in  the  Supplements  to  Stay's  Philosophy ;  the  distinction 
being  very  necessary  to  avoid  equivocations.  The  former  cannot  be  obtained  in  an  instant 
of  time,  but  requires  continuous  time  for  the  motion  to  take  place  ;  it  also  requires  uniform 
motion  in  order  to  measure  it  accurately.  The  latter  can  be  determined  at  any  given 
instant ;  &  it  is  this  kind  that  is  everywhere  intended  by  Mechanicians,  when  they  make 
geometrical  measured  diagrams  for  any  non-uniform  velocities  whatever.  In  which,  if 
the  abscissa  represents  time  &  the  ordinate  velocity,  no  matter  how  it  is  varied,  then 
the  area  will  express  the  distance  passed  over ;  or  again,  if  the  abscissa  represents  time 
&  the  ordinate  force,  then  the  area  will  represent  the  velocity  already  produced.  This 
is  always  the  case,  for  other  scales  of  the  same  kind,  whenever  algebraical  formulae  & 
this  potential  velocity  are  employed ;  the  latter  being  taken  to  be  but  the  propensity  for 
actual  velocity,  such  indeed  as  I  understand  it  to  be,  when  in  collision  of  bodies  I  deny 
from  the  foregoing  argument  that  there  can  be  any  sudden  change. 

65.  Now  it  is  quite  clear  that  there  cannot  be  two  actual  velocities  at  one  &  the  same  I4    is     impossible 
time  in  the  same  moving  body.     For,  then  it  would  be  necessary  that  the  moving  body,  have  two  velocities" 
which  at  the  beginning  of  a  certain  time  occupied  a  certain  given  point  of  space,  should  at  either    actual    or 
all  times  afterwards  occupy  two  points  of  that  space  ;  so  that  the  space  traversed  would  be  ^given)  or  we  are 
twofold,  the  one  space  being  determined  by  the  one  velocity  &  the  other  by  the  other,  forced    to    admit, 
Thus  an  actual  replication  would  be  required  ;   &  this  we  can  clearly  prove  in  a  perfectly  penetration 'S 
simple  way  from  the  principle  of  induction.     Because,  for  instance,  we  never  see  the  same 

movable  body  departing  from  the  same  place  in  two  directions,  nor  being  in  two  places  at 
the  same  time  in  such  a  way  that  it  is  clear  to  us  that  it  is  in  both.  Again,  it  can  be  easily 
proved  that  it  is  also  impossible  that  there  should  be  two  potential  velocities  at  the  same 
time.  For  potential  velocity  is  the  propensity  that  the  body  has,  at  the  end  of  any  given 
continuous  time,  for  existing  at  a  certain  given  point  of  space  that  has  a  given  distance 
from  that  point  of  space,  which  the  moving  body  occupied  at  the  instant  of  time  in  which 
it  is  said  to  have  the  prescribed  potential  velocity.  Wherefore  to  have  at  one  &  the  same 
time  two  potential  velocities  is  the  same  thing  as  being  prescribed  to  occupy  at  the  same 
instant  of  time  two  points  of  space  ;  each  of  which  has  its  own  distinct  distance  from  that 
point  of  space  that  the  body  occupied  at  the  start ;  &  this  is  the  same  thing  as  prescribing 
that  there  should  be  replication  at  all  subsequent  instants  of  time.  It  is  commonly  said 
that  a  movable  body  acquires  from  different  causes  several  velocities  simultaneously ;  but 
these  velocities  are  compounded  into  one  in  such  a  way  that  each  produces  a  state  of  the 
moving  body  ;  &  this  state,  with  regard  to  the  dispositions  that  it  has  at  that  instant  (these 
include  all  circumstances  both  past  &  present),  is  only  conditional,  not  absolute.  That  is 
to  say,  each  involves  the  propensity  which  the  body,  on  account  of  all  past  &  present 
circumstances,  would  have  for  occupying  that  prescribed  point  of  space  at  that  particular 


72  PHILOSOPHISE  NATURALIS  THEORIA 

temporis ;  nisi  aliunde  ejusmodi  determinatio  per  conjunctionem  alterius  causae,  quae  turn 
agat,  vel  jam  egerit,  mutaretur,  &  loco  ipsius  alia,  quae  composita  dicitur,  succederet.  Sed 
status  absolutus  resultans  ex  omnibus  eo  momento  praasentibus,  &  prseteritis  circumstantiis 
ipsius  mobilis,  est  unica  determinatio  ad  existendum  pro  quovis  determinato  momento 
temporis  sequentis  in  quodam  determinato  puncto  spatii,  qui  quidem  status  pro  circum- 
stantiis omnibus  praeteritis,  &  prsesentibus  est  absolutus,  licet  sit  itidem  conditionatus  pro 
futuris  :  si  nimirum  esedem,  vel  alias  causa;  agentes  sequentibus  momentis  non  mutent 
determinationem,  &  punctum  illud  loci,  ad  quod  revera  deveniri  deinde  debet  dato  illo 
momento  temporis,  &  actu  devenitur ;  si  ipsae  nihil  aliud  agant.  Porro  patet  ejusmodi 
status  ex  omnibus  prseteritis,  &  praesentibus  circumstantiis  absolutes  non  posse  eodem 
momento  temporis  esse  duos  sine  determinatione  ad  replicationem,  quam  ille  conditionatus 
status  resultans  e  singulis  componentibus  velocitatibus  non  inducit  ob  id  ipsum,  quod 
conditionatus  est.  Jam  vero  si  haberetur  saltus  a  velocitate  ex  omnibus  prsteritis,  & 
praesentibus  circumstantiis  exigente,  ex.  gr.  post  unum  minutum,  punctum  spatii  distans 
per  palmos  6  ad  exigentem  punctum  distans  per  palmos  9  ;  deberet  eo  momento  temporis, 
quo  fieret  saltus,  haberi  simul  utraque  determinatio  absoluta  respectu  circumstantiarum 
omnium  ejus  momenti,  &  omnium  praeteritarum  ;  nam  toto  prsecedenti  tempore  habita 
fuisset  realis  series  statuum  cum  ilia  priore,  &  toto  sequenti  deberet  haberi  cum  ilia 
posteriore,  adeoque  eo  momento,  simul  utraque,  cum  neutra  series  realis  sine  reali  suo 
termino  stare  possit. 


Quovis     momento          66.  Praeterea  corporis,  vel  puncti  existentis  potest  utique  nulla  esse  velocitas  actualis, 

denbeUre  hTbeTe  saltern  accurate  talis ;   si  nimirum  difformem  habeat  motum,  quod  ipsum  etiam  semper  in 

statum  reaiem   ex  Natura  accidit,  ut  demonstrari  posse  arbitror,  sed  hue  non  pertinet ;    at  semper  utique 

potentialis'6    li!itlS  haberi  debet  aliqua  velocitas  potentialis,  vel  saltern  aliquis  status,  qui  licet  alio  vocabulo 

appellari  soleat,  &  dici  velocitas  nulla,  est  tamen  non  nihilum  quoddam,  sed  realis  status, 

nimirum  determinatio  ad  quietem,  quanquam  hanc  ipsam,  ut  &  quietem,  ego  quidem 

arbitrer  in  Natura  reapse  haberi  nullam,  argumentis,  quae  in  Stayanis  Supplementis  exposui 

in  binis  paragraphis  de  spatio,  ac  tempore,  quos  hie  addam  in  fine  inter  nonnulla,  quae  hie 

etiam  supplementa  appellabo,  &  occurrent  primo,  ac  secundo  loco.     Sed  id  ipsum  itidem 

nequaquam  hue  pertinet.     lis  etiam  penitus  praetermissis,  eruitur  e  reliquis,  quae  diximus, 

admisso  etiam  ut  existente,  vel  possibili  in  Natura  motu  uniformi,  &  quiete,  utramque 

velocitatem  habere  conditiones  necessarias  ad  [31]  hoc,  ut  secundum  argumentum  pro 

continuitatis  lege  superius  allatum  vim  habeat  suam,  nee  ab  una  velocitate  ad  alteram  abiri 

possit  sine  transitu  per  intermedias. 


ento  te^oris'trari"  ^7'  Patet  autenij  nmc  illud  evinci,  nee  interire  momento  temporis  posse,  nee  oriri 

sin  ab  una  veioci-  velocitatem  totam  corporis,  vel  puncti  non  simul  intereuntis,  vel  orientis,  nee  hue  transferri 
demonstratliai&  Posse»  quod  de  creatione,  &  morte  diximus ;  cum  nimirum  ipsa  velocitas  nulla  corporis,  vel 
vindicatur.  puncti  existentis,  sit  non  purum  nihil,  ut  monui,  sed  realis  quidam  status,  qui  simul  cum 

alio  reali  statu  determinatae  illius  intereuntis,  vel  orientis  velocitatis  deberet  conjungi ;  unde 
etiam  fit,  ut  nullum  effugium  haberi  possit  contra  superiora  argumenta,  dicendo,  quando  a 
12  gradibus  velocitatis  transitur  ad  9,  durare  utique  priores  9,  &  interire  reliquos  tres,  in 
quo  nullum  absurdum  sit,  cum  nee  in  illorum  duratione  habeatur  saltus,  nee  in  saltu  per 
interitum  habeatur  absurdi  quidpiam,  ejus  exemplo,  quod  superius  dictum  fuit,  ubi  ostensum 
est,  non  conjungi  non  esse  simul,  &  esse.  Nam  in  primis  12  gradus  velocitatis  non  sunt  quid 
compositum  e  duodecim  rebus  inter  se  distinctis,  atque  disjunctis,  quarum  9  manere  possint, 
3  interire,  sed  sunt  unica  determinatio  ad  existendum  in  punctis  spatii  distantibus  certo 
intervallo,  ut  palmorumi2,  elapsis  datis  quibusdam  temporibus  aequalibus  quibusvis.  Sic 
etiam  in  ordinatis  GD,  HE,  quae  exprimunt  velocitates  in  fig.  6,  revera,  in  mea  potissimuim 
Theoria,  ordinata  GD  non  est  quaedam  pars  ordinatae  HE  communis  ipsi  usque  ad  D,  sed 
sunt  duae  ordinatae,  quarum  prima  constitit  in  relatione  distantiaa,  puncti  curvae  D  a  puncto 
axis  G,  secunda  in  relatione  puncti  curvae  E  a  puncto  axis  H,  quod  estibi  idem,  ac  punctum  G. 


A  THEORY  OF  NATURAL  PHILOSOPHY  73 

instant  of  time  ;  were  it  not  for  the  fact  that  that  particular  propensity  is  for  other  reasons 
altered  by  the  conjunction  of  another  cause,  which  acts  at  the  time,  or  has  already  done  so  ; 
&  then  another  propensity,  which  is  termed  compound,  will  take  the  place  of  the  former. 
But  the  absolute  propensity,  which  arises  from  the  combination  of  all  the  past  &  present 
circumstances  of  the  moving  body  for  that  instant,  is  but  a  single  propensity  for  existing  at 
any  prescribed  instant  of  subsequent  time  in  a  certain  prescribed  point  of  space  ;  &  this 
state  is  absolute  for  all  past  &  present  circumstances,  although  it  may  be  conditional  for 
future  circumstances.  That  is  to  say,  if  the  same  or  other  causes,  acting  during  subsequent 
instants,  do  not  change  that  propensity,  &  the  point  of  space  to  which  it  ought  to  get 
thereafter  at  the  given  instant  of  time,  &  which  it  actually  does  reach  if  these  causes  have 
no  other  effect.  Further,  it  is  clear  that  we  cannot  have  two  such  absolute  states,  arising 
from  all  past  &  present  circumstances,  at  the  same  time  without  prescribing  replication  ; 
&  this  the  conditional  state  arising  from  each  of  the  component  velocities  does  not  induce 
because  of  the  very  fact  that  it  is  conditional.  If  now  there  should  be  a  jump  from  the 
velocity,  arising  out  of  all  the  past  &  present  circumstances,  which,  after  one  minute  for 
example,  compels  a  point  of  space  to  move  through  6  palms,  to  a  velocity  that  compels  the 
point  to  move  through  9  palms ;  then,  at  the  instant  of  time,  in  which  the  sudden  change 
takes  place,  there  would  be  each  of  two  absolute  propensities  in  respect  of  all  the  circum- 
stances of  that  instant  &  all  that  had  gone  before,  existing  simultaneously.  For  in  the 
whole  of  the  preceding  time  there  would  have  been  a  real  series  of  states  having  the  former 
velocity  as  a  term,  &  in  the  whole  of  the  subsequent  time  there  must  be  one  having  the 
latter  velocity  as  a  term  ;  hence  at  that  particular  instant  each  of  them  must  occur  at  one 
&  the  same  time,  since  neither  real  series  can  stand  good  without  each  having  its  own 
real  end  term. 

66.  Again,  it  is  at  least  possible  that  the  actual  velocity  of  a  body,  or  of  an  existing  At  any  instant  an 
point,  may  be  nothing  ;    that  is  to  say,  if  the  motion  is  non-uniform.     Now,  this  always  ^^l  *?££  ""** 
is  the  case  in  Nature  ;  as  I  think  can  be  proved,  but  it  does  not  concern  us  at  present.     But,  arising  from  a  kind 
at  any  rate,  it  is  bound  to  have  some  potential  velocity,  or  at  least  some  state,  which,  °£yP°tentlal  vel°- 
although  usually  referred  to  by  another  name,  &  the  velocity  stated  to  be  nothing,  yet  is 

not  definitely  nothing,  but  is  a  real  state,  namely,  a  propensity  for  rest.  I  have  come  to 
the  conclusion,  however,  that  in  Nature  there  is  not  really  such  a  thing  as  this  state,  or 
absolute  rest,  from  arguments  that  I  gave  in  the  Supplements  to  Stay's  Philosophy  in 
two  paragraphs  concerning  space  &  time  ;  &  these  I  will  add  at  the  end  of  the  work,  amongst 
some  matters,  that  I  will  call  by  the  name  of  supplements  in  this  work  as  well ;  they  will 
be  placed  first  &  second  amongst  them.  But  that  idea  also  does  not  concern  us  at  present. 
Now,  putting  on  one  side  these  considerations  altogether,  it  follows  from  the  rest  of  what 
I  have  said  that,  if  we  admit  both  uniform  motion  &  rest  as  existing  in  Nature,  or  even 
possible,  then  each  velocity  must  have  conditions  that  necessarily  lead  to  the  conclusion 
that  according  to  the  argument  given  above  in  support  of  the  Law  of  Continuity  it  has  its 
own  corresponding  force,  &  that  no  passage  from  one  velocity  to  another  can  be  made 
except  through  intermediate  stages. 

67.  Further,  it  is  quite  clear  that  from  this  it  can  be  rigorously  proved  that  the  whole  Rigorous  proof  that 

e        i  .  i  .        .  .  ,    9  J    r,  .  it  is   impossible  to 

velocity  of  a  body  cannot  perish  or  arise  in  an  instant  of  time,  nor  for  a  point  that  does  pass  from  one  veio- 
not  perish  or  arise  along  with  it ;    nor  can  our  arguments  with  regard  to  production  &  city  to  a™*11?1  in 

1-1  i  r  i  •         T-i          •  i.*«««°»«*»  an  instant  of  time. 

destruction  be  made  to  refer  to  this.  For,  since  that  no  velocity  of  a  body,  or  of  an 
existing  point,  is  not  absolutely  nothing,  as  I  remarked,  but  is  some  real  state  ;  &  this  real 
state  is  bound  to  be  connected  with  that  other  real  state,  namely,  that  of  the  prescribed 
velocity  that  is  being  created  or  destroyed.  Hence  it  comes  about  that  there  can  be  no 
escape  from  the  arguments  I  have  given  above,  by  saying  that  when  the  change  from  twelve 
degrees  of  velocity  is  made  to  nine  degrees,  the  first  nine  at  least  endure,  whilst  the 
remaining  three  are  destroyed  ;  &  then  by  asserting  that  there  is  nothing  absurd  in  this, 
since  neither  in  the  duration  of  the  former  has  there  been  any  sudden  change,  nor  is  there 
anything  absurd  in  the  jump  caused  by  the  destruction  of  the  latter,  according  to  the  instance 
of  it  given  above,  where  it  was  shown  that  non-existence  &  existence  must  be  disconnected. 
For  in  the  first  place  those  twelve  degrees  of  velocity  are  not  something  compounded  of 
twelve  things  distinct  from,  &  unconnected  with,  one  another,  of  which  nine  can  endure 
&  three  can  be  destroyed  ;  but  are  a  single  propensity  for  existing,  after  the  lapse  of  any 
given  number  of  equal  times  of  any  given  length,  in  points  of  space  at  a  certain  interval, 
say  twelve  palms,  away  from  the  original  position.  So  also,  with  regard  to  the  ordinates 
GD,  HE,  which  in  Fig.  6.  express  velocities,  it  is  the  fact  that  (most  especially  in  my  Theory) 
the  ordinate  GD  is  not  some  part  of  the  ordinate  HE,  common  with  it  as  far  as  the  point 
D  ;  but  there  are  two  ordinates,  of  which  the  first  depends  upon  the  relation  of  the  distance 
of  the  point  D  of  the  curve  from  the  point  G  on  the  axis,  &  the  second  upon  the  relation 
of  the  distance  of  point  E  on  the  curve  from  the  point  H  on  the  axis,  which  is  here  the 


74 


PHILOSOPHIC  NATURALIS  THEORIA 


Relationem  distantiae  punctorum  D,  &  G  constituunt  duo  reales  modi  existendi  ipsorum, 
relationem  distantias  punctorum  D.  &  E  duo  reales  modi  existendi  ipsorum,  &  relationem 
distantiae  punctorum  H,  &  E  duo  reales  modi  existendi  ipsorum.  Haec  ultima  relatio 
constat  duobus  modis  realibus  tantummodo  pertinentibus  ad  puncta  E,  &  H,  vel  G,  & 
summa  priorum  constat  modis  realibus  omnium  trium,  E,  D,  G.  Sed  nos  indefinite  con- 
cipimus  possibilitatem  omnium  modorum  realium  intermediorum,  ut  infra  dicemus,  in  qua 
praecisiva,  &  indefinita  idea  stat  mini  idea  spatii  continui ;  &  intermedii  modi  possibles  inter 
G,  &  D  sunt  pars  intermediorum  inter  E,  &  H.  Praeterea  omissis  etiam  hisce  omnibus  ipse 
ille  saltus  a  velocitate  finita  ad  nullam,  vel  a  nulla  ad  finitam,  haberi  non  potest. 


Cur  adhibita   col-  68.  Atque  hinc  ego  quidem  potuissem  etiam  adhibere  duos  globos  asquales,  qui  sibi 

eaiuicm^aKanTpro  mv*cem  occurrant  cum  velocitatibus  sequalibus,  quae  nimirum  in  ipso  contactu  deberent 

Thcoria  deducenda.  momento  temporis  intcrirc  ;  sed  ut  hasce  ipsas  considerationes  evitarem  de  transitu  a  statu 

reali  ad  statum  itidem  realem,  ubi  a  velocitate  aliqua  transitur  ad  velocitatem  nullam  ; 

adhibui  potius  [32]  in  omnibus  dissertationibus  meis  globum,  qui  cum  12  velocitatis  gradibus 

assequatur  alterum  praecedentem  cum  6  ;    ut   nimirum  abeundo  ad  velocitatem  aliam 

quamcunque  haberetur  saltus  ab  una  velocitate  ad  aliam,  in  quo  evidentius  esset  absurdum. 


Quo  pacto  mutata 
velocitate  poten- 
tial! per  saltum, 
non  mutetur  per 
saltum  actualis. 


69.  Jam  vero  in  hisce  casibus  utique  haberi  deberet  saltus  quidam,  &  violatio  legis 
continuitatis,  non  quidem  in  velocitate  actuali,  sed  in  potentiali,  si  ad  contactum  deveniretur 
cum  velocitatum  discrimine  aliquo  determinato  quocunque.  In  velocitate  actuali,  si  earn 
metiamur  spatio,  quod  conficitur,  diviso  per  tempus,  transitus  utique  fieret  per  omnes 
intermedias,  quod  sic  facile  ostenditur  ope  Geometriae.  In  fig.  10  designent  AB,  BC  bina 
tempora  ante  &  post  contactum,  &  momento  quolibet  H  sit  velocitas  potentialis  ilia  major 
HI,  quae  aequetur  velocitati  primae  AD  :  quovis  autem  momento  Q  posterioris  temporis  sit 


velocitas  potentialis  minor  QR,  quae  aequetur 
velocitati  cuidam  data:  CG.  Assumpto  quovis 
tempore  HK  determinatae  magnitudinis,  area 
IHKL  divisa  per  tempus  HK,  sive  recta  HI, 
exhibebit  velocitatem  actualem.  Moveatur 
tempus  HK  versus  B,  &  donee  K  adveniat  ad 
B,  semper  eadem  habebitur  velocitatis  men- 
sura  ;  eo  autem  progressoin  O  ultra  B,  sed  adhuc 
H  existente  in  M  citra  B,  spatium  illi  tem- 
pori  respondens  componetur  ex  binis  MNEB, 
BFPO,  quorum  summa  si  dividatur  per  MO  ; 
jam  nee  erit  MN  aequalis  priori  AD,  nee  BF, 
ipsa  minor  per  datam  quantitatem  FE  ;  sed 
facile  demonstrari  potest  (&),  capta  VE  asquali 


D!  ~     L  V  N  E  Y 


Irrcgularitas  alia 
in  cxpressione  act- 
ualis velocitatis. 


\ 

"1 

1 

X 

\ 

p;  R  T  G 

1 

1 

1 

1 

AH      K 


M  B  OQ    S  C 

FIG.  10. 

IL,  vel  HK,  sive  MO,  &  ducta  recta  VF,  quae  secet  MN  in  X,  quotum  ex  illo  divisione 
prodeuntem  fore  MX,  donee,  abeunte  toto  illo  tempore  ultra  B  in  QS,  jam  area  QRTS 
divisa  per  tempus  QS  exhibeat  velocitatem  constantem  QR. 

70.  Patet  igitur  in  ea  consideratione  a  velocitate  actuali  praecedente  HI  ad  sequentem 
QR  transiri  per  omnes  intermedias  MX,  quas  continua  recta  VF  definiet ;  quanquam  ibi 
etiam  irregulare  quid  oritur  inde,  quod  velocitas  actualis  XM  diversa  obvenire  debeat  pro 
diversa  magnitudine  temporis  assumpti  HK,  quo  nimirum  assumpto  majore,  vel  minore 
removetur  magis,  vel  minus  V  ab  E,  &  decrescit,  vel  crescit  XM.  Id  tamen  accidit  in 
motibus  omnibus,  in  quibus  velocitas  non  manet  eadem  toto  tempore,  ut  nimirum  turn 
etiam,  si  velocitas  aliqua  actualis  debeat  agnosci,  &  determinari  spatio  diviso  per  tempus ; 
pro  aliis,  atque  aliis  temporibus  assumptis  pro  mensura  alias,  atque  alias  velocitatis  actualis 
mensuras  ob-[33]-veniant,  secus  ac  accidit  in  motu  semper  aequabili,  quam  ipsam  ob  causam, 
velocitatis  actualis  in  motu  difformi  nulla  est  revera  mensura  accurata,  quod  supra  innui 
sed  ejus  idea  praecisa,  ac  distincta  aequabilitatem  motus  requirit,  &  idcirco  Mechanic!  in 
difformibus  motibus  ad  actualem  velocitatem  determinandam  adhibere  solent  spatiolum 
infinitesimo  tempusculo  percursum,  in  quo  ipso  motum  habent  pro  aequabili. 


(b)    Si  enim  producatur  OP  usque  ad  NE  in  T,  erit  ET  =  VN,  ob  VE  =  MO  =NT.     Est  autem 

VE  :  VN  :  :  EF  :  NX  ;   quart  VN  X  EF  =  VE  X  NX,    sive  posito  ET  pro  VN,  W  MO  pro  VE,  erit 
ET  XEF  =MO  X  NX.  Totum  MNTO  est  MO  X  MN,  pars  FETP  est  =  EY  X  EF.     Quafe  residuus 
gnomon  NMOPFE  est  MOx(MN-NX),  sive  est  MO  X  MX,  quo  diviso  per  MO  babetur  MX. 


A  THEORY  OF  NATURAL  PHILOSOPHY  75 

same  as  the  point  G.  The  relation  of  the  distance  between  the  points  D  &  G  is  determined 
by  the  two  real  modes  of  existence  peculiar  to  them,  the  relation  of  the  distance  between 
the  points  D  &  E  by  the  two  real  modes  of  existence  peculiar  to  them,  &  the  relation  of 
the  distance  between  the  points  H  &  E  by  the  two  real  modes  of  existence  peculiar  to  them. 
The  last  of  these  relations  depends  upon  the  two  real  modes  of  existence  that  pertain  to  the 
points  E  &  H  (or  G),  &  upon  these  alone  ;  the  sum  of  the  first  &  second  depends  upon  all 
three  of  the  modes  of  the  points  E,  D,  &  G.  But  we  have  some  sort  of  ill-defined  conception 
of  the  possibility  of  all  intermediate  real  modes  of  existence,  as  I  will  remark  later  ;  &  on 
this  disconnected  &  ill-defined  idea  is  founded  my  conception  of  continuous  space  ;  also 
the  possible  intermediate  modes  between  G  &  D  form  part  of  those  intermediate  between 
E  &  H.  Besides,  omitting  all  considerations  of  this  sort,  -that  sudden  change  from  a  finite 
velocity  to  none  at  all,  or  from  none  to  a  finite,  cannot  happen. 

68.  Hence  I  might  just  as  well  have  employed  two  equal  balls,  colliding  with  one  why  the  collision 
another  with  equal  velocities,  which  in  truth  at  the  moment  of  contact  would  have  to  be  thebsameTirecfion 
destroyed  in  an  instant  of  time.     But,  in  order  to  avoid  the  very  considerations  just  stated  is  employed  for  the 
with  regard  to  the  passage  from  a  real  state  to  another  real  state  (when  we  pass  from  a  In  " 
definite  velocity  to  none),  I  have  preferred  to  employ  in  all  my  dissertations  a  ball  having 

1 2  degrees  of  velocity,  which  follows  another  ball  going  in  front  of  it  with  6  degrees ; 
so  that,  by  passing  to  some  other  velocity,  there  would  be  a  sudden  change  from  one 
velocity  to  another ;  &  by  this  means  the  absurdity  of  the  idea  would  be  made  more 
evident. 

69.  Now,  at  least  in  such  cases  as  these,  there  is  bound  to  be  some  sudden  change  & 

a  breach  of  the  Law  of  Continuity,  not  indeed  in  the  actual  velocity,  but  in  the  potential  sudden  change  in 
velocity,  if  the  collision  occurs  with  any  given  difference  of  velocities  whatever.  In  the  ^  ^^T^  might 
actual  velocity,  measured  by  the  space  traversed  divided  by  the  time,  the  change  will  at  any  not 'be  a  sudden 
rate  be  through  all  intermediate  stages ;  &  this  can  easily  be  shown  to  be  50  by  the  aid  of  ^^veioclty16  ***" 
Geometry. 

In  Fig.  10  let  AB,  BC  represent  two  intervals  of  time,  respectively  before  &  after 
contact ;  &  at  any  instant  let  the  potential  velocity  be  the  greater  velocity  HI,  equal  to  the  . 
first  velocity  AD  ;  &  at  any  instant  Q  of  the  time  subsequent  to  contact  let  the  potential 
velocity  be  the  less  velocity  QR,  equal  to  some  given  velocity  CG.  If  any  prescribed  interval 
of  time  HK  be  taken,  the  area  IHKL  divided  by  the  time  HK,  i.e.,  the  straight  line  HI, 
will  represent  the  actual  velocity.  Let  the  time  HK  be  moved  towards  B  ;  then  until 
K  comes  to  B,  the  measure  of  the  velocity  will  always  be  the  same.  If  then,  K  goes  on 
beyond  B  to  O,  whilst  H  still  remains  on  the  other  side  of  B  at  M  ;  then  the  space  corre- 
sponding to  that  time  will  be  composed  of  the  two  spaces  MNEB,  BFPO.  Now,  if  the 
sum  of  these  is  divided  by  MO,  the  result  will  not  be  equal  to  either  MN  (which  is  equal 
to  the  first  AD),  or  BF  (which  is  less  than  MN  by  the  given  quantity  FE).  But  it  can 
easily  be  proved  (  )  that,  if  VE  is  taken  equal  to  IL,  or  HK,  or  MO,  &  the  straight  line 
VF  is  drawn  to  cut  MN  in  X ;  then  the  quotient  obtained  by  the  division  will  be  MX. 
This  holds  until,  when  the  whole  of  the  interval  of  time  has  passed  beyond  B  into  the 
position  QS,  the  area  QRTS  divided  by  the  time  QS  now  represents  a  constant  velocity 
equal  to  QR. 

70.  From    the    foregoing    reasoning   it    is  therefore  clear  that  the  change  from  the  A   further   irregu- 
preceding  actual  velocity  HI  to  the  subsequent  velocity  QR  is  made  through  all  intermediate  larity  m  the  repre- 

r  ,      .  .  TV/TTT-       i  •   i        MI  i       i  •       i  i        i  ••>•>•  sentation  of  actual 

velocities  such  as  MX,  which  will  be  determined  by  the  continuous  straight  line  VF.  There  velocity, 
is,  however,  some  irregularity  arising  from  the  fact  that  the  actual  velocity  XM  must  turn 
out  to  be  different  for  different  magnitudes  of  the  assumed  interval  of  time  HK.  For, 
according  as  this  is  taken  to  be  greater  or  less,  so  the  point  V  is  removed  to  a  greater  or 
less  distance  from  E  ;  &  thereby  XM  will  be  decreased  or  increased  correspondingly.  This 
is  the  case,  however,  for  all  motions  in  which  the  velocity  does  not  remain  the  same  during 
the  whole  interval ;  as  for  instance  in  the  case  where,  if  any  actual  velocity  has  to  be  found 
&  determined  by  the  quotient  of  the  space  traversed  divided  by  the  time  taken,  far  other 
&  different  measures  of  the  actual  velocities  will  arise  to  correspond  with  the  different 
intervals  of  time  assumed  for  their  measurement  ;  which  is  not  the  case  for  motions  that 
are  always  uniform.  For  this  reason  there  is  no  really  accurate  measure  of  the  actual 
velocity  in  non-uniform  motion,  as  I  remarked  above  ;  but  a  precise  &  distinct  idea  of  it 
requires  uniformity  of  motion.  Therefore  Mechanicians  in  non-uniform  motions,  as  a 
means  to  the  determination  of  actual  velocity,  usually  employ  the  small  space  traversed  in 
an  infinitesimal  interval  of  time,  &  for  this  interval  they  consider  that  the  motion  is  uniform. 

(b)    For  if  OP  be  produced  to  meet  NE  in  T,  then  EY  =  VN  ;  for  VE  =  MO  =  NT.     Moreover 

VE  :  VN=EF :  NX  ;  and  therefore  VN.EF=VE.NX.  Hence,  replacing  VN  hy  EY,  and.  VE  hy  MO,  we  have 
EYEF=MO.NX.  Now,  the  whole  MNYO  =  MO.MN,  and  the  part  FEYP=  ET.EF.  Hence  the  remainder 
(the  gnomon  NMOPFE)  =  MO.(MN  —  NX)  =  MO.MX .-  and  this,  on  division  by  MO,  will  give  MX. 


76  PHILOSOPHIC  NATURALIS  THEORIA 

"  \mmc  yi-  At  velocitas  potcntialis,  quas  singulis  momentis  temporis  respondet  sua,  mutaretur 


citatum  non  posse  utique  per  saltum  ipso  momento  B,  quo  deberet  haberi  &  ultima  velocitatum  praecedentium 
entianivciodtatumr"  ^'  ^  P"ma  sequentium  BF,  quod  cum  haberi  nequeat,  uti  demonstratum  est,  fieri  non 
potest  per  secundum  ex  argumentis,  quae  adhibuimus  pro  lege  continuitatis,  ut  cum  ilia 
velocitatum  inasqualitate  deveniatur  ad  immediatum  contactum  ;  atque  id  ipsum  excludit 
etiam  inductio,  quam  pro  lege  continuitatis  in  ipsis  quoque  velocitatibus,  atque  motibus 
primo  loco  proposui. 

Prpmovenda    ana-  72.  Atque  hoc  demum  pacto  illud  constitit  evidenter,  non  licere  continuitatis  legem 

deserere  in  collisione  corporum,  &  illud  admittere,  ut  ad  contactum  immediatum  deveniatur 
cum  illaesis  binorum  corporum  velocitatibus  integris.  Videndum  igitur,  quid  necessario 
consequi  debeat,  ubi  id  non  admittatur,  &  haec  analysis  ulterius  promovenda. 


ifaberimu-  73'  Quoniam  a^  immediatum  contactum  devenire  ea  corpora  non  possunt  cum  praece- 

tationem  veiocita-  dentibus  velocitatibus  ;  oportet,  ante  contactum  ipsum  immediatum  incipiant  mutari 
auk  mutat  Ue  Vlm>  velocitates  ipsae,  &  vel  ea  consequentis  corporis  minui,  vel  ea  antecedentis  augeri,  vel 
utrumque  simul.  Quidquid  accidat,  habebitur  ibi  aliqua  mutatio  status,  vel  in  altero 
corpore,  vel  in  utroque,  in  ordine  ad  motum,  vel  quietem,  adeoque  habebitur  aliqua 
mutationis  causa,  quaecunque  ilia  sit.  Causa  vero  mutans  statum  corporis  in  ordine  ad 
motum,  vel  quietem,  dicitur  vis  ;  habebitur  igitur  vis  aliqua,  quae  effectum  gignat,  etiam 
ubi  ilia  duo  corpora  nondum  ad  contactum  devenerint. 

Earn    vim    debere  74.  Ad   impediendam  violationem  continuitatis  satis  esset,  si  ejusmodi  vis  ageret  in 


.  iSf-SSi    &  alterum  tantummodo  e  binis  corporibus,  reducendo  praecedentis  velocitatem  ad  gradus  12, 

agere  m  panes  op-  .  r.  .  '  . 

positas.  vel  sequentis  ad  6.     Videndum  igitur  aliunde,  an  agere  debeat  in  alterum  tantummodo,  an 

in  utrumque  simul,  &  quomodo.  Id  determinabitur  per  aliam  Naturae  legem,  quam  nobis 
inductio  satis  ampla  ostendit,  qua  nimirum  evincitur,  omnes  vires  nobis  cognitas  agere 
utrinque  &  aequaliter,  &  in  partes  oppositas,  unde  provenit  principium,  quod  appellant 
actionis,  &  reactionis  aequalium  ;  est  autem  fortasse  quaedam  actio  duplex  semper  aequaliter 
agens  in  partes  oppositas.  Ferrum,  &  magnes  aeque  se  mutuo  trahunt  ;  elastrum  binis 
globis  asqualibus  interjectum  aeque  utrumque  urget,  &  aequalibus  velocitatibus  propellit  ; 
gravitatem  ipsam  generalem  mutuam  esse  osten-[34]-dunt  errores  Jovis,  ac  Saturni  potissi- 
mum,  ubi  ad  se  invicem  accedunt,  uti  &  curvatura  orbitae  lunaris  orta  ex  ejus  gravitate  in 
terram  comparata  cum  aestu  maris  orto  ex  inaequali  partium  globi  terraquei  gravitate  in 
Lunam.  Ipsas  nostrae  vires,  quas  nervorum  ope  exerimus,  semper  in  partes  oppositas  agunt, 
nee  satis  valide  aliquid  propellimus,  nisi  pede  humum,  vel  etiam,  ut  efficacius  agamus, 
oppositum  parietem  simul  repellamus.  En  igitur  inductionem,  quam  utique  ampliorem 
etiam  habere  possumus,  ex  qua  illud  pro  eo  quoque  casu  debemus  inferre,  earn  ibi  vim  in 
utrumque  corpus  agere,  quae  actio  ad  aequalitatem  non  reducet  inaequales  illas  velocitates, 
nisi  augeat  praecedentis,  minuat  consequentis  corporis  velocitatem  ;  nimirum  nisi  in  iis 
producat  velocitates  quasdam  contrarias,  quibus,  si  solae  essent,  deberent  a  se  invicem 
recedere  :  sed  quia  eae  componuntur  cum  praecedentibus  ;  hasc  utique  non  recedunt,  sed 
tantummodo  minus  ad  se  invicem  accedunt,  quam  accederent. 


Hinc    dicendam  75.  Invenimus  igitur  vim  ibi  debere  esse  mutuam,  quae  ad  partes  oppositas  agat,  &  quae 


esse 


sua  natura  determinet  per  sese  ilia  corpora  ad  recessum  mutuum  a  se  invicem.     Hujusmodi 

quaerendam      ejus    .    .  .  .    .      ./„    .   .  11      •  •  i  •  /~»  j  •  i 

legem.  igitur  vis  ex  nomims  denmtione  appellari  potest  vis  repulsiva.      Uuaerendum  jam  ulterius, 

qua  lege  progredi  debeat,  an  imminutis  in  immensum  distantiis  ad  datam  quandam  mensuram 
deveniat,  an  in  infinitum  excrescat  ? 

Ea  vi  debere  totum  76.  Ut  in  illo  casu  evitetur  saltus ;  satis  est  in  allato  exemplo  ;  si  vis  repulsiva,  ad  quam 

crimenateHdi   ante  delati  sumus,  extinguat  velocitatum  differentiam  illam  6  graduum,  antequam  ad  contactum 

contactum.  immediatum  corpora  devenirent  :   quamobrem  possent  utique  devenire  ad  eum  contactum 

eodem  illo  momento,  quo  ad  aequalitatem  velocitatum  deveniunt.     At  si  in  alio  quopiam 

casu  corpus  sequens  impellatur  cum  velocitatis  gradibus  20,  corpore  praecedente  cum  suis  6  ; 


A  THEORY  OF  NATURAL  PHILOSOPHY  77 

71.  The  potential  velocity,  each  corresponding  to  its  own  separate  instant  of  time,  The  conclusion  is 

ij  •    f     i  j         jj      i  ^.t    i.  •  t     •          n        a  i  •  •  tnat     immediate 

would  certainly  be  changed  suddenly  at  that  instant  ot  time  r>  ;    &  at  this  point  we  are  contact  with  a  dif- 

bound  to  have  both  the  last  of  the  preceding  velocities,  BE,  &  the  first  of  the  subsequent  ference  of  velocities 

velocities,  BF.     Now,  since  (as  has  been  already  proved)  this  is  impossible,  it  follows  from 

the  second  of  the  arguments  that  I  used  to  prove  the  Law  of  Continuity,  that  it  cannot 

come  about  that  the  bodies  come  into  immediate  contact  with  the  inequality  of  velocities 

in  question.     This  is  also  excluded  by  induction,  such  as  I  gave  in  the  first  place  for  the 

Law  of  Continuity,  in  the  case  also  of  these  velocities  &  motions. 

72.  In  this  manner  it  is  at  length  clearly  established  that  it  is  not  right  to  neglect  the  immediate  contact 
Law  of  Continuity  in  the  collision  of  bodies,  &  admit  the  idea  that  they  can  come  into  ^Sysis^tobe ca^ 
immediate  contact  with  the  whole  velocities  of  both  bodies  unaltered.     Hence,  we  must  ried  further, 
now  investigate  the  consequences  that  necessarily  follow  when  this  idea  is  not  admitted ; 

&  the  analysis  must  be  carried  further. 

73.  Since  the  bodies  cannot  come  into  immediate  contact  with  the  velocities  they  had  There  must  be  then, 
at  first,  it  is  necessary  that  those  velocities  should  commence  to  change  before  that  immediate  change  in  the  v'eioa 
contact ;   &  either  that  of  the  body  that  follows  should  be  diminished,  or  that  of  the  one  city  '•   &  therefore 
going  in  front  should  be  increased,  or  that  both  these  changes  should  take  place  together,  causes  the  change!1 
Whatever  happens,  there  will  be  some  change  of  state  at  the  time,  in  one  or  other  of  the 

bodies,  or  in  both,  with  regard  to  motion  or  rest ;  &  so  there  must  be  some  cause  for  this 
change,  whatever  it  is.  But  a  cause  that  changes  the  state  of  a  body  as  regards  motion  or 
rest  is  called  force.  Hence  there  must  be  some  force,  which  gives  the  effect,  &  that  too 
whilst  the  two  bodies  have  not  as  yet  come  into  contact. 

74.  It  would  be  enough,  to  avoid  a  breach  of  the  Law  of  Continuity,  if  a  force  of  The  f°rce omust  V6 

i  •     i.ii        11  r     -I  IT  i         i       •          i  i      •  <•     i       i      i      •       mutual,   &  act    m 

this  kind  should  act  on  one  of  the  two  bodies  only,  altering  the  velocity  of  the  body  in  opposite  directions, 
front  to  12  degrees,  or  that  of  the  one  behind  to  6  degrees.  Hence  we  must  find  out, 
from  other  considerations,  whether  it  should  act  on  one  of  the  two  bodies  only,  or  on  both 
of  them  at  the  same  time,  &  how.  This  point  will  be  settled  by  another  law  of  Nature, 
which  sufficiently  copious  induction  brings  before  us ;  that  is,  the  law  in  which  it  is  estab- 
lished that  all  forces  that  are  known  to  us  act  on  both  bodies,  equally,  and  in  opposite 
directions.  From  this  comes  the  principle  that  is  called  '  the  principle  of  equal  action 
&  reaction  '  ;  perchance  this  may  be  a  sort  of  twofold  action  that  always  produces  its 
effect  equally  in  opposite  directions.  Iron  &  a  loadstone  attract  one  another  with  the 
same  strength ;  a  spring  introduced  between  two  balls  exerts  an  equal  action  on  either 
ball,  &  generates  equal  velocities  in  them.  That  universal  gravity  itself  is  mutual  is  proved 
by  the  aberrations  of  Jupiter  &  of  Saturn  especially  (not  to  mention  anything  else) ;  that 
is  to  say,  the  way  in  which  they  err  from  their  orbits  &  approach  one  another  mutually. 
So  also,  when  the  curvature  of  the  lunar  orbit  arising  from  its  gravitation  towards  the 
Earth  is  compared  with  the  flow  of  the  tides  caused  by  the  unequal  gravitation  towards 
the  Moon  of  different  parts  of  the  land  &  water  that  make  up  the  Earth.  Our  own  bodily 
forces,  which  produce  their  effect  by  the  help  of  our  muscles,  always  act  in  opposite  direc- 
tions ;  nor  have  we  any  power  to  set  anything  in  motion,  unless  at  the  same  time  we  press 
upon  the  earth  with  our  feet  or,  in  order  to  get  a  better  purchase,  upon  something  that 
will  resist  them,  such  as  a  wall  opposite.  Here  then  we  have  an  induction,  that  can  be 
made  indeed  more  ample  still ;  &  from  it  we  are  bound  in  this  case  also  to  infer  that  the 
force  acts  on  each  of  the  two  bodies.  This  action  will  not  reduce  to  equality  those  two 
unequal  velocities,  unless  it  increases  that  of  the  body  which  is  in  front  &  diminishes  that 
of  the  one  which  follows.  That  is  to  say,  unless  it  produces  in  them  velocities  that  are 
opposite  in  direction  ;  &  with  these  velocities,  if  they  alone  existed,  the  bodies  would 
move  away  from  one  another.  But,  as  they  are  compounded  with  those  they  had  to  start 
with,  the  bodies  do  not  indeed  recede  from  one  another,  but  only  approach  one  another 
less  quickly  than  they  otherwise  would  have  done. 

75.  We  have  then  found  that  the  force  must  be  a  mutual  force  which  acts  in  opposite  Hence    the   force 
directions ;    one  which  from  its  very  nature  imparts  to  those  bodies  a  natural  propensity  ™pu*sive*r  ^"1^ 
for  mutual  recession  from  one  another.     Hence  a  force  of  this  kind,  from  the  very  meaning  governing  it  is  now 
of  the  term,  may  be  called  a  repulsive  force.     We  have  now  to  go  further  &  find  the  law  to  ^  found- 
that  it  follows,  &  whether,  when  the  distances  are  indefinitely  diminished,  it  attains  any 

given  measure,  or  whether  it  increases  indefinitely. 

76.  In  this  case,  in  order  that  any  sudden  change  may  be  avoided,  it  is  sufficient,  in  The    whole  differ- 
the  example  under  consideration,  if  the  repulsive  force,  to  which  our  arguments  have  led  veiocitiesWmust  *be 
us,  should  destroy  that  difference  of  6  degrees  in  the  velocities  before  the  bodies  should  destroyed  by    the 
have  come  into  immediate  contact.     Hence  they  might  possibly  at  least  come  into  contact  t°^e 

at  the  instant  in  which  they  attained  equality  between  the  velocities.     But  if  in  another 
case,  say,  the  body  that  was  behind  were  moving  with  20  degrees  of  velocity,  whilst  the 


78  PHILOSOPHL/E   NATURALIS  THEORIA 

turn  vero  ad  contactum  deveniretur  cum  differentia  velocitatum  majore,  quam  graduum  8. 
Nam  illud  itidem  amplissima  inductione  evincitur,  vires  omnes  nobis  cognitas,  quas  aliquo 
tempore  agunt,  ut  velocitatem  producant,  agere  in  ratione  temporis,  quo  agunt,  &  sui 
ipsius.  Rem  in  gravibus  oblique  descendentibus  experimenta  confirmant ;  eadem  &  in 
elastris  institui  facile  possunt,  ut  rem  comprobent ;  ac  id  ipsum  est  fundamentum  totius 
Mechanicae,  quae  inde  motuum  leges  eruit,  quas  experimenta  in  pendulis,  in  projectis 
gravibus,  in  aliis  pluribus  comprobant,  &  Astronomia  confirmat  in  caelestibus  motibus. 
Quamobrem  ilia  vis  repulsiva,  quae  in  priore  casu  extinxit  6  tantummodo  gradus  discriminis, 
si  agat  breviore  tempore  in  secundo  casu,  non  poterit  extinguere  nisi  pauciores,  minore 
nimirum  velocitate  producta  utrinque  ad  partes  contrarias.  At  breviore  utique  tempore 
aget  :  nam  cum  majore  velocitatum  discrimine  velocitas  respectiva  est  major,  ac  proinde 
accessus  celerior.  [35]  Extingueret  igitur  in  secundo  casu  ilia  vis  minus,  quam  6  discriminis 
gradus,  si  in  primo  usque  ad  contactum  extinxit  tantummodo  6.  Superessent  igitur  plures, 
quam  8  ;  nam  inter  20  &  6  erant  14,  ubi  ad  ipsum  deveniretur  contactum,  &  ibi  per  saltum 
deberent  velocitates  mutari,  ne  compenetratio  haberetur,  ac  proinde  lex  continuitatis 
violari.  Cum  igitur  id  accidere  non  possit ;  oportet,  Natura  incommodo  caverit  per 
ejusmodi  vim,  quae  in  priore  casu  aliquanto  ante  contactum  extinxerit  velocitatis  discrimen, 
ut  nimirum  imminutis  in  secundo  casu  adhuc  magis  distantiis,  vis  ulterior  illud  omne 
discrimen  auferat,  elisis  omnibus  illis  14  gradibus  discriminis,  qui  habebantur. 


Earn  vim  debere 
augeri  in  infinitum, 
imminutis,  &  qui- 
dem  in  infinitum, 
distantiis  :  habente 
virium  curva  ali- 
quam  asymptotum 
in  origine  abscissa- 
rum. 


77.  Quando  autem  hue  jam  delati  sumus,  facile  est  ulterius  progredi,  &  illud  con- 
siderare,  quod  in  secundo  casu  accidit  respectu  primi,  idem  accidere  aucta  semper  velocitate 
consequentis  corporis  in  tertio  aliquo  respectu  secundi,  &  ita  porro.  Debebit  igitur  ad 
omnem  pro  omni  casu  evitandum  saltum  Natura  cavisse  per  ejusmodi  vim,  quae  imminutis 
distantiis  crescat  in  infinitum,  atque  ita  crescat,  ut  par  sit  extinguendas  cuicunque  velocitati, 
utcunque  magnae.  Devenimus  igitur  ad  vires  repulsivas  imminutis  distantiis  crescentes 
in  infinitum,  nimirum  ad  arcum  ilium  asymptoticum  ED  curae  virium  in  fig.  i  propositum. 
Illud  quidem  ratiocinatione  hactenus  instituta  immediate  non  deducitur,  hujusmodi 
incrementa  virium  auctarum  in  infinitum  respondere  distantiis  in  infinitum  imminutis. 
Posset  pro  hisce  corporibus,  quae  habemus  prae  manibus,  quasdam  data  distantia  quascunque 
esse  ultimus  limes  virium  in  infinitum  excrescentium,  quo  casu  asymptotus  AB  non  transiret 
per  initium  distantiae  binorum  corporum,  sed  tanto  intervallo  post  ipsum,  quantus  esset 
ille  omnium  distantiarum,  quas  remotiores  particulse  possint  acquirere  a  se  invicem,  limes 
minimus  ;  sed  aliquem  demum  esse  debere  extremum  etiam  asymptoticum  arcum  curvas 
habentem  pro  asymptote  rectam  transeuntem  per  ipsum  initium  distantiae,  sic  evincitur  ; 
si  nullus  ejusmodi  haberetur  arcus  ;  particulae  materiae  minores,  &  primo  collocatae  in 
distantia  minore,  quam  esset  ille  ultimus  limes,  sive  ilia  distantia  asymptoti  ab  initio 
distantias  binorum  punctorum  materiae,  in  mutuis  incursibus  velocitatem  deberent  posse 
mutare  per  saltum,  quod  cum  fieri  nequeat,  debet  utique  aliquis  esse  ultimus  asymptoticus 
arcus,  qui  asymptotum  habeat  transeuntem  per  distantiarum  initium,  &  vires  inducat 
imminutis  in  infinitum  distantiis  crescentes  in  infinitum  ita,  ut  sint  pares  velocitati  extin- 
guendae  cuivis,  utcunque  magnae.  Ad  summum  in  curva  virium  haberi  possent  plures 
asymptotici  arcus,  alii  post  alios,  habentes  ad  exigua  intervalla  asymptotes  inter  se  parallelas, 
qui  casus  itidem  uberrimum  aperit  contemplationibus  fcecundissimis  campum,  de  quo 
aliquid  inferius ;  sed  aliquis  arcus  asympto-[36]-ticus  postremus,  cujusmodi  est  is,  quern 
in  figura  i  proposui,  haberi  omnino  debet.  Verum  ea  perquisitione  hie  omissa,  pergendum 
est  in  consideratione  legis  virium,  &  curvae  earn  exprimentis,  quae  habentur  auctis  distantiis. 


vim  in  majoribus 
tractfvam,  ^ 


78.  In  primis  gravitas  omnium  corporum  in  Terram,  quam  quotidie  experimur,  satis 
,  evmcit>  repulsionem  illam,  quam  pro  minimis  distantiis  invenimus,  non  extendi  ad  distantias 

secante    axem    in  quascunque,  sed  in  magnis  jam  distantiis  haberi  determinationem  ad  accessum,  quam  vim 
aliquo  hmite.  attractivam   nominavimus.     Quin  immo  Keplerianae  leges  in  Astronomia  tarn  feliciter  a 

Newtono  adhibitae  ad  legem  gravitatis  generalis  deducendam,  &  ad  cometas  etiam  traductas, 


A  THEORY  OF  NATURAL  PHILOSOPHY 


79 


I? 
3 


0 


O 


8o 


PHILOSOPHIC  NATURALIS  THEORIA 


o 


A  THEORY  OF  NATURAL  PHILOSOPHY  81 

body  in  front  still  had  its'  original  6  degrees  ;  then  they  would  come  into  contact  with 
a  difference  of  velocity  greater  than  8  degrees.  For,  it  can  also  be  proved  by  the  fullest 
possible  induction  that  all  forces  known  to  us,  which  act  for  any  intervals  of  time  so  as  to 
produce  velocity,  give  effects  that  are  proportional  to  the  times  for  which  they  act,  &  also 
to  the  magnitudes  of  the  forces  themselves.  This  is  confirmed  by  experiments  with  heavy 
bodies  descending  obliquely  ;  the  same  things  can  be  easily  established  in  the  case  of  springs 
so  as  to  afford  corroboration.  Moreover  it  is  the  fundamental  theorem  of  the  whole  of 
Mechanics,  &  from  it  are  derived  the  laws  of  motion  ;  these  are  confirmed  by  experiments 
with  pendulums,  projected  weights,  &  many  other  things  ;  they  are  corroborated  also  by 
astronomy  in  the  matter  of  the  motions  of  the  heavenly  bodies.  Hence  the  repulsive  force, 
which  in  the  first  case  destroyed  only  6  degrees  difference  of  velocity,  if  it  acts  for  a  shorter 
time  in  the  second  case,  will  not  be  able  to  destroy  aught  but  a  less  number  of  degrees,  as 
the  velocity  produced  in  the  two  bodies  in  opposite  directions  is  less.  Now  it  certainly 
will  act  for  a  shorter  time  ;  for,  owing  to  the  greater  difference  of  velocities,  the  relative 
velocity  is  greater  &  therefore  the  approach  is  faster.  Hence,  in  the  second  case  the  force 
would  destroy  less  than  6  degrees  of  the  difference,  if  in  the  first  case  it  had,  just  at  contact, 
destroyed  6  degrees  only.  There  would  therefore  be  more  than  8  degrees  left  over  (for, 
between  20  &  6  there  are  14)  when  contact  happened,  &  then  the  velocities  would  have 
to  be  changed  suddenly  unless  there  was  compenetration  ;  &  thereby  the  Law  of  Continuity 
would  be  violated.  Since,  then,  this  cannot  be  the  case,  Nature  would  be  sure  to  guard 
against  this  trouble  by  a  force  of  such  a  kind  as  that  which,  in  the  former  case,  extinguished 
the  difference  of  velocity  some  time  before  contact  ;  that  is  to  say,  so  that,  when  the 
distances  are  still  further  diminished  in  the  second  case,  a  further  force  eliminates  all 
that  difference,  all  of  the  14  degrees  of  difference  that  there  were  originally  being 
destroyed. 

77.  Now,  after  that  we  have  been  led  so  far,  it  is  easy  to  go  on  further  still  &  to  consider  'nie  fon:e  mus*  "*• 
that,  what  happens  in  the  second  case  when  compared  with  the  first,  will  happen  also  in  SThe  distances  Ire 
a  third  case,  in  which  the  velocity  of  the  body  that  follows  is  once  more  increased,  when  diminished,  also 
compared  with  the  second  case  ;  &  so  on,  &  so  on.  Hence,  in  order  to  guard  against  any  Sn-ve"^*6  forces  has 


sudden  change  at  all  in  every  case  whatever,  Nature  will  necessarily  have  taken  measures  an  asymptote  at  the 

for  this  purpose  by  means  of  a  force  of  such  a  kind  that,  as  the  distances  are  diminished  the  ongm 

force  increases  indefinitely,  &  in  such  a  manner  that  it  is  capable  of  destroying  any  velocity, 

however  great  it  may  be.     We  have  arrived  therefore  at  repulsive  forces  that  increase  as 

the  distances  diminish,  &  increase  indefinitely  ;   that  is  to  say,  to  the  asymptotic  arc,  ED, 

of  the  curve  of  forces  exhibited  in  Fig.  i  .     It  is  indeed  true  that  by  the  reasoning  given  so 

far  it  is  not  immediately  deduced  that  increments  of  the  forces  when  increased  to  infinity 

correspond  with  the  distances  diminished  to  infinity.     There  may  be  for  these  bodies, 

such  as  we  have  in  consideration,  some  fixed  distance  that  acts  as  a  boundary  limit  to  forces 

that    increase    indefinitely  ;    in  this  case   the   asymptote  AB  will  not    pass    through  the 

beginning  of  the  distance  between  the  two  bodies,  but  at  an  interval  after  it  as  great  as  the 

least  limit  of  all  distances  that  particles,  originally  more  remote,  might  acquire  from  one 

another.     But,  that  there  is  some  final  asymptotic  arc  of  the  curve  having  for  its  asymptote 

the  straight  line  passing  through  the  very  beginning  of  the  distance,  is  proved  as  follows. 

If  there  were  no  arc  of  this  kind,  then  the  smaller  particles  of  matter,  originally  collected 

at  a  distance  less  than  this  final  limit  would  be,  i.e.,  less  than  the  distance  of  the  asymptote 

from  the  beginning  of  the  distance  between  the  two  points  of  matter,  must  be  capable  of 

having-  their  velocities,  on  collision  with  one  another,  suddenly  changed.     Now,  as  this  is 

impossible,  then  at  any  rate  there  must  be  some  asymptotic  arc,  which  has  an  asymptote 

passing  through  the  very  beginning  of  the  distances  ;   &  this  leads  us  to  forces  that,  as  the 

distances  are  indefinitely  diminished,  increase  indefinitely  in  such  a  way  that  they  are 

capable  of  destroying  any  velocity,  no  matter  how  large  it  may  be.     In  general,  in  a  curve 

of  forces  there  may  be  several  asymptotic  arcs,  one  after  the  other,  having  at  short  intervals 

asymptotes  parallel  to  one  another  ;   &  this  case  also  opens  up  a  very  rich  field  for  fruitful 

investigations,  about  which  I  will  say  something  later.     But  there  must  certainly  be  some 

one  final  asymptotic   arc  of  the  kind  that   I   have  given  in  Fig.  i.     However,  putting 

this    investigation  on   one  side,   we   must   get   on  with  the   consideration   of  the    law 

of   forces,   &  the   curve  that  represents   them,   which  are  obtained  when   the  distances 

are  increased. 

78.  First  of  all,  the  gravitation  of  all  bodies  towards  the  Earth,  which  is  an  everyday  The  force  at  greater 
experience,  proves  sufficiently  that  the  repulsion  that  we  found  for  very  small  distances  fv^he^curve^cut- 
does  not  extend  to  all  distances  ;    but  that  at  distances  that  are  now  great  there  is  a  ting  the  axis  at 
propensity  for  approach,  which  we  have  called  an  attractive  force.     Moreover  the  Keplerian  s 
Laws  in  astronomy,  so  skilfully  employed  by  Newton  to  deduce  the  law  of  universal 
gravitation,  &  applied  even  to  the  comets,  show  perfectly  well  that  gravitation  extends, 


82  PHILOSOPHIC  NATURALIS  THEORIA 

satis  ostendunt,  gravitatem  vel  in  infinitum,  vel  saltern  per  totum  planetarium,  &  come- 
tarium  systema  extendi  in  ratione  reciproca  duplicata  distantiarum.  Quamobrem  virium 
curva  arcum  habet  aliquem  jacentem  ad  partes  axis  oppositas,  qui  accedat,  quantum  sensu 
percipi  possit,  ad  earn  tertii  gradus  hyperbolam,  cujus  ordinatae  sunt  in  ratione  reciproca 
duplicata  distantiarum,  qui  nimirum  est  ille  arcus  STV  figuras  I.  Ac  illud  etiam  hinc 
patet,  esse  aliquem  locum  E,  in  quo  curva  ejusmodi  axem  secet,  qui  sit  limes  attractionum, 
&  repulsionum,  in  quo  ab  una  ad  alteram  ex  iis  viribus  transitus  fiat. 

Plures  esse  debere,  79.  Duos  alios  nobis  indicat  limites  ejusmodi,  sive  alias  duas  intersectiones,  ut  G  &  I, 

linStes3  Pn3enomenum  vaporum,  qui  oriuntur  ex  aqua,  &  aeris,  qui  a  fixis  corporibus  gignitur  ; 
cum  in  iis  ante  nulla  particularum  repulsio  fuerit,  quin  immo  fuerit  attractio,  ob 
cohaerentiam,  qua,  una  parte  retracta,  altera  ipsam  consequebatur,  &  in  ilia  tanta  expansione, 
&  elasticitatis  vi  satis  se  manifesto  prodat  repulsio,  ut  idcirco  a  repulsione  in  minimis  distantiis 
ad  attractionem  alicubi  sit  itum,  turn  inde  iterum  ad  repulsionem,  &  iterum  inde  ad  generalis 
gravitatis  attractiones.  Effervescentiae,  &  fermentationes  adeo  diversae,  in  quibus  cum 
adeo  diversis  velocitatibus  eunt,  ac  redeunt,  &  jam  ad  se  invicem  accedunt,  jam  recedunt 
a  se  invicem  particulae,  indicant  utique  ejusmodi  limites,  atque  transitus  multo  plures  ; 
sed  illos  prorsus  evincunt  substantise  molles,  ut  cera,  in  quibus  compressiones  plurimse 
acquiruntur  cum  distantiis  admodum  adversis,  in  quibus,  tamen  omnibus  limites  haberi 
debent ;  nam,  anteriore  parte  ad  se  attracta,  posteriores  earn  sequuntur,  eadem  propulsa, 
illae  recedunt,  distantiis  ad  sensum  non  mutatis,  quod  ob  illas  repulsiones  in  minimis 
distantiis,  quae  contiguitatem  impediunt,  fieri  alio  modo  non  potest,  nisi  si  limites  ibidem 
habeantur  in  iis  omnibus  distantiis  inter  attractiones,  &  repulsiones,  quae  nimirum  requi- 
runtur  ad  hoc,  ut  pars  altera  alteram  consequatur  retractam,  vel  prsecedat  propulsam. 


Hinc    tota   curvae  80.  Habentur  igitur  plurimi  limites,  &  plurimi  flexus  curvse  hinc,  &  inde  ab  axe  prseter 

a°yroptotLm&  bTu-  ^uos  arcus>  quorum  prior  ED  in  infinitum  protenditur,  &  asymptoticus  est,  alter  STV, 
ribus  flexibus,  ac  [37]  si  gravitas  generalis  in  infinitum  protenditur,  est  asymptoticus  itidem,  &  ita  accedit 
ad  crus  illud  hyperbolae  gradus  tertii,  ut  discrimen  sensu  percipi  nequeat  :  nam  cum  ipso 
penitus  congruere  omnino  non  potest ;  non  enim  posset  ab  eodem  deinde  discedere,  cum 
duarum  curvarum,  quarum  diversa  natura  est,  nulli  arcus  continui,  utcunque  exigui,  possint 
penitus  congruere,  sed  se  tantummodo  secare,  contingere,  osculari  possint  in  punctis 
quotcunque,  &  ad  se  invicem  accedere  utcumque.  Hinc  habetur  jam  tota  forma  curvae 
virium,  qualem  initio  proposui,  directa  ratiocinatione  a  Naturae  phsenomenis,  &  genuinis 
principiis  deducta.  Remanet  jam  determinanda  constitutio  primorum  elementorum 
materiae  ab  iis  viribus  deducta,  quo  facto  omnis  ilia  Theoria,  quam  initio  proposui,  patebit, 
nee  erit  arbitraria  quaedam  hypothesis,  ac  licebit  progredi  ad  amovendas  apparentes  quasdam 
difHcultates,  &  ad  uberrimam  applicationem  ad  omnem  late  Physicam  qua  exponendam, 
qua  tantummodo,  ne  hoc  opus  plus  aequo  excrescat,  indicandam. 


Hinc  elementorum  81.  Quoniam,  imminutis  in  infinitum  distantiis,  vis  repulsiva  augetur  in  infinitum  ; 

m  facile  patet,  nullam  partem  materias  posse  esse  contiguam  alteri  parti  :  vis  enim  ilia  repulsiva 


carens 


partibus.  protinus  alteram  ab  altera  removeret.     Quamobrem  necessario  inde  consequitur,  prima 

materiae  elementa  esse  omnino  simplicia,  &  a  nullis  contiguis  partibus  composita.  Id 
quidem  immediate,  &  necessario  fluit  ex  ilia  constitutione  virium,  quae  in  minimis  distantiis 
sunt  repulsivae,  &  in  infinitum  excrescunt. 

Soiutio  objectionis  82.  Objicit  hie  fortasse  quispiam  illud,  fieri  posse,  ut  particulae  primigenias  materias 

petitaeex  eo  quod  sjnt  compOsitae  quidem,  sed  nulla  Naturae  vi  divisibiles  a  se  invicem,  quarum  altera  tota 

vires     repulsivas  r.       .      ^       .  ....          .    .     .      ,.  ..  i  •  i 

habere  possent  non  respectu  altenus  totius  habeat  vires  illas  in  minimis  distantiis  repulsivas,  vel  quarum  pars 
puncta  smguia,  se  qu3evis  respectu  reliquarum  partium  eiusdem  particulae  non  solum  nullam  habeat  repulsivam 

particulae    primi-    T-.  1,1  '-.,,  J.r  ,.  ,.,.  r   .  . 

geniae.  vim,  sed  habeat  maximam  illam  attractivam,  qua;  ad  ejusmodi  cohaesionem  requintur  : 

eo  pacto  evitari  debere  quemvis  immediatum  impulsum,  adeoque  omnem  saltum,  &  con- 
tinuitatis  laesionem.  At  in  primis  id  esset  contra  homogeneitatem  materiae,  de  qua  agemus 
infra  :  nam  eadem  materiae  pars  in  iisdem  distantiis  respectu  quarundam  paucissimarum 
partium,  cum  quibus  particulam  suam  componit,  haberet  vim  repulsivam,  respectu  autem 


A  THEORY  OF  NATURAL  PHILOSOPHY  83 

either  to  infinity  or  at  least  to  the  limits  of  the  system  including  all  the  planets  &  comets, 
in  the  inverse  ratio  of  the  squares  of  the  distances.  Hence  the  curve  will  have  an  arc 
lying  on  the  opposite  side  of  the  axis,  which,  as  far  as  can  be  perceived  by  our  senses, 
approximates  to  that  hyperbola  of  the  third  degree,  of  which  the  ordinates  are  in  the  inverse 
ratio  of  the  squares  of  the  distances ;  &  this  indeed  is  the  arc  STV  in  Fig.  i.  Now  from 
this  it  is  evident  that  there  is  some  point  E,  in  which  a  curve  of  this  kind  cuts  the  axis  ; 
and  this  is  a  limit-point  for  attractions  and  repulsions,  at  which  the  passage  from  one  to 
the  other  of  these  forces  is  made. 

79.  The  phenomenon  of  vapour  arising  from  water,  &  that  of  gas   produced   from  There  are  bound  to 
fixed  bodies  lead  us  to  admit  two  more  of  these  limit-points,  i.e.,  two  other  intersections,  ^Syof'tiSep^ 
say,  at  G  &  I.      Since  in  these   there  would  be  initially  no  repulsion,  nay  rather  there  sages,  with  corre- 
would  be  an  attraction  due  to  cohesion,  by  which,  when  one  part  is  retracted,  another  1"6  hmit 
generally  followed  it  :    &  since  in   the    former,   repulsion  is   clearly   evidenced  by  the 

greatness  of  the  expansion,  &  by  the  force  of  its  elasticity ;  it  therefore  follows  that 
there  is,  somewhere  or  other,  a  passage  from  repulsion  at  very  small  distances  to  attraction, 
then  back  again  to  repulsion,  &  from  that  back  once  more  to  the  attractions  of  universal 
gravitation.  Effervescences  &  fermentations  of  many  different  kinds,  in  which  the 
particles  go  &  return  with  as  many  different  velocities,  &  now  approach  towards  & 
now  recede  from  one  another,  certainly  indicate  many  more  of  these  limit-points  & 
transitions.  But  the  existence  of  these  limit-points  is  perfectly  proved  by  the  case  of 
soft  substances  like  wax  ;  for  in  these  substances  a  large  number  of  compressions  are  acquired 
with  very  different  distances,  yet  in  all  of  these  there  must  be  limit-points.  For,  if  the 
front  part  is  drawn  out,  the  part  behind  will  follow  ;  or  if  the  former  is  pushed  inwards, 
the  latter  will  recede  from  it,  the  distances  remaining  approximately  unchanged.  This,  on 
account  of  the  repulsions  existing  at  very  small  distances,  which  prevent  contiguity,  can- 
not take  place  in  any  way,  unless  there  are  limit-points  there  in  all  those  distances  between 
attractions  &  repulsions ;  namely,  those  that  are  requisite  to  account  for  the  fact  that  one 
part  will  follow  the  other  when  the  latter  is  drawn  out,  &  will  recede  in  front  of  the 
latter  when  that  is  pushed  in. 

80.  Therefore  there  are  a  large  number  of  limit-points,  &  a  large  number  of  flexures  Hence  we  get  the 
on  the  curve,  first  on  one  side  &  then  on  the  other  side  of  the  axis,  in  addition  to  two  whole  for™hof  t^e 
arcs,  one  of  which,  ED,  is  continued  to  infinity  &  is  asymptotic,  &  the  other,  STV,  is  asymptotes,  many 
asymptotic  also,  provided  that  universal  gravitation  extends  to  infinity.     It  approximates  flexures    &   many 

J    ir    j-  r      i       i  r      i         i  •    i     i  -11  111  i       intersections     with 

to  the  form  of  the  hyperbola  of  the  third  degree  mentioned  above  so  closely  that  the  the  axis, 
difference  from  it  is  imperceptible ;  but  it  cannot  altogether  coincide  with  it,  because,  in 
that  case  it  would  never  depart  from  it.  For,  of  two  curves  of  different  nature,  there 
cannot  be  any  continuous  arcs,  no  matter  how  short,  that  absolutely  coincide  ;  they  can 
only  cut,  or  touch,  or  osculate  one  another  in  an  indefinitely  great  number  of  points,  & 
approximate  to  one  another  indefinitely  closely.  Thus  we  now  have  the  whole  form  of 
the  curve  of  forces,  of  the  nature  that  I  gave  at  the  commencement,  derived  by  a  straight- 
forward chain  of  reasoning  from  natural  phenomena,  &  sound  principles.  It  only  remains 
for  us  now  to  determine  the  constitution  of  the  primary  elements  of  matter,  derived  from 
these  forces ;  £:  in  this  manner  the  whole  of  the  Theory  that  I  enunciated  at  the  start 
will  become  quite  clear,  &  it  will  not  appear  to  be  a  mere  arbitrary  hypothesis.  We 
can  proceed  to  remove  certain  apparent  difficulties,  &  to  apply  it  with  great  profit  to 
the  whole  of  Physics  in  general,  explaining  some  things  fully  &,  to  prevent  the  work 
from  growing  to  an  unreasonable  size,  merely  mentioning  others. 

81.  Now,  because  the  repulsive  force  is  indefinitely  increased  when  the  distances  are  The  simplicity   of 
indefinitely  diminished,  it  is  quite  easy  to  see  clearly  that  no  part  of  matter  can  be  contiguous  ments^oT^att^r " 
to  any  other  part ;    for  the  repulsive  force  would  at  once  separate  one  from  the  other,  they  are  altogether 
Therefore  it  necessarily  follows  that  the  primary  elements  of  matter  are  perfectly  simple,  w^110"*  Parts. 

&  that  they  are  not  composed  of  any  parts  contiguous  to  one  another.  This  is  an 
immediate  &  necessary  deduction  from  the  constitution  of  the  forces,  which  are  repulsive 
at  very  small  distances  &  increase  indefinitely. 

82.  Perhaps  someone  will  here  raise  the  objection  that  it  may  be  that  the  primary  Solution  of  the  ob- 
particles  of  matter  are  composite,  but  that  they  cannot  be  disintegrated  by  any  force  in  jnetlo^SSertiond  that 
Nature;    that  one  whole  with  regard  to  another  whole  may  possibly  have  those  forces  single  points  can- 
that  are  repulsive  at  very  small  distances,  whilst  any  one  part  with  regard  to  any  other  part  ?OTces,a™u7Pt  h'at 
of  the  same  particle  may  not  only  have  no  repulsive  force,  but  indeed  may  have  a  very  primary     particles 
great  attractive  force  such  as  is  required  for  cohesion  of  this  sort ;    that,  in  this  way,  we  can  have  them- 
are  bound  to  avoid  all  immediate  impulse,  &  so  any  sudden  change  or  breach  of  continuity. 

But,  in  the  first  place,  this  would  be  in  opposition  to  the  homogeneity  of  matter,  which 
we  will  consider  later  ;  for  the  same  part  of  matter,  at  the  same  distances  with  regard  to 
those  very  few  parts,  along  with  which  it  makes  up  the  particle,  would  have  a  repulsive 


84 


PHILOSOPHISE  NATURALIS  THEORIA 


aliarum  omnium  attractivam  in  iisdem  distantiis,  quod  analogic  adversatur.  Deinde  si  a 
Deo  agente  supra  vires  Naturae  sejungerentur  illas  partes  a  se  invicem,  turn  ipsius  Naturae 
vi  in  se  invicem  incurrerent ;  haberetur  in  earum  collisione  saltus  naturalis,  utut  praesup- 
ponens  aliquid  factum  vi  agente  supra  Naturam.  Demum  duo  turn  cohaesionum  genera 
deberent  haberi  in  Natura  admodum  diversa,  alterum  per  attractionem  in  minimis  distantiis, 
alterum  vero  longe  alio  pacto  in  elementarium  particularum  massis,  nimirum  per  limites 
cohaesionis ;  adeoque  multo  minus  simplex,  &  minus  uniformis  evaderet  Theoria. 


An  elementa    sint   [38] 
extensa :  argumen- 
ta  pro  virtual!  eor- 
um  extensione. 


83.  Simplicitate  &  incompositione  elementorum  defmita,  dubitari  potest,  an  ea 
sint  etiam  inextensa,  an  aliquam,  utut  simplicia,  extensionem  habeant  ejus  generis,  quam 
virtualem  extensionem  appellant  Scholastici.  Fuerunt  enim  potissimum  inter  Peripateticos, 
qui  admiserint  elementa  simplicia,  &  carentia  partibus,  atque  ex  ipsa  natura  sua  prorsus 
indivisibilia,  sed  tamen  extensa  per  spatium  divisibile  ita,  ut  alia  aliis  ma  jus  etiam  occupent 
spatium,  ac  eo  loco,  quo  unum  stet,  possint,  eo  remote,  stare  simul  duo,  vel  etiam  plura  ; 
ac  sunt  etiamnum,  qui  ita  sentiant.  Sic  etiam  animam  rationalem  hominis  utique  prorsus 
indivisibilem  censuerunt  alii  per  totum  corpus  diffusam  :  alii  minori  quidem  corporis  parti, 
sed  utique  parti  divisibili  cuipiam,  &  extensae,  praesentem  toti  etiamnum  arbitrantur. 
Deum  autem  ipsum  praesentem  ubique  credimus  per  totum  utique  divisibile  spatium, 
quod  omnia  corpora  occupant,  licet  ipse  simplicissimus  sit,  nee  ullam  prorsus  compositionem 
admittat.  Videtur  autem  sententia  eadem  inniti  cuidam  etiam  analogiae  loci,  ac  temporis. 
Ut  enim  quies  est  conjunctio  ejusdem  puncti  loci  cum  serie  continua  omnium  moment- 
orum  ejus  temporis,  quo  quies  durat  :  sic  etiam  ilia  virtualis  extensio  est  conjunctio  unius 
momenti  temporis  cum  serie  continua  omnium  punctorum  spatii,  per  quod  simplex  illud 
ens  virtualiter  extenditur ;  ut  idcirco  sicut  ilia  quies  haberi  creditur  in  Natura,  ita  &  haec 
virtualis  extensio  debeat  admitti,  qua  admissa  poterunt  utique  ilia  primse  materiae  elementa 
esse  simplicia,  &  tamen  non  penitus  inextensa. 


Exciuditur    virtu- 


rite  appiicato. 


84.  At  ego  quidem  arbitror,  hanc  itidem  sententiam  everti  penitus  eodem  inductionis 
principio,  ex  quo  alia  tarn  multa  hucusque,  quibus  usi  sumus,  deduximus.  Videmus  enim 
in  his  corporibus  omnibus,  quae  observare  possumus,  quidquid  distinctum  occupat  locum, 
distinctum  esse  itidem  ita,  ut  etiam  satis  magnis  viribus  adhibitis  separari  possint,  quae 
diversas  occupant  spatii  partes,  nee  ullum  casum  deprehendimus,  in  quo  magna  haec  corpora 
partem  aliquam  habeant,  quae  eodem  tempore  diversas  spatii  partes  occupet,  &  eadem 
sit.  Porro  haec  proprietas  ex  natura  sua  ejus  generis  est,  ut  aeque  cadere  possit  in 
magnitudines,  quas  per  sensum  deprehendimus,  ac  in  magnitudines,  quae  infra  sensuum 
nostrorum  limites  sunt  ;  res  nimirum  pendet  tantummodo  a  magnitudine  spatii,  per  quod 
haberetur  virtualis  extensio,  quae  magnitudo  si  esset  satis  ampla,  sub  sensus  caderet.  Cum 
igitur  nunquam  id  comperiamus  in  magnitudinibus  sub  sensum  cadentibus,  immo  in 
casibus  innumeris  deprehendamus  oppositum  :  debet  utique  res  transferri  ex  inductionis 
principio  supra  exposito  ad  minimas  etiam  quasque  materiae  particulas,  ut  ne  illae  quidem 
ejusmodi  habeant  virtualem  extensionem. 


Responsioadexem-  [39]  85.  Exempla,  quae  adduntur,  petita  ab  anima  rational},  &  ab  omnipraesentia 
plum  anima  &  Dei.  j)ej}  nj^  positive  evincunt,  cum  ex  alio  entium  genere  petita  sint  ;  praeterquam  quod  nee 
illud  demonstrari  posse  censeo,  animam  rationalem  non  esse  unico  tantummodo,  simplici, 
&  inextenso  corporis  puncto  ita  praesentem,  ut  eundem  locum  obtineat,  exerendo  inde 
vires  quasdam  in  reliqua  corporis  puncta  rite  disposita,  in  quibus  viribus  partim  necessariis, 
&  partim  liberis,  stet  ipsum  animae  commercium  cum  corpore.  Dei  autem  praesentia 
cujusmodi  sit,  ignoramus  omnino  ;  quem  sane  extensum  per  spatium  divisibile  nequaquam 
dicimus,  nee  ab  iis  modis  omnem  excedentibus  humanum  captum,  quibus  ille  existit, 
cogitat,  vult,  agit,  ad  humanos,  ad  materiales  existendi,  agendique  modos,  ulla  esse  potest 
analogia,  &  deductio. 


itidem   ad   analo-  86.  Quod  autem  pertinet  ad  analogiam  cum  quiete,  sunt  sane  satis  valida  argumenta, 

giam  cum  quiete.     quibus,  ut  supra  innui,  ego  censeam,  in  Natura  quietem  nullam  existere.     Ipsam  nee  posse 


A  THEORY  OF  NATURAL  PHILOSOPHY  85 

» 

force  ;  but  it  would  have  an  attractive  force  with  regard  to  all  others,  at  the  very  same 
distances ;  &  this  is  in  opposition  to  analogy.  Secondly,  if,  due  to  the  action  of  GOD 
surpassing  the  forces  of  Nature,  those  parts  are  separated  from  one  another,  then  urged 
by  the  forces  of  Nature  they  would  rush  towards  one  another  ;  &  we  should  have,  from 
their  collision,  a  sudden  change  appertaining  to  Nature,  although  conveying  a  presumption 
that  something  was  done  by  the  action  of  a  supernatural  force.  Lastly,  with  this  idea, 
there  would  have  to  be  two  kinds  of  cohesion  in  Nature  that  were  altogether  different  in 
constitution  ;  one  due  to  attraction  at  very  small  distances,  &  the  other  coming  about 
in  a  far  different  way  in  the  case  of  masses  of  elementary  particles,  that  is  to  say,  due  to 
the  limit-points  of  cohesion.  Thus  a  theory  would  result  that  is  far  less  simple  &  less 
uniform  than  mine. 

83.  Taking    it    for    granted,  then,  that  the  elements  are  simple   &   non-composite,  whether  the  ele- 
there  can  be  no  doubt  as  to  whether  they  are  also  non-extended  or  whether,  although  ments  are  extended; 

,  ,      ,        ,  .'  ,  1-1  •          i          V      certain    arguments 

simple,  they  have  an  extension  of   the  kind   that   is   termed  virtual    extension   by  the  m  favour  of  virtual 

Scholastics.     For  there  were  some,  especially  among  the  Peripatetics,  who  admitted  elements  extension. 

that  were  simple,  lacking   in    all    parts,  &   from  their  very  nature  perfectly  indivisible  ; 

but,  for  all  that,  so  extended  through  divisible  space  that  some  occupied  more  room  than 

others ;    &  such  that  in  the  position  once  occupied  by  one  of  them,  if  that  one  were 

removed,  two  or  even  more  others  might  be  placed  at  the  same  time  ;   &  even  now  there 

are  some  who  are  of  the  same  opinion.     So  also  some  thought  that  the  rational  soul  in 

man,  which  certainly  is  altogether  indivisible,  was  diffused  throughout  the  whole  of  the 

body  ;    whilst  others  still  consider  that  it  is  present  throughout  the  whole  of,  indeed,  a 

smaller    part    of    the  body,  but  yet    a  part  that  is  at   any  rate  divisible  &  extended. 

Further  we  believe  that  GOD  Himself  is  present  everywhere  throughout  the  whole  of  the 

undoubtedly  divisible  space  that  all  bodies  occupy ;   &  yet  He  is  onefold  in  the  highest 

degree  &  admits  not  of  any  composite  nature  whatever.     Moreover,  the  same  idea  seems 

to  depend  on  an  analogy  between  space  &  time.     For,  just  as  rest  is  a  conjunction  with 

a  continuous  series  of  all  the  instants  in  the  interval  of  time  during  which  the  rest  endures ; 

so  also  this  virtual  extension  is  a  conjunction  of  one  instant  of  time  with  a  continuous  series 

of  all  the  points  of  space  throughout  which  this  one-fold  entity  extends  virtually.     Hence, 

just  as  rest  is  believed  to  exist  in  Nature,  so  also  are  we  bound  to  admit  virtual  extension  ; 

&  if  this  is  admitted,  then  it  will  be  possible  for  the  primary  elements  of  matter  to  be 

simple,  &  yet  not  absolutely  non-extended. 

84.  But  I  have  come  to  the  conclusion  that  this  idea  is  quite  overthrown  by  that  same  virtual    extension 
principle  of  induction,  by  which  we  have  hitherto  deduced  so  many  results  which  we  have  isr .excluded^ by  the 
employed.     For  we  see,  in  all   those  bodies  that  we  can  bring  under  observation,  that  auction6  correctly 
whatever  occupies  a  distinct  position  is  itself  also  a  distinct  thing  ;  so  that  those  that  occupy  aPPlied- 
different  parts  of  space  can  be  separated  by  using  a  sufficiently  large  force ;    nor  can  we 

detect  a  case  in  which  these  larger  bodies  have  any  part  that  occupies  different  parts  of 
space  at  one  &  the  same  time,  &  yet  is  the  same  part.  Further,  this  property  by  its  very 
nature  is  of  the  sort  for  which  it  is  equally  probable  that  it  happens  in  magnitudes  that  we  can 
detect  by  the  senses  &  in  magnitudes  which  are  below  the  limits  of  our  senses.  In  fact, 
the  matter  depends  only  upon  the  size  of  the  space,  throughout  which  the  virtual  extension 
is  supposed  to  exist ;  &  this  size,  if  it  were  sufficiently  ample,  would  become  sensible 
to  us.  Since  then  we  never  find  this  virtual  extension  in  magnitudes  that  fall  within  the 
range  of  our  senses,  nay  rather,  in  innumerable  cases  we  perceive  the  contrary  ;  the  matter 
certainly  ought  to  be  transferred  by  the  principle  of  induction,  as  explained  above,  to 
any  of  the  smallest  particles  of  matter  as  well ;  so  that  not  even  they  are  admitted  to  have 
such  virtual  extension. 

85.  The  illustrations  that  are  added,  derived  from  a  consideration  of  the  rational  Reply   to    the 
soul  &  the  omnipresence  of  GOD,  prove  nothing  positively ;    for  they  are  derived  from  s^uf&'cot)6  ' 
another  class  of  entities,  except  that,  I  do  not  think  that  it  can  even  be  proved  that  the 

rational  soul  does  not  exist  in  merely  a  single,  simple,  &  non-extended  point  of  the  body ; 
so  that  it  maintains  the  same  position,  &  thence  it  puts  forth  some  sort  of  force  into  the 
remaining  points  of  the  body  duly  disposed  about  it ;  &  the  intercommunication  between 
the  soul  &  the  body  consists  of  these  forces,  some  of  which  are  involuntary  whilst  others 
are  voluntary.  Further,  we  are  absolutely  ignorant  of  the  nature  of  the  presence  of  GOD  ; 
&  in  no  wise  do  we  say  that  He  is  really  extended  throughout  divisible  space  ;  nor  from 
those  modes,  surpassing  all  human  intelligence,  by  which  HE  exists,  thinks,  wills  &  acts, 
can  any  analogy  or  deduction  be  made  which  will  apply  to  human  or  material  modes  of 
existence  &  action. 

86.  Again,  as  regards  the  analogy  with  rest,  we  have  arguments  that  are  sufficiently  Again  with  regard 

IT  T  i     j      i  i  i_          •  t  ^v        •      vr  ..     '    to  the  analogy  with 

strong  to  lead  us  to  believe,  as  I  remarked  above,  that  there  is  no  such  thing  m  Nature  rest. 
as  absolute  rest.     Indeed,  I  proved  that  such  a  thing  could  not  be,  by  a  direct  argument 


86  PHILOSOPHISE  NATURALIS  THEORIA 

existere,  argumento  quodam  positive  ex  numero  combinationum  possibilium  infinite 
contra  alium  finitum,  demonstravi  in  Stayanis  Supplementis,  ubi  de  spatio,  &  tempore 
quae  juxta  num.  66  occurrent  infra  Supplementorum  §  i,  &  §  2  ;  numquam  vero  earn 
existere  in  Natura,  patet  sane  in  ipsa  Newtoniana  sententia  de  gravitate  generali,  in  qua  in 
planetario  systemate  ex  mutuis  actionibus  quiescit  tantummodo  centrum  commune  gravi- 
tatis,  punctum  utique  imaginarium,  circa  quod  omnia  planetarum,  cometarumque  corpora 
moventur,  ut  &  ipse  Sol ;  ac  idem  accidit  fixis  omnibus  circa  suorum  systematum  gravitatis 
centra  ;  quin  immo  ex  actione  unius  systematis  in  aliud  utcunque  distans,  in  ipsa  gravitatis 
centra  motus  aliquis  inducetur  ;  &  generalius,  dum  movetur  quaecunque  materiae  particula, 
uti  luminis  particula  qusecunque  ;  reliquae  omnes  utcunque  remotae,  quas  inde  positionem 
ab  ilia  mutant,  mutant  &  gravitatem,  ac  proinde  moventur  motu  aliquo  exiguo,  sed  sane 
motu.  In  ipsa  Telluris  quiescentis  sententia,  quiescit  quidem  Tellus  ad  sensum,  nee  tota 
ab  uno  in  alium  transfertur  locum  ;  at  ad  quamcunque  crispationem  maris,  rivuli  decursum, 
muscae  volatum,  asquilibrio  dempto,  trepidatio  oritur,  perquam  exigua  ilia  quidem,  sed 
ejusmodi,  ut  veram  quietem  omnino  impediat.  Quamobrem  analogia  inde  petita  evertit 
potius  virtualem  ejusmodi  simplicium  elementorum  extensionem  positam  in  conjunctione 
ejusdem  momenti  temporis  cum  serie  continua  punctorum  loci,  quam  comprobet. 


in  quo  deficiat  ana-  87.  Sed  nee  ea  ipsa  analogia,  si  adesset,  rem  satis  evinceret ;  cum  analogiam  inter  tempus, 

logia  loci,  &  tem-  £  locum  videamus  in  aliis  etiam  violari  :  nam  in  iis  itidem  paragraphis  Supplementorum 
demonstravi,  nullum  materiae  punctum  unquam  redire  ad  punctum  spatii  quodcunque, 
in  quo  semel  fuerit  aliud  materiae  punctum,  ut  idcirco  duo  puncta  materiae  nunquam 
conjungant  idem  [40]  punctum  spatii  ne  cum  binis  quidem  punctis  temporis,  dum  quam- 
plurima  binaria  punctorum  materiae  conjungunt  idem  punctum  temporis  cum  duobus 
punctis  loci ;  nam  utique  coexistunt  :  ac  praeterea  tempus  quidem  unicam  dimensionem 
habet  diuturnitatis,  spatium  vero  habet  triplicem,  in  longum,  latum,  atque  profundum. 

inextensio  utilis  88.  Quamobrem  illud  jam  tuto  inferri  potest,  haec  primigenia  materiae  elementa,  non 

ad     exciudendum  soium  esse  simplicia,  ac  indivisibilia,  sed  etiam  inextensa.     Et  quidem  haec  ipsa  simplicitas, 

transitum    momen-  ,        t  ;  .  i      •  «i_  j  ji_ 

taneum  a  densitate  &  inextensio  elementorum  praestabit  commoda  sane  plunma,  quibus  eadem  adnuc  magis 
nuiia  ad  summam.  fuicitur,  ac  comprobatur.  Si  enim  prima  elementa  materiae  sint  quaedam  partes  solidse, 
ex  partibus  compositae,  vel  etiam  tantummodo  extensae  virtualiter,  dum  a  vacuo  spatio 
motu  continue  pergitur  per  unam  ejusmodi  particulam,  fit  saltus  quidam  momentaneus 
a  densitate  nulla,  quae  habetur  in  vacuo,  ad  densitatem  summam,  quae  habetur,  ubi  ea 
particula  spatium  occupat  totum.  Is  vero  saltus  non  habetur,  si  elementa  simplicia  sint, 
&  inextensa,  ac  a  se  invicem  distantia.  Turn  enim  omne  continuum  est  vacuum  tantum- 
modo, &  in  motu  continue  per  punctum  simplex  fit  transitus  a  vacuo  continue  ad  vacuum 
continuum.  Punctum  illud  materiae  occupat  unicum  spatii  punctum,  quod  punctum 
spatii  est  indivisibilis  limes  inter  spatium  praecedens,  &  consequens.  Per  ipsum  non 
immoratur  mobile  continue  motu  delatum,  nee  ad  ipsum  transit  ab  ullo  ipsi  immediate 
proximo  spatii  puncto,  cum  punctum  puncto  proximum,  uti  supra  diximus,  nullum  sit  ; 
sed  a  vacuo  continue  ad  vacuum  continuum  transitur  per  ipsum  spatii  punctum  a  materiae 
puncto  occupatum. 


itidem  ad  hoc,  ut  go,.  Accedit,  quod  in  sententia  solidorum,  extensorumque  elementorum  habetur  illud, 

possit,  ut  p"test  densitatem  corporis  minui  posse  in  infinitum,  augeri  autem  non  posse,  nisi  ad  certum  limitem 
minui  in  infinitum.  in  quo  increment!  lex  necessario  abrumpi  debeat.  Primum  constat  ex  eo,  quod  eadem 
particula  continua  dividi  possit  in  particulas  minores  quotcunque,  quae  idcirco  per  spatium 
utcunque  magnum  diffundi  potest  ita,  ut  nulla  earum  sit,  quae  aliquam  aliam  non  habeat 
utcunque  libuerit  parum  a  se  distantem.  Atque  eo  pacto  aucta  mole,  per  quam 
eadem  ilia  massa  diffusa  sit,  eaque  aucta  in  ratione  quacunque  minuetur  utique 
densitas  in  ratione  itidem  utcunque  magna.  Patet  &  alterum  :  ubi  enim  omnes 
particulae  ad  contactum  devenerint ;  densitas  ultra  augeri  non  poterit.  Quoniam 
autem  determinata  quaedam  erit  utique  ratio  spatii  vacui  ad  plenum,  nonnisi  in  ea  ratione 
augeri  poterit  densitas,  cujus  augmentum,  ubi  ad  contactum  deventum  fuerit,  adrumpetur. 
At  si  elementa  sint  puncta  penitus  indivisibilia,  &  inextensa  ;  uti  augeri  eorum  distantia 
poterit  in  infinitum,  ita  utique  poterit  etiam  minui  pariter  in  ratione  quacunque  ;  cum 


A  THEORY  OF  NATURAL  PHILOSOPHY  87 

founded  upon  the  infiniteness  of  a  number  of  possible  combinations  as  against  the  finiteness 
of  another  number,  in  the  Supplements  to  Stay's  Philosophy,  in  connection  with  space 
&  time ;  these  will  be  found  later  immediately  after  Art.  14  of  the  Supplements,  §§  I 
and  II.  That  it  never  does  exist  in  Nature  is  really  clear  in  the  Newtonian  theory  of 
universal  gravitation  ;  according  to  this  theory,  in  the  planetary  system  the  common  centre 
of  gravity  alone  is  at  rest  under  the  action  of  the  mutual  forces ;  &  this  is  an  altogether 
imaginary  point,  about  which  all  the  bodies  of  the  planets  &  comets  move,  as  also  does 
the  sun  itself.  Moreover  the  same  thing  happens  in  the  case  of  all  the  fixed  stars  with  regard 
to  the  centres  of  gravity  of  their  systems ;  &  from  the  action  of  one  system  on  another 
at  any  distance  whatever  from  it,  some  motion  will  be  imparted  to  these  very  centres  of 
gravity.  More  generally,  so  long  as  any  particle  of  matter,  so  long  as  any  particle  of  light, 
is  in  motion,  all  other  particles,  no  matter  how  distant,  which  on  account  of  this  motion 
have  their  distance  from  the  first  particle  altered,  must  also  have  their  gravitation  altered, 
&  consequently  must  move  with  some  very  slight  motion,  but  yet  a  true  motion.  In 
the  idea  of  a  quiescent  Earth,  the  Earth  is  at  rest  approximately,  nor  is  it  as  a  whole  translated 
from  place  to  place  ;  but,  due  to  any  tremulous  motion  of  the  sea,  the  downward  course 
of  rivers,  even  to  the  fly's  flight,  equilibrium  is  destroyed  &  some  agitation  is  produced, 
although  in  truth  it  is  very  slight ;  yet  it  is  quite  enough  to  prevent  true  rest  altogether. 
Hence  an  analogy  deduced  from  rest  contradicts  rather  than  corroborates  virtual  extension 
of  the  simple  elements  of  Nature,  on  the  hypothesis  of  a  conjunction  of  the  same  instant 
of  time  with  a  continuous  series  of  points  of  space. 

87.  But  even  if  the  foregoing  analogy  held  good,  it  would  not  prove  the  matter  Where  the  analogy 
satisfactorily  ;  since  we  see  that  in  other  ways  the  analogy  between  space  &  time  is  impaired.  2^pace  and  tlme 
For  I  proved,  also  in  those  paragraphs  of  the  Supplements  that  I  have  mentioned,  that 

no  point  of  matter  ever  returned  to  any  point  of  space,  in  which  there  had  once  been  any 
other  point  of  matter ;  so  that  two  points  of  matter  never  connected  the  same  point  of 
space  with  two  instants  of  time,  let  alone  with  more ;  whereas  a  huge  number  of  pairs  of 
points  connect  the  same  instant  of  time  with  two  points  of  space,  since  they  certainly  coexist. 
Besides,  time  has  but  one  dimension,  duration  ;  whilst  space  has  three,  length,  breadth 
&  depth. 

88.  Therefore  it  can  now  be  safely  accepted  that  these  primary  elements  of  matter  Non-extension  use- 
are  not    only  simple    &   indivisible,  but  also  that    they  are  non-extended.     Indeed  this  aun   \nstanTaneous 
very  simplicity  &  non-extension  of  the  elements  will  prove  useful  in  a  really  large  number  passage  from  •  no  • 
of   cases   for   still   further   strengthening   &  corroborating  the  results  already  obtained.  J^-one.0  a  Very 
For  if  the  primary  elements  were  certain  solid  parts,  themselves  composed  of  parts  or  even 

virtually  extended  only,  then,  whilst  we  pass  by  a  continuous  motion  from  empty  space 
through  one  particle  of  this  kind,  there  would  be  a  sudden  change  from  a  density  that  is 
nothing  when  the  space  is  empty,  to  a  density  that  is  very  great  when  the  particle  occupies 
the  whole  of  the  space.  But  there  is  not  this  sudden  change  if  we  assume  that  the  elements 
are  simple,  non-extended  &  non-adjacent.  For  then  the  whole  of  space  is  merely  a 
continuous  vacuum,  &,  in  the  continuous  motion  by  a  simple  point,  the  passage  is  made 
from  continuous  vacuum  to  continuous  vacuum.  The  one  point  of  matter  occupies  but 
one  point  of  space  ;  &  this  point  of  space  is  the  indivisible  boundary  between  the  space 
that  precedes  &  the  space  that  follows.  There  is  nothing  to  prevent  the  moving  point 
from  being  carried  through  it  by  a  continuous  motion,  nor  from  passing  to  it  from  any 
point  of  space  that  is  in  immediate  proximity  to  it  :  for,  as  I  remarked  above,  there 
is  no  point  that  is  the  next  point  to  a  given  point.  But  from  continuous  vacuum 
to  continuous  vacuum  the  passage  is  made  through  that  point  of  space  which  is  occupied 
by  the  point  of  matter. 

89.  There  is  also  the  point,  that  arises  in  the  theory  of  solid  extended  elements,  namely  Also  for  the  idea 
that  the  density  of  a  body  can  be  diminished  indefinitely,  but  cannot  be  increased  except  j^^a'^ ^can 
up  to  a  certain  fixed  limit,  at  which  the  law  of  increase  must  be  discontinuous.     The  first  be    decreased, 
comes  from  the  fact  that  this  same  continuous  particle  can  be  divided  into  any  number  mdefinltely- 

of  smaller  particles  ;  these  can  be  diffused  through  space  of  any  size  in  such  a  way  that 
there  is  not  one  of  them  that  does  not  have  some  other  one  at  some  little  (as  little  as  you 
will)  distance  from  itself.  In  this  way  the  volume  through  which  the  same  mass  is  diffused 
is  increased  ;  &  when  that  is  increased  in  any  ratio  whatever,  then  indeed  the  density 
will  be  diminished  in  the  same  ratio,  no  matter  how  great  the  ratio  may  be.  The  second 
thing  is  also  evident ;  for  when  the  particles  have  come  into  contact,  the  density  cannot 
be  increased  any  further.  Moreover,  since  there  will  undoubtedly  be  a  certain  determinate 
ratio  for  the  amount  of  space  that  is  empty  compared  with  the  amount  of  space  that  is 
full,  the  density  can  only  be  increased  in  that  ratio ;  &  the  regular  increase  of  density 
will  be  arrested  when  contact  is  attained.  But  if  the  elements  are  points  that  are  perfectly 
indivisible  &  non-extended,  then,  just  as  their  distances  can  be  increased  indefinitely, 


88  PHILOSOPHIC  NATURALIS  THEORIA 

in  [41]  ratione  quacunque  lineola  quaecunque  secari  sane  possit  :    adeoque  uli  nullus  est 
limes  raritatis  auctae,  ita  etiam  nullus  erit  auctae  densitatis. 

Et  ad  excludendum  9°-  Sed  &  illud  commodum  accidet,  quod  ita  omne  continuum  coexistens  eliminabitur 

continuum    extcn-  e  Natura,  in  quo  explicando  usque  adeo  dcsudarunt,  &  fere  incassum,  Philosophi,  ncc  idcirco 

sum,  &  in  infinitum    j«    •  •        «          **•  •     •      •     r    •  j       •  •  i  •  i  • 

in  existentibus.  divisio  ulla  realis  entis  in  innmtum  produci  potent,  nee  naerebitur,  ubi  quaeratur,  an  numerus 
partium  actu  distinctarum,  &  separabilium,  sit  finitus,  an  infinitus  ;  nee  alia  ejusmodi 
sane  innumera,  quae  in  continui  compositione  usque  adeo  negotium  facessunt  Philosophis, 
jam  habebuntur.  Si  enim  prima  materiae  elementa  sint  puncta  penitus  inextensa,  & 
indivisibilia,  a  se  invicem  aliquo  intervallo  disjuncta  ;  jam  erit  finitus  punctorum  numerus 
in  quavis  massa  :  nam  distantiae  omnes  finitae  erunt ;  infinitesimas  enim  quantitates  in  se 
determinatas  nullas  esse,  satis  ego  quidem,  ut  arbitror,  luculcnter  demonstravi  &  in  disser- 
tatione  De  Natura,  t$  Usu  infinitorum,  ac  infinite  parvorum,  &  in  dissertatione  DC  Lege 
Continuitatis,  &  alibi.  Intervallum  quodcunque  finitum  erit,  &  divisibile  utique  in 
infinitum  per  interpositionem  aliorum,  atque  aliorum  punctorum,  quae  tamen  singula, 
ubi  fuerint  posita,  finita  itidem  erunt,  &  aliis  pluribus,  finitis  tamen  itidem,  ubi  extiterint, 
locum  reliquent,  ut  infinitum  sit  tantummodo  in  possibilibus,  non  autem  in  existentibus, 
in  quibus  possibilibus  ipsis  omnem  possibilium  seriem  idcirco  ego  appellare  soleo  constantem 
terminis  finitis  in  infinitum,  quod  quaecunque,  quae  existant,  finita  esse  debeant,  sed  nullus 
sit  existentium  finitus  numerus  ita  ingens,  ut  alii,  &  alii  majores,  sed  itidem  finiti,  haberi 
non  possint,  atque  id  sine  ullo  limite,  qui  nequeat  praeteriri.  Hoc  autcm  pacto,  sublato 
ex  existentibus  omni  actuali  infinite,  innumerae  sane  difficultates  auferentur. 


inextensionem  91.  Cum  igitur  &  positive  argumento.  a  lege  virium  positive  demonstrata  desumpto, 

qua'rend^m^e  simplicitas,  &  inextensio  primorum  materiae  elementorum  deducatur,  £  tam  multis  aliis 

homogeneitate.         vel  indiciis  fulciatur,  vel  emolumentis  inde  derivatis  confirmetur  ;    ipsa  itidem  admitti 

jam  debet,  ac  supererit  quaerendum  illud  tantummodo,  utrum  haec  elementa  homogenca 

censeri  debeant,  &  inter  se  prorsus  similia,  ut  ea  initio  assumpsimus,  an  vero  heterogenea, 

ac  dissimilia. 

Homogeneitatem  92.  Pro  homogeneitate  primorum  materiae  elementorum  illud  est  quoddani  veluti 

genefta1teaprim°i!n(&  Prmcipium,  quod  in  simplicitate,  &  inextensione  conveniant,  ac  etiam  vires  quasdam  habeant 
uitimi  asymptotici  utique  omnia.  Deinde  curvam  ipsam  virium  eandem  esse  omnino  in  omnibus  illud  indicat, 
omnibus'0  P"'  S  ve^  etiani  evincit,  quod  primum  crus  repulsivum  impenetrabilitatem  secum  trahens,  & 
postremum  attractivum  gravitatem  definiens,  omnino  communia  in  omnibus  sint  :  nam 
corpora  omnia  aeque  impenetrabilia  sunt,  &  vero  etiam  aeque  gravia  pro  quantitate  materiae 
suae,  uti  satis  [42]  evincit  aequalis  velocitas  auri,  &  plumse  cadentis  in  Boyliano  recipiente 
Si  reliquus  curvae  arcus  intermedius  esset  difformis  in  diversis  materiae  punctis  ;  infinities 
probabilius  esset,  difformitatem  extendi  etiam  ad  crus  primum,  &  ultimum,  cum  infinities 
plures  sint  curvae,  quae,  cum  in  reliquis  differant  partibus,  differant  plurimum  etiam  in 
hisce  extremis,  quam  quae  in  hisce  extremis  tantum  modo  tam  arete  consentiant.  Et  hoc 
quidem  argumento  illud  etiam  colligitur,  curvam  virium  in  quavis  directione  ab  eodem 
primo  materiae  elemento,  nimirum  ab  eodem  materiae  puncto  eandem  esse,  cum  &  primum 
impenetrabilitatis,  &  postremum  gravitatis  crus  pro  omnibus  directionibus  sit  ad  sensum 
idem.  Cum  primum  in  dissertatione  De  Firibus  Vivis  hanc  Theoriam  protuli,  suspicabar 
diversitatem  legis '  virium  respondentis  diversis  directionibus ;  sed  hoc  argumento  adi 
majorem  simplicitatem,  &  uniformitatem  deinde  adductus  sum.  Diversitas  autem  legum 
virium  pro  diversis  particulis,  &  pro  diversis  respectu  ejusdem  particulae  directionibus, 
habetur  utique  ex  diverso  numero,  &  positione  punctorum  earn  componentium,  qua  de 
re  inferius  aliquid. 


i  contra  deduci          93-  Nee   vero   huic   homogeneitati    opponitur   inductionis    principium,    quo    ipsam 
ex  principio  indis-  Leibnitiani  oppugnare  solent,  nee  principium  rationis  sufficients,  atque  indiscernibilium, 

cermbUium,  &  rati-  .     •          °  .  T    £    •  TV-    •    •   /"•       j-        •  -j 

onis  sufficients.       quod  supenus  innui  numero  3.     Innmtam  Divini  v_onditons  mentem,  ego  quidem  omnino. 
arbitror,  quod  &  tam  multi  Philosophi  censuerunt,  ejusmodi  perspicacitatem  habere,  atque 
intuitionem  quandam,  ut  ipsam  etiam,  quam  individuationem  appellant,  omnino  similium 
individuorum  cognoscat,  atque  ilia  inter  se  omnino  discernat.     Rationis  autem  sufficientis 


A  THEORY  OF  NATURAL  PHILOSOPHY  89 

so  also  can  they  just  as  well  be  diminished  in  any  ratio  whatever.  For  it  is  certainly  possible 
that  a  short  line  can  be  divided  into  parts  in  any  ratio  whatever ;  &  thus,  just  as  there 
is  no  limit  to  increase  of  rarity,  so  also  there  is  none  to  increase  of  density. 

qo.  The  theory  of  non-extension  is  also  convenient  for  eliminating  from  Nature  all  ^lso-/or  excludms 

7  /  1     •  1    •     1  1    •!  1  1  Ml  11  1  6    *"ea   °      a     C011" 

idea  of  a  coexistent  continuum — to  explain  which  philosophers  have  up  till  now  laboured  tinuum  in  existing 


so  very  hard  &  generally  in  vain.     Assuming  non-extension,  no  division  of  a  real  entity  thmRs-    that 


can  be  carried  on  indefinitely ;  we  shall  not  be  brought  to  a  standstill  when  we  seek  to 
find  out  whether  the  number  of  parts  that  are  actually  distinct  &  separable  is  finite  or 
infinite  ;  nor  with  it  will  there  come  in  any  of  those  other  truly  innumerable  difficulties 
that,  with  the  idea  of  continuous  composition,  have  given  so  much  trouble  to  philosophers. 
For  if  the  primary  elements  of  matter  are  perfectly  non-extended  &  indivisible  points 
separated  from  one  another  by  some  definite  interval,  then  the  number  of  points  in  any 
given  mass  must  be  finite  ;  because  all  the  distances  are  finite.  I  proved  clearly  enough, 
I  think,  in  the  dissertation  De  Natura,  &  Usu  infinitorum  ac  infinite  parvorum,  &  in  the 
dissertation  De  Lege  Continuitatis,  &  in  other  places,  that  there  are  no  infinitesimal 
quantities  determinate  in  themselves.  Any  interval  whatever  will  be  finite,  &  at  least 
divisible  indefinitely  by  the  interpolation  of  other  points,  &  still  others  ;  each  such  set 
however,  when  they  have  been  interpolated,  will  be  also  finite  in  number,  &  leave  room 
for  still  more  ;  &  these  too,  when  they  existed,  will  also  be  finite  in  number.  So  that 
there  is  only  an  infinity  of  possible  points,  but  not  of  existing  points ;  &  with  regard 
to  these  possible  points,  I  usually  term  the  whole  series  of  possibles  a  series  that  ends  at 
finite  limits  at  infinity.  This  for  the  reason  that  any  of  them  that  exist  must  be  finite 
in  number  ;  but  there  is  no  finite  number  of  things  that  exist  so  great  that  other  numbers, 
greater  &  greater  still,  but  yet  all  finite,  cannot  be  obtained  ;  &  that  too  without  any 
limit,  which  cannot  be  surpassed.  Further,  in  this  way,  by  doing  away  with  all  idea  of 
an  actual  infinity  in  existing  things,  truly  countless  difficulties  are  got  rid  of. 

91.  Since  therefore,  by  a  direct  argument  derived  from  a  law  of  forces  that  has  been  Non-extension 
directly  proved,  we  have  both  deduced  the  simplicity  &  non-extension  of  the  primary  w"5  havea  now e  to 
elements  of  matter,  &  also  we  have  strengthened  the  theory  by  evidence  pointing  towards  investigate    homo- 
it,  or  corroborated  it  by  referring  to  the  advantages  to  be  derived  from  it ;    this  theory  gen 

ought  now  to  be  accepted  as  true.  There  only  remains  the  investigation  as  to  whether 
these  elements  ought  to  be  considered  to  be  homogeneous  &  perfectly  similar  to  one 
another,  as  we  assumed  at  the  start,  or  whether  they  are  really  heterogeneous  &  dissimilar. 

92.  In  favour  of  the  homogeneity  of  the  primary  elements  of  matter  we  have  so  to  Homogeneity   for 
speak  some  foundation  derived  from  the  fact  that  all  of  them  agree  in  simplicity  &  non-  Voca!tedStf0romaa 
extension,  &  also  that  they  are  all  endowed  with  forces  of   some  sort.     Now,  that  this  consideration    of 
curve  of  forces  is  exactly  the  same  for  all  of  them  is  indicated  or  even  proved  by  the  fact  Of  6the  °fir?t86™  last 
that  the  first  repulsive  branch  necessitating  impenetrability,  &  the  last  attractive  branch  a  s  y"m  p  t  o  t  i  c 
determining  gravitation,  are  exactly  the  same  in  all  respects.     For  all  bodies  are  equally  c^l  S  forces* 
impenetrable ;  &  also   all   are   equally  heavy  in   proportion   to   the   amount   of   matter 

contained  in  them,  as  is  sufficiently  proved  by  the  equal  velocity  of  the  piece  of  gold  & 
the  feather  when  falling  in  Boyle's  experiment.  If  the  remaining  intermediate  arc  of  the 
curve  were  non-uniform  for  different  points  of  matter,  it  would  be  infinitely  more  probable 
that  the  non-uniformity  would  extend  also  to  the  first  &  last  branches  also  ;  for  there 
are  infinitely  more  curves  which,  when  they  differ  in  the  remaining  parts,  also  differ  to 
the  greatest  extent  in  the  extremes,  than  there  are  curves,  which  agree  so  closely  only  in 
these  extremes.  Also  from  this  argument  we  can  deduce  that  the  curve  of  forces  is  indeed 
exactly  the  same  from  the  same  point  of  matter,  in  any  direction  whatever  from  the  same 
primary  element  of  matter  ;  for  both  the  first  branch  of  impenetrability  &  the  last  branch 
of  gravitation  are  the  same,  so  far  as  we  can  perceive,  for  all  directions.  When  I  first 
published  this  Theory  in  my  dissertation  De  Firibus  Fivis,  I  was  inclined  to  believe  that 
there  was  a  diversity  in  the  law  of  forces  corresponding  to  diversity  of  direction  ;  but  I 
was  led  by  the  argument  given  above  to  the  greater  simplicity  &  the  greater  uniformity 
derived  therefrom.  Further,  diversity  of  the  laws  of  forces  for  diverse  particles,  &  for 
different  directions  with  the  same  particle,  is  certainly  to  be  obtained  from  the  diverse 
number  &  position  of  the  points  composing  it ;  about  which  I  shall  have  something 
to  say  later. 

93.  Nor  indeed  is  there  anything  opposed  to  this  idea  of  homogeneity  to  be  derived  Notl?ins  t?  b« 

r  i  •      •    i        r  •     i  J  i        o      rr  t  o  /_  .        brought  against 

from  the  principle  of  induction,  by  means  of  which  the  followers  of  Leibniz  usually  raise  this  from  the  doc- 
an  objection  to  it ;   nor  from  the  principle  of  sufficient  reason,  &  of  indiscernibles,  that  fc™es°f  .indj.scern: 

J    .          ,       .  .        .  -rr  •     i       i          •  -TO  i_        ibles  &     sufficient 

1    mentioned   above   in  Art.  3.     I    am   indeed   quite   convinced,  &  a  great  many  other  reason.1 
philosophers  too  have  thought,   that  the  Infinite   Will  of    the   Divine   Founder  has   a 
perspicacity  &  an  intuition  of   such  a  nature  that  it  takes  cognizance  of   that  which  is 
called    individuation    amongst     individuals    that    are    perfectly    similar,    &    absolutely 


90 


PHILOSOPHIC  NATURALIS  THEORIA 


principium  falsum  omnino  esse  censeo,  ac  ejusmodi,  ut  omnem  verse  libertatis  ideam  omnino 
tollat  ;  nisi  pro  ratione,  ubi  agitur  de  voluntatis  determinatione,  ipsum  liberum  arbitrium, 
ipsa  libera  determinatio  assumatur,  quod  nisi  fiat  in  voluntate  divina,  quaccunque  existunt, 
necessario  existunt,  &  qusecunque  non  existunt,  ne  possibilia  quidem  erunt,  vera  aliqua 
possibilitate,  uti  facile  admodum  demonstratur  ;  quod  tamen  si  semel  admittatur,  mirum 
sane,  quam  prona  demum  ad  fatalem  necessitatem  patebit  via.  Quamobrem  potest  divina 
voluntas  determinari  ex  toto  solo  arbitrio  suo  ad  creandum  hoc  individuum  potius,  quam 
illud  ex  omnibus  omnino  similibus,  &  ad  ponendum  quodlibet  ex  iis  potius  eo  loco,  quo 
ponit,  quam  loco  alterius.  Sed  de  rationis  sufficientis  principio  haec  ipsa  fusius  pertractavi 
turn  in  aliis  locis  pluribus,  turn  in  Stayanis  Supplementis,  ubi  etiam  illud  ostendi,  id  prin- 
cipium nullum  habere  usum  posse  in  iis  ipsis  casibus,  in  quibus  adhibetur,  &  praedicari  solet 
tantopere,  atque  id  idcirco,  quod  nobis  non  innotescant  rationes  omnes,  quas  tamen 
oporteret  utique  omnes  nosse  ad  hoc,  ut  eo  principio  uti  possemus,  amrmando,  nullam 
esse  rationem  sufncientem  pro  hoc  potius,  quam  pro  illo  [43]  alio  :  sane  in  exemplo  illo 
ipso,  quod  adhiberi  solet,  Archimedis  hoc  principio  aequilibrium  determinantis,  ibidem 
ostendi,  ex  ignoratione  causarum,  sive  rationum,  quse  postea  detectae  sunt,  ipsum  in  suae 
investigationis  progressu  errasse  plurimum,  deducendo  per  abusum  ejus  principii  sphsericam 
figuram  marium,  ac  Telluris. 


combinatiombus. 


Posse  etiam  puncta  94.  Accedit  &  illud,  quod  ilia  puncta  materiae,  licet  essent  prorsus  similia  in  simplicitate, 

dlfierrTin  aiiis  11S>  &  extensione,  ac  mensura  virium,  pendentium  a  distantia,  possent  alias  habere  proprietates 

metaphysicas  diversas  inter  se,  nobis  ignotas,  quae  ipsa  etiam  apud  ipsos    Leibnitianos 

discriminarent. 

Non  vaierehicprin-  95.  Quod  autem  attinet  ad  inductionem,  quam  Leibnitiani  desumunt  a  dissimilitudine, 

a^ma^sis^eas^de!  quam  observamus  in  rebus  omnibus,  cum  nimirum  nusquam  ex.  gr.  in  amplissima  silva  reperire 
ferre  ex  diversis  sit  duo  folia  prorsus  similia  ;  ea  sane  me  nihil  movet  ;  cum  nimirum  illud  discrimen  sit 
prOprietas  relativa  ad  rationem  aggregati,  &  nostros  sensus,  quos  singula  materiae  elementa 
non  afficiunt  vi  sufficiente  ad  excitandam  in  animo  ideam,  nisi  multa  sint  simul,  &  in  molem 
majorem  excrescant.  Porro  scimus  utique  combinationes  ejusdem  numeri  terminorum 
in  immensum  excrescere,  si  ille  ipse  numerus  sit  aliquanto  major.  Solis  24  litterulis 
Alphabet!  diversimodo  combinatis  formantur  voces  omnes,  quibus  hue  usque  usa  sunt 
omnia  idiomata,  quae  extiterunt,  &  quibus  omnia  ilia,  quae  possunt  existere,  uti  possunt. 
Quid  si  numerus  earum  existeret  tanto  major,  quanto  major  est  numerus  puuctorum 
materiae  in  quavis  massa  sensibili  ?  Quod  ibi  diversus  est  litterarum  diversarum  ordo,  id 
in  punctis  etiam  prorsus  homogeneis  sunt  positiones,  &  distantia,  quibus  variatis,  variatur 
utique  forma,  &  vis,  qua  sensus  afficitur  in  aggregatis.  Quanto  major  est  numerus 
combinationum  diversarum  possibilium  in  massis  sensibilibus,  quam  earum  massarum,  quas 
possumus  observare,  &  inter  se  conferre  (qui  quidem  ob  distantias,  &  directiones  in  infinitum 
variabiles  praescindendo  ab  aequilibrio  virium,  est  infinitus,  cum  ipso  aequilibrio  est  immen- 
sus)  ;  tanto  major  est  improbabilitas  duarum  massarum  omnino  similium,  quam  omnium 
aliquantisper  saltern  inter  se  dissimilium. 


Physica  ratio  dis-  96.  Et  quidem  accedit  illud  etiam,  quod  alicujus  dissimilitudinis  in  aggregatis  physicam 

massU1ut1in1foriiuUS  I1100!116  rationem  cernimus  in  iis  etiam  casibus,  in  quibus  maxime  inter  se  similia  esse 
deberent.  Cum  enim  mutuae  vires  ad  distantias  quascunque  pertineant ;  status  uniuscu- 
jusque  puncti  pendebit  saltern  aliquantisper  a  statu  omnium  aliorum  punctorum,  quae 
sunt  in  Mundo.  Porro  utcunque  puncta  quaedam  sint  parum  a  se  invicem  remota,  uti 
sunt  duo  folia  in  eadem  silva,  &  multo  magis  in  eodem  ramo ;  adhuc  tamen  non  eandem 
prorsus  relationem  distantiae,  &  virium  habent  ad  reliqua  omnia  materiae  puncta,  quae 
[44]  sunt  in  Mundo,  cum  non  eundem  prorsus  locum  obtineant ;  &  inde  jam  in  aggregate 
discrimen  aliquod  oriri  debet,  quod  perfectam  similitudinem  omnino  impediat.  Sed  illud 
earn  inducit  magis,  quod  quae  maxime  conferunt  ad  ejusmodi  dispositionem,  necessario 
respectu  diversarum  frondium  diversa  non  nihil  esse  debeant.  Omissa  ipsa  earum  forma 
in  semine,  solares  radii,  humoris  ad  nutritionem  necessarii  quantitas,  distantia,  a  qua  debet 
is  progredi,  ut  ad  locum  suum  deveniat,  aura  ipsa,  &  agitatio  inde  orta,  non  sunt  omnino 
similia,  sed  diversitatem  aliquam  habent,  ex  qua  diversitas  in  massas  inde  efformatas 
redundat. 


A  THEORY  OF  NATURAL  PHILOSOPHY  91 

distinguishes  them  one  from  the  other.  Moreover,  I  consider  that  the  principle  of  sufficient 
reason  is  altogether  false,  &  one  that  is  calculated  to  take  away  all  idea  of  true  freewill. 
Unless  free  choice  or  free  determination  is  assumed  as  the  basis  of  argument,  in  discussing 
the  determination  of  will,  unless  this  is  the  case  with  the  Divine  Will,  then,  whatever 
things  exist,  exist  because  they  must  do  so,  &  whatever  things  do  not  exist  will  not  even 
be  possible,  i.e.,  with  any  real  possibility,  as  is  very  easily  proved.  Nevertheless,  once  this 
idea  is  accepted,  it  is  truly  wonderful  how  it  tends  to  point  the  way  finally  to  fatalistic 
necessity.  Hence  the  Divine  Will  is  able,  of  its  own  pleasure  alone,  to  be  determined 
to  the  creation  of  one  individual  rather  than  another  out  of  a  whole  set  of  exactly  similar 
things,  &  to  the  setting  of  any  one  of  these  in  the  place  in  which  it  puts  it  rather  than  in 
the  place  of  another.  But  I  have  discussed  these  very  matters  more  at  length,  besides  several 
other  places,  in  the  Supplements  to  Stay's  Philosophy ;  where  I  have  shown  that  the 
principle  cannot  be  employed  in  those  instances  in  which  it  is  used  &  generally  so  strongly 
asserted.  The  reason  being  that  all  possible  reasons  are  not  known  to  us ;  &  yet  they 
should  certainly  be  known,  to  enable  us  to  employ  the  principle  by  stating  that  there  is 
no  sufficient  reason  in  favour  of  this  rather  than  that  other.  In  truth,  in  that  very  example 
of  the  principle  generally  given,  namely,  that  of  Archimedes'  determination  of  equilibrium 
by  means  of  it,  I  showed  also  that  Archimedes  himself  had  made  a  very  big  mistake  in  following 
out  his  investigation  because  of  his  lack  of  knowledge  of  causes  or  reasons  that  were  discovered 
in  later  days,  when  he  deduced  a  spherical  figure  for  the  seas  &  the  Earth  by  an  abuse 
of  this  principle. 

94.  There  is  also  this,  that  these  points  of  matter,  although  they  might  be  perfectly  it  is   possible   for 
similar  as  regards  simplicity  &  extension,  &  in  having  the  measure  of  their  forces  depen-  ^"^ese^ro  erties 
dent  on  their  distances,  might  still  have  other  metaphysical  properties  different  from  one  but  to  disagree  in 
another,  &    unknown  to    us ;    &    these  distinctions   also    are  made    by  the  followers   of  others- 
Leibniz. 

95.  As  regards  the  induction  which  the  followers  of  Leibniz  make  from  the  lack  of  The  principle  does 
similitude  that  we  see  in  all  things,  (for  instance  such  as  that  there  never  can  be  found  in  n°t.hold  g°°d  here 

T_      i  j  i  i        vi    \       i     •  i  •  .        ,        °*   induction   from 

the  largest  wood  two   leaves  exactly  alike),  their  argument  does  not   impress  me  in  the  masses;  they  differ 

slightest  degree.     For  that  distinction  is  a  property  that  is  concerned  with  reasoning  for  °.n  account   of 

an  aggregate,  &  also  with  our  senses  ;   &  these  senses   single  elements  of  matter  cannot  tionsof  their parta. 

influence  with  sufficient  force  to  excite  an  idea  in  the  mind,  except  when  there  are  many 

of  them  together  at  a  time,  &  they  develop  into  a  mass  of  considerable  size.     Further 

it  is  well  known  that  combinations  of  the  same  number  of  terms  increase  enormously,  if 

that  number  itself  increase  a  little.      From  the  24  letters  of  the  alphabet  alone,  grouped 

together  in  different  ways,  are  formed  all  the  words  that  have  hitherto  been  used  in  all 

expressions  that  have  existed,  or  can  possibly  come  into  existence.     What  then  if  their 

number  were  increased  to  equal  the  number  of  points  of  matter  in  any  sensible  mass  ? 

Corresponding  to  the  different  order  of  the  several  letters  in  the  one,  we  have  in  perfectly 

homogeneous  points  also  different  positions  &  distances ;   &  if    these  are  altered  at  least 

the  form  &  the  force,  which  affect  our  senses  in  the  groups,  are  altered  as  well.     How 

much  greater  is  the  number  of  different  combinations  that  are  possible  in  sensible  masses 

than  the  number  of  those  masses  that  we  can  observe  &  compare  with  one  another  (& 

this  number,  on  account  of  the  infinitely  variable   distances  &  directions  of  the  forces, 

when  equilibrium  is  precluded,  is  infinite,  since  including  equilibrium  it  is  very  great) ; 

just  so  much  greater  is  the  improbability  of  two   masses   being  exactly  similar  than  of 

their  being  all  at  least  slightly  different  from  one  another. 

96.  There  is  also  this  point  in  addition  ;   we  discern  a  physical  reason  as  well  for  some  Physical  reason  for 
dissimilarity  in  groups  for  those  cases  too,  in  which  they  ought  to  be  especially  similar  to  the    difference   in 

.1  -n          •  i    f  •  11  -11       T  r   i        '  several  masses,   as 

one  another,  ror  since  mutual  forces  pertain  to  all  possible  distances,  the  state  of  any  in  leaves. 
one  point  will  depend  upon,  at  least  in  some  slight  degree,  the  state  of  all  other  points 
that  are  in  the  universe.  Further,  however  short  the  distance  between  certain  points  may 
be,  as  of  two  leaves  in  the  same  wood,  much  more  so  on  the  same  branch,  still  for  all 
that  they  do  not  have  quite  the  same  relation  as  regards  distance  &  forces  as  all  the  rest 
of  the  points  of  matter  that  are  in  the  universe,  because  they  do  not  occupy  quite  the 
same  place.  Hence  in  a  group  some  distinction  is  bound  to  arise  which  will  entirely  prevent 
perfect  similarity.  Moreover  this  tendency  is  all  the  stronger,  because  those  things  which 
especially  conduce  to  this  sort  of  disposition  must  necessarily  be  somewhat  different  with 
regard  to  different  leaves.  For  the  form  itself  being  absent  in  the  seed,  the  rays  of  the 
sun,  the  quantity  of  moisture  necessary  for  nutrition,  the  distance  from  which  it  has  to 
proceed  to  arrive  at  the  place  it  occupies,  the  air  itself  &  the  continual  motion  derived 
from  this,  these  are  not  exactly  similar,  but  have  some  diversity ;  &  from  this  diversity 
there  proceeds  a  diversity  in  the  masses  thus  formed. 


92  PHILOSOPHIC  NATURALIS  THEORIA 

simiiitudine  quaii-  97.  Patet  igitur,  varietatem  illam  a  numero  pendere  combinationum  possibilium  in 

^  numero  punctorum  necessario  ad  sensationem,  &  circumstantiarum,  quae  ad  formationem 


geneitatem,    quam  massze  sunt  neccssariae,  adeoque  ejusmodi  inductionem  extend!  ad  elementa  non    posse. 

*  ' 


Quin  immo  ilia  tanta  similitude,  quae  cum  exigua  dissimilitudine  commixta  invenitur  in 
tarn  multis  corporibus,  indicat  potius  similitudinem  ingentem  in  elementis.  Nam  ob 
tantum  possibilium  combinationum  numerum,  massae  elementorum  etiam  penitus  homo- 
geneorum  debent  a  se  invicem  differre  plurimum,  adeoque  si  elementa  heterogenea  sint, 
in  immensum  majorem  debent  habere  dissimilitudinem,  quam  ipsa  prima  elementa,  ex 
quibus  idcirco  nullae  massas,  ne  tantillum  quidem,  similes  provenire  deberent.  Cum 
elementa  multo  minus  dissimilia  esse  debeant,  quam  aggregata  elementorum,  multo 
magis  ad  elementorum  homogeneitatem  valere  debet  ilia  quaecunque  similitudo,  quam 
in  corporibus  observamus,  potissimum  in  tarn  multis,  quae  ad  eandem  pertinent  speciem, 
quam  ad  homogeneitatem  eorundem  tarn  exiguum  illud  discrimen,  quod  in  aliis  tarn 
multis  observatur.  Rem  autem  penitus  conficit  ilia  tanta  similitudo,  qua  superius  usi 
sumus,  in  primo  crure  exhibente  impenetrabilitatem,  &  in  postremo  exhibente  gravitatem 
generalem,  quae  crura  cum  ob  hasce  proprietates  corporibus  omnibus  adeo  generales,  adeo 
inter  se  in  omnibus  similia  sint,  etiam  reliqui  arcus  curvae  exprimentis  vires  omnimodam 
similitudinem  indicant  pro  corporibus  itidem  omnibus. 

Homogeneitatem  98.  Superest,  quod  ad  hanc  rem  pertinet,  illud  unum  iterum  hie  monendum,  quod 

insinuarr'  ^xem*  ipsum  etiam  initio  hujus  Operis  innui,  ipsam  Naturam,  &  ipsum  analyseos  ordinem  nos 

plum  a  libris,  lit-  ducere    ad    simplicitatem  &  homogeneitatem  elementorum,  cum  nimirum,  quo  analysis 

ns>  pul  promovetur  magis,  eo  ad  pauciora,  &  inter  se  minus  discrepantia  principia  deveniatur,  uti 

patet  in  resolutionibus  Chemicis.     Quam  quidem  rem  ipsum  litterarum,  &  vocum  exemplum 

multo  melius  animo  sistet.     Fieri  utique  possent  nigricantes  litteras,  non  ductu  atramenti 

continue,  sed  punctulis  rotundis  nigricantibus,  &  ita  parum  a  se  invicem  remotis,  ut  inter- 

valla  non  nisi  ope  microscopii  discerni  possent,  &  quidem  ipsae  litterarum  formae  pro  typis 

fieri  pos-[45]-sent  ex  ejusmodi  rotundis  sibi  proximis  cuspidibus  constantes.     Concipiatur 

ingens  quaedam  bibliotheca,  cujus  omnes  libri  constent  litteris  impressis,  ac  sit  incredibilis 

in  ea  multitude  librorum  conscriptorum  linguis  variis,  in  quibus  omnibus  forma  charac- 

terum  sit  eadem.     Si  quis  scripturae  ejusmodi,  &  linguarum  ignarus  circa  ejusmodi  libros, 

quos  omnes  a  se  invicem  discrepantes  intueretur,  observationem  institueret  cum  diligenti 

contemplatione  ;    primo  quidem  inveniret  vocum  farraginem  quandam,   quae   voces   in 

quibusdam  libris  occurrerent  saepe,  cum  eaedem  in  aliis  nusquam  apparent,  &  inde  lexica 

posset  quaedam  componere  totidem  numero,  quot  idiomata  sunt,  in  quibus  singulis  omnes 

ejusdem  idiomatis  voces  reperirentur,  quae  quidem  numero  admodum  pauca  essent,  discri- 

mine  illo  ingenti  tot,  tarn  variorum  librorum  redacto  ad  illud  usque  adeo  minus  discrimen, 

quod  contineretur  lexicis  illis,  &  haberetur  in  vocibus  ipsa  lexica  constituentibus.     At 

inquisitione   promota,   facile   adverteret,    omnes   illas   tarn   varias   voces   constare   ex    24 

tantummodo  diversis  litteris,  discrimen  aliquod  inter  se  habentibus  in  ductu  linearum, 

quibus  formantur,  quarum  combinatio  diversa  pareret  omnes  illas  voces  tarn  varias,  ut 

earum  combinatio  libros  efformaret  usque  adeo  magis  a  se  invicem  discrepantes.     Et  ille 

quidem  si  aliud  quodcunque  sine  microscopic  examen  institueret,  nullum  aliud  inveniret 

magis  adhuc  simile  elementorum  genus,  ex  quibus  diversa  ratione  combinatis  orirentur 

ipsae  litterse  ;    at  microscopic  arrepto,  intueretur  utique  illam  ipsam  litterarum  composi- 

tionem  e  punctis    illis    rotundis  prorsus    homogeneis,  quorum    sola    diversa    positio,  ac 

distributio  litteras  exhiberet. 


Appiicatio  exempli  99.  Haec  mihi  quaedam  imago  videtur  esse  eorum,  quae  cernimus  in  Natura.     Tarn 

a<^  Naturae  analy-  mu\t{}  tam  var;j  fift  ijbrj  corpora  sunt,  &  quae  ad  diversa  pertinent  regna,  sunt  tanquam 
diversis  conscripta  linguis.  Horum  omnium  Chemica  analysis  principia  quaedam  invenit 
minus  inter  se  difrormia,  quam  sint  libri,  nimirum  voces.  Hae  tamen  ipsae  inter  se  habent 
discrimen  aliquod,  ut  tam  multas  oleorum,  terrarum,  salium  species  eruit  Chemica  analysis 
e  diversis  corporibus.  Ulterior  analysis  harum,  veluti  vocum,  litteras  minus  adhuc  inter 
se  difformes  inveniret,  &  ultima  juxta  Theoriam  meam  deveniret  ad  homogenea  punctula, 
quae  ut  illi  circuli  nigri  litteras,  ita  ipsa  diversas  diversorum  corporum  particulas  per  solam 
dispositionem  diversam  efformarent  :  usque  adeo  analogia  ex  ipsa  Naturae  consideratione 


A  THEORY  OF  NATURAL  PHILOSOPHY  93 

97.  It  is  clear  then  that  this  variety  depends  on  the  number  of  possible  combinations  Homogeneity  is  to 
to  be  found  for  the  number  of   points  that  are  necessary  to  make  the  mass  sensible,  &  ^m  d<^° ™ort *  ot 
of  the  circumstances  that  arenecessary    for  the  formation  of  the  mass ;    &  so  it  is  not  similitude  in  some 
possible  that  the  induction  should  be  extended  to  the  elements.     Nay  rather,  the  great  heterogeneity  from" 
similarity  that  is  found  accompanied  by  some  very  slight  dissimilarity  in  so  many  bodies  dissimilarity. 
points  more  strongly  to  the  greatest  possible  similarity  of  the  elements.     For  on  account 

of  the  great  number  of  the  possible  combinations,  even  masses  of  elements  that  are  perfectly 
homogeneous  must  be  greatly  different  from  one  another  ;  &  thus  if  the  elements  are 
heterogeneous,  the  masses  must  have  an  immensely  greater  dissimilarity  than  the  primary 
elements  themselves ;  &  therefore  no  masses  formed  from  these  ought  to  come  out  similar, 
not  even  in  the  very  slightest  degree.  Since  the  elements  are  bound  to  be  much  less 
dissimilar  than  aggregates  formed  from  these  elements,  homogeneity  of  the  elements  must 
be  indicated  by  that  certain  similarity  that  we  observe  in  bodies,  especially  in  so  many 
of  those  that  belong  to  the  same  species,  far  more  strongly  than  heterogeneity  of  the  elements 
is  indicated  by  the  slight  differences  that  are  observed  in  so  many  others.  The  whole 
discussion  is  made  perfectly  complete  by  that  great  similarity,  which  we  made  use  of  above, 
that  exists  in  the  first  branch  representing  impenetrability,  &  in  the  last  branch  representing 
universal  gravitation  ;  for  since  these  branches,  on  account  of  properties  that  are  so  general 
to  all  bodies,  are  so  similar  to  one  another  in  all  cases,  they  indicate  complete  similarity 
of  the  remaining  arc  of  the  curve  expressing  the  forces  for  all  bodies  as  well. 

98.  Naught  that  concerns  this  subject  remains  but  for  me  to  once  more  mention  in  Homogeneity      is 
this  connection  that  one  thing,  which  I  have  already  remarked  at  the  beginning  of  this  anftysis  of  Nature" 
work,  namely,  that  Nature  itself  &  the  method  of  analysis  lead  us  towards  simplicity  &  example      taken 
homogeneity  of  the  elements ;   since  in  truth  the  farther  the  analysis  is  pushed,  the  fewer  ancj  dots°   ' 

the  fundamental  substances  we  arrive  at  &  the  less  they  differ  from  one  another ;  as  is 
to  be  seen  in  chemical  experiments.  This  will  be  presented  to  the  mind  far  more  clearly 
by  an  illustration  derived  from  letters  &  words.  Suppose  we  have  made  black  letters, 
not  by  drawing  a  continuous  line  with  ink,  but  by  means  of  little  black  dots  which  are  at 
such  small  distances  from  one  another  that  the  intervals  cannot  be  perceived  except  with 
the  aid  of  a  microscope — &  indeed  such  forms  of  letters  may  be  made  as  types  from  round 
points  of  this  sort  set  close  to  one  another.  Now  imagine  that  we  have  a  huge  library, 
all  the  books  in  it  consisting  of  printed  letters,  &  let  there  be  an  incredible  multitude 
of  books  printed  in  various  languages,  in  all  which  the  form  of  the  characters  is  the  same. 
If  anyone,  who  was  ignorant  of  such  compositions  or  languages,  started  on  a  careful  study 
of  books  of  this  kind,  all  of  which  he  would  perceive  differed  from  one  another ;  then  first 
of  all  he  would  find  a  medley  of  words,  some  of  which  occurred  frequently  in  certain  books 
whilst  they  never  appeared  at  all  in  others.  Hence  he  could  compose  lexicons,  as  many 
in  number  as  there  are  languages ;  in  each  of  these  all  words  of  the  same  language  would 
be  found,  &  these  would  indeed  be  very  few  in  number ;  for  the  immense  multiplicity 
of  words  in  this  numerous  collection  of  books  of  so  many  kinds  is  now  reduced  to  what 
is  still  a  multiplicity,  but  smaller,  than  is  contained  in  the  lexicons  &  the  words  forming 
these  lexicons.  Now  if  he  continued  his  investigation,  he  would  easily  perceive  that  the 
whole  of  these  words  of  so  many  different  kinds  were  formed  from  24  letters  only  ;  that 
these  differed  in  some  sort  from  one  another  in  the  manner  in  which  the  lines  forming 
them  were  drawn  ;  that  the  different  combinations  of  these  would  produce  the  whole  of 
that  great  variety  of  words,  &  that  combinations  of  these  words  would  form  books  differing 
from  one  another  still  more  widely.  Now  if  he  made  yet  another  examination  without  the 
aid  of  a  microscope,  he  would  not  find  any  other  kind  of  elements  that  were  more  similar 
to  one  another  than  these  letters,  from  a  combination  of  which  in  different  ways  the  letters 
themselves  could  be  produced.  But  if  he  took  a  microscope,  then  indeed  would  he  see 
the  mode  of  formation  of  the  letters  from  the  perfectly  homogeneous  round  points,  by 
the  different  position  &  distribution  of  which  the  letters  were  depicted. 

99.  This  seems  to  me  to  be  a  sort  of  picture  of  what  we  perceive  in  Nature.     Those  Application  of  the 

i,-7-7  .  ,..,.  r.,  i      T        n      i  1-111  illustration  to    the 

books,  so  many  m  number  &  so  different  in  character  are  bodies,  &  those  which  belong  analysis  of  Nature. 

to  the  different  kingdoms  are  written  as  it  were  in  different  tongues.     Of  all  of  these, 

chemical  analysis  finds  out  certain  fundamental  constituents  that  are  less  unlike  one  another 

than  the  books ;    these  are  the  words.     Yet  these  constituent  substances  have  some  sort 

of  difference  amongst  themselves,  &  thus  chemical  analysis   produces  a  large  number  of 

species  of  oils,  earths  &  salts  from  different  bodies.     Further  analysis  of  these,  like  that 

of  the  words,  would  disclose  the  letters  that  are  still  less  unlike  one  another ;    &  finally, 

according  to  my  Theory,  the  little  homogeneous  points  would  be  obtained.     These,  just 

as  the  little  black  circles  formed  the  letters,  would  form  the  diverse  particles  of  diverse 

bodies  through  diverse  arrangement  alone.     So  far  then  the  analogy  derived  from  such  a 


94  PHILOSOPHIC  NATURALIS  THEORIA 

derivata  non  ad  difformitatem,  sed  ad  conformitatem  elementorum  nos  ducit. 

Transitus    a    pro-  ioo.  Atque  hoc  demum  pacto  ex  principiis  certis  &  vulgo  receptis,  per  legitimam, 

ad  consectariorum  seriem  devenimus  ad  omnem  illam,  quam  initio  proposui,  Theoriam, 
nimirum  ad  legem  virium  mutuarum,  &  ad  constitutionem  primorum  materiae  elementorum 
ex  ilia  ipsa  virium  lege  derivatorum.  [46]  Videndum  jam  superest,  quam  uberes  inde 
fructus  per  universam  late  Physicam  colligantur,  explicatis  per  earn  unam  praecipuis  cor- 
porum  proprietatibus,  &  Naturae  phaenomenis.  Sed  antequam  id  aggredior,  praecipuas 
quasdam  e  difficultatibus,  quae  contra  Theoriam  ipsam  vel  objectae  jam  sunt,  vel  in  oculos 
etiam  sponte  incurrunt,  dissolvam,  uti  promisi. 

Legem  virium  non  ioi.  Contra  vires  mutuas  illud  sclent  objicere,  illas  esse  occultas  quasdam  qualitates, 

in  distans,anec0esse  ve^  etiam  actionem  in  distans  inducere.  His  satisfit  notione  virium  exhibita  numero  8, 
occuitam  quaiita-  &  9.  Illud  unum  praeterea  hie  addo,  admodum  manifestas  eas  esse,  quarum  idea  admodum 
facile  efformatur,  quarum  existentia  positive  argumento  evincitur,  quarum  effectus  multi- 
plices  continue  oculis  observantur.  Sunt  autem  ejusmodi  hae  vires.  Determinationis 
ad  accessum,  vel  recessum  idea  efformatur  admodum  facile.  Constat  omnibus,  quid  sit 
accedere,  quid  recedere  ;  constat,  quid  sit  esse  indifferens,  quid  determinatum  ;  adeoque 
&  determinationis  ad  accessum,  vel  recessum  habetur  idea  admodum  sane  distincta. 
Argumenta  itidem  positiva,  quae  ipsius  ejusmodi  determinationis  existentiam  probant, 
superius  prolata  sunt.  Demum  etiam  motus  varii,  qui  ab  ejusmodi  viribus  oriuntur,  ut 
ubi  corpus  quoddam  incurrit  in  aliud  corpus,  ubi  partem  solidi  arreptam  pars  alia  sequitur, 
ubi  vaporum,  vel  elastrorum  particulae  se  invicem  repellunt,  ubi  gravia  descendunt,  hi 
motus,  inquam,  quotidie  incurrant  in  oculos.  Patet  itidem  saltern  in  genere  forma  curvae 
ejusmodi  vires  exprimentis.  Haec  omnia  non  occuitam,  sed  patentem  reddunt  ejusmodi 
virium  legem. 


Quid  adhuc  lateat :  IO2.  Sunt  quidem  adhuc  quaedam,  quae  ad  earn  pertinent,  prorsus  incognita,  uti  est 

admittendam   om-  numerus,  &  distantia  intersectionum  curvae  cum  axe,  forma  arcuum  intermediorum,  atque 

nino :     quo    pacto      ..         .  ,.  -11  i  -11          i        i     i      • 

evitetur  hie  actio  alia  ejusmodi,  quae  quidem  longe  superant  humanum  captum,  &  quas  me  solus  habuit 
in  distans.  omnia  simul  prae  oculis,  qui  Mundum  condidit ;    sed  id  omnino   nil   officit.      Nee   sane 

id  ipsum  in  causa  esse  debet,  ut  non  admittatur  illud,  cujus  existentiam  novimus,  &  cujus 
proprietates  plures,  &  effectus  deprehendimus ;  licet  alia  multa  nobis  incognita  eodem 
pertinentia  supersint.  Sic  aurum  incognitam,  occultamque  substantiam  nemo  appellant, 
&  multo  minus  ejusdem  existentiam  negabit  idcirco,  quod  admodum  probabile  sit,  plures 
alias  latere  ipsius  proprietates,  olim  forte  detegendas,  uti'aliae  tarn  multae  subinde  detectae 
sunt,  &  quia  non  patet  oculis,  qui  sit  particularum  ipsum  componentium  textus,  quid,  & 
qua  ratione  Natura  ad  ejus  compositionem  adhibeat.  Quod  autem  pertinet  ad  actionem 
in  distans,  id  abunde  ibidem  praevenimus,  cum  inde  pateat  fieri  posse,  ut  punctum  quodvis 
in  se  ipsum  agat,  &  ad  actionis  directionem,  ac  energiam  determinetur  ab  altero  puncto, 
vel  ut  Deus  juxta  liberam  sibi  legem  a  se  in  Natura  condenda  stabilitam  motum  progignat 
in  utroque  pun-[47]-cto.  Illud  sane  mihi  est  evidens,  nihilo  magis  occuitam  esse,  vel  explicatu, 
&  captu  difficilem  productionem.  motus  per  hasce  vires  pendentes  a  certis  distantiis,  quam 
sit  productio  motus  vulgo  concepta  per  immediatum  impulsum,  ubi  ad  motum  determinat 
impenetrabilitas,  quae  itidem  vel  a  corporum  natura,  vel  a  libera  conditoris  lege  repeti 
debet. 


sine  impuisione  103.  Et  quidem  hoc  potius  pacto,  quam  per  impulsionem,  in  motuum  causas,  &  leges 

Mst'hucus^'u^N™  inquirendum  esse,  illud  etiam  satis  indicat,  quod  ubi  hue  usque,  impuisione  omissa,  vires 

turam,  &  menus  ex-  adhibitae  sunt  a  distantiis  pendentes,  ibi  sane  tantummodo  accurate  definita  sunt  omnia, 

phcajidam.  impost-  atque  determinata,  &  ad  calculum  redacta  cum  phaenomenis  congruunt  ultra,  quam  sperare 

liceret,  accuratissime.     Ego  quidem  ejusmodi  in  explicando,  ac  determinando  felicitatem 

nusquam  alibi  video  in  universa  Physica,  nisi  tantummodo  in  Astronomia  mechanica,  quae 

abjectis  vorticibus,  atque  omni  impuisione  submota,  per  gravitatem  generalem  absolvit 

omnia,  ac  in  Theoria  luminis,  &  colorum,  in  quibus  per  vires  in  aliqua  distantia  agentes, 

&  reflexionem,  &  refractionem,  &  diffractionem  Newtonus  exposuit,  ac  priorum  duarum, 

potissimum  leges  omnes  per  calculum,  &  Geometriam  determinavit,  &  ubi  ilia  etiam,  quae 

ad  diversas  vices  facilioris  transmissus,  &  facilioris  reflexionis,  quas  Physici  passim  relinquunt 


A  THEORY  OF  NATURAL  PHILOSOPHY  95 

consideration  of  Nature  leads  us  not  to  non-uniformity  but  to  uniformity  of  the 
elements. 

100.  Thus   at   length,   from  known   principles   that   are   commonly  accepted,   by  a  Pa^g  ,,fro™  the 

...  ,   ,     ,  °     .'  .  .r  r  ,          i     i      %    i      n-<i  i         T  •         i    proof  of  the  Theory 

legitimate  series  of  deductions,  we  have  arrived  at  the  whole  of  the  I  heory  that  I  enunciated  to   the  considera- 
at  the  start ;   that  is  to  say.  at  a  law  of  mutual  forces  &  the  constitution  of  the  primary  tion  °f.  objections 

, '  i     •       i    ,-  i  re  XT  •  i  r   i         '     against  it. 

elements  of  matter  derived  from  that  law  of  forces.  Now  it  remains  to  be  seen  what  a 
bountiful  harvest  is  to  be  gathered  throughout  the  wide  field  of  general  physics ;  for  from 
this  one  theory  we  obtain  explanations  of  all  the  chief  properties  of  bodies,  &  of  the 
phenomena  of  Nature.  But  before  I  go  on  to  that,  I  will  give  solutions  of  a  few  of  the 
principal  difficulties  that  have  been  raised  against  the  Theory  itself,  as  well  as  some  that 
naturally  meet  the  eye,  according  to  the  promise  I  made. 

10 1.  The  objection  is  frequently  brought  forward  against  mutual  forces  that  they  The  law  o{  forces 

,J  .  * «.  .     «  ,  .  .  j.  mi  •     does    not    necessi- 

are  some  sort  of  mysterious  qualities  or  that  they  necessitate  action  at  a  distance.     This  tate  action   at  a 

is  answered  by  the  idea  of  forces  outlined  in  Art.  8,  &  9.     In  addition,  I  will  make  just  distance,  nor  is  it 

one  remark,  namely,  that  it  is  quite  evident  that  these  forces  exist,  that  an  idea  of  them  quTuty.  " 

can  be  easily  formed,  that  their  existence  is    demonstrated   by  direct  reasoning,  &  that 

the  manifold  results  that  arise  from  them  are  a  matter  of   continual  ocular  observation. 

Moreover  these  forces  are  of  the  following  nature.     The  idea  of  a  propensity  to  approach 

or  of  a  propensity  to  recede  is  easily  formed.     For  everybody  knows  what  approach  means, 

and  what  recession  is ;   everybody  knows  what  it  means  to  be  indifferent,  &  what  having 

a  propensity  means ;  &  thus  the  idea  of  a  propensity  to  approach,  or  to  recede,  is  perfectly 

distinctly  obtained.     Direct  arguments,  that  prove  the  existence  of  this  kind  of  propensity, 

have  been  given  above.     Lastly  also,  the  various  motions  that  arise  from  forces  of  this 

kind,  such  as  when  one  body  collides  with  another  body,  when  one  part  of  a  solid  is  seized 

&  another  part  follows  it,  when  the  particles  of  gases,  &  of  springs,  repel  one  another, 

when  heavy  bodies  descend,  these  motions,  I  say,  are  of  everyday  occurrence  before  our 

eyes.     It  is  evident  also,  at  least  in  a  general  way,  that  the  form  of  the  curve  represents 

forces  of  this  kind.     In  all  of  these  there  is  nothing  mysterious ;   on  the  contrary  they  all 

tend  to  make  the  law  of  forces  of  this  kind  perfectly  plain. 

102.  There  are  indeed  certain  things  that  relate  to  the  law  of  forces  of  which  we  are  What  is  so  far  un 
altogether  ignorant,  such  as  the   number  &  distances  of   the  intersections  of   the  curve  J^0^  idmitted°m 
with  the  axis,  the  shape  of  the  intervening  arcs,  &  other  things  of  that  sort ;  these  indeed  ail  detail ;  the  way 
far  surpass  human  understanding,  &  He  alone,  Who  founded  the  universe,  had  the  whole  ^  action1  atha  to* 
before  His  eyes.     But  truly  there  is  no  reason  on  that  account,  why  a  thing,  whose  existence  tance  is  eliminated, 
we  fully  recognize,  &  many  of  the  properties  &  results  of  which  are  readily  understood, 

should  not  be  accepted  ;  although  certainly  there  do  remain  many  other  things  pertaining 
to  it  that  are  unknown  to  us.  For  instance,  nobody  would  call  gold  an  unknown  & 
mysterious  substance,  &  still  less  would  deny  its  existence,  simply  because  it  is  quite 
probable  that  many  of  its  properties  are  unknown  to  us,  to  be  discovered  perhaps  in  the 
future,  as  so  many  others  have  been  already  discovered  from  time  to  time,  or  because  it 
is  not  visually  apparent  what  is  the  texture  of  the  particles  composing  it,  or  why  &  in 
what  way  Nature  adopts  that  particular  composition.  Again,  as  regards  action  at  a  distance, 
we  amply  guard  against  this  by  the  same  means ;  for,  if  this  is  admitted,  then  it  would 
be  possible  for  any  point  to  act  upon  itself,  &  to  be  determined  as  to  its  direction  of  action 
&  energy  apart  from  another  point,  or  that  God  should  produce  in  either  point  a  motion 
according  to  some  arbitrary  law  fixed  by  Him  when  founding  the  universe.  To  my  mind 
indeed  it  is  clear  that  motions  produced  by  these  forces  depending  on  the  distances  are 
not  a  whit  more  mysterious,  involved  or  difficult  of  understanding  than  the  production 
of  motion  by  immediate  impulse  as  it  is  usually  accepted ;  in  which  impenetrability 
determines  the  motion,  &  the  latter  has  to  be  derived  just  the  same  either  from  the  nature 
of  solid  bodies,  or  from  an  arbitrary  law  of  the  founder  of  the  universe. 

103.  Now,  that  the  investigation  of  the  causes  &  laws  of   motion  are   better   made  As  far  as  we  have 
by  my  method,  than  through  the  idea  of  impulse,  is  sufficiently  indicated  by  the  fact  that,  f^'  m0reUIciearJy 
where  hitherto  we  have  omitted  impulse  &  employed  forces  depending  on  the  distances,  explained    without 
only  in  this  way  has  everything  been  accurately  defined  &  determined,  &  when  reduced  g^1  what 

to  calculation  everything  agrees  with  the  phenomena  with  far  more  accuracy  than  we  will  be  so  too. 
could  possibly  have  expected.  Indeed  I  do  not  see  anywhere  such  felicity  in  explaining 
&  determining  the  matters  of  general  physics,  except  only  in  celestial  mechanics ;  in 
which  indeed,  rejecting  the  idea  of  vortices,  &  doing  away  with  that  of  impulse  entirely, 
Newton  gave  a  solution  of  everything  by  means  of  universal  gravitation  ;  &  in  the  theory 
of  light  &  colours,  where  by  means  of  forces  acting  at  some  distance  he  explained  reflection, 
refraction  &  diffraction  ;  &,  especially  in  the  two  first  mentioned,  he  determined  all 
the  laws  by  calculus  &  Geometry.  Here  also  those  things  depending  on  alternate  fits 
of  easier  transmission  &  easier  reflection,  which  physicists  everywhere  leave  almost 


96 


PHILOSOPHISE  NATURALIS  THEORIA 


fere  intactas,  ac  alia  multa  admodum  feliciter  determinantur,  explicanturque,  quod  &  ego 
praestiti  in  dissertatione  De  Lumine,  &  praestabo  hie  in  tertia  parte  ;  cum  in  ceteris  Physicae 
partibus  plerumque  explicationes  habeantur  subsidariis  quibusdam  principiis  innixae  & 
vagas  admodum.  Unde  jam  illud  conjectare  licet,  si  ab  impulsione  immediata  penitus 
recedatur,  &  sibi  constans  ubique  adhibeatur  in  Natura  agendi  ratio  a  distantiis  pendens, 
multo  sane  facilius,  &  certius  explicatum  iri  cetera  ;  quod  quidem  mihi  omnino  successit, 
ut  patebit  inferius,  ubi  Theoriam  ipsam  applicavero  ad  Naturam. 


Non  fieri  saltum  in 

tracfiva  ad  repui- 
sivam. 


104.  Solent  &  illud  objicere,  in  hac  potissimo  Theoria  virium  committi  saltum  ilium, 
a(^  quern  evitandum  ea  inprimis  admittitur  ;  fieri  enim  transitum  ab  attractionibus  ad 
repulsiones  per  saltum,  ubi  nimirum  a  minima  ultima  repulsione  ad  minimam  primam 
attractionem  transitur.  At  isti  continuitatis  naturam,  quam  supra  exposuimus,  nequaquam 
intelligunt.  Saltus,  cui  evitando  Theoria  inducitur,  in  eo  consistit,  quod  ab  una  magnitudine 
ad  aliam  eatur  sine  transitu  per  intermedias.  Id  quidem  non  accidit  in  casu  exposito. 
Assumatur  quaecunque  vis  repulsiva  utcunque  parva  ;  turn  quaecunque  vis  attractiva. 
Inter  eas  intercedunt  omnes  vires  repulsivae  minores  usque  ad  zero,  in  quo  habetur  deter- 
minatio  ad  conservandum  praecedentum  statum  quietis,  vel  motus  uniformis  in  directum  : 
turn  omnes  vires  attractivae  a  z^-[48]-ro  usque  ad  earn  determinatam  vim,  &  omnino  nullus 
erit  ex  hisce  omnibus  intermediis  statibus,  quern  aliquando  non  sint  habitura  puncta,  quae 
a  repulsione  abeunt  ad  attractionem.  Id  ipsum  facile  erit  contemplari  in  fig.  i,  in  qua  a 
vi  repulsiva  br  ad  attractionem  dh  itur  utique  continue  motu  puncti  b  ad  d  transeundo 
per  omnes  intermedias,  &  per  ipsum  zero  in  E  sine  ullo  saltu  ;  cum  ordinata  in  eo  motu 
habitura  sit  omnes  magnitudines  minores  priore  br  usque  ad  zero  in  E  ;  turn  omnes  oppositas 
majores  usque  ad  posteriorem  dh.  Qui  in  ea  veluti  imagine  mentis  oculos  defigat,  is  omnem 
apparentem  difficultatem  videbit  plane  sibi  penitus  evanescere. 


Nuiium  esse  post-  JOP    Quod  autem  additur  de  postremo  repulsionis  gradu,  &  primo  attractionis  nihil 

remum  attractions,  -1.  ...  r,....r,  ., 

A:  primum  repuisio-  sane  probaret,  quando  etiam  essent  aliqui  n  gradus  postrerm,  &  primi ;    nam  ab  altero 
ms  gradum,  qm  si  eorum  transiretur  ad  alterum  per  intermedium  illud  zero,  &  ex  eo  ipso,  quod  illi  essent 

essent,  adhuc  tran-  .....*.  .  ,.  .     . 

sire  per  omnes  in-  postremus,  ac  primus,  mhil  omitteretur  mtermeaium,  quae  tamen  sola  intermedn  omissio 
termedios.  continuitatis  legem  evertit,  &  saltum  inducit.     Sed  nee  habetur  ullus  gradus  postremus, 

aut  primus,  sicut  nulla  ibi  est  ordinata  postrema,  aut  prima,  nulla  lineola  omnium  minima. 
Data  quacunque  lineola  utcunque  exigua,  aliae  ilia  breviores  habentur  minores,  ac  minores 
ad  infinitum  sine  ulla  ultima,  in  quo  ipso  stat,  uti  supra  etiam  monuimus,  continuitatis 
natura.  Quamobrem  qui  primum,  aut  ultimum  sibi  confingit  in  lineola,  in  vi,  in  celeritatis 
gradu,  in  tempusculo,  is  naturam  continuitatis  ignorat,  quam  supra  hie  innui,  &  quam  ego 
idcirco  initio  meae  dissertationis  De  Lege  Continuitatis  abunde  exposui. 


potest  cuipiam  saltern  illud,  ejusmodi  legem  virium,  &  curvam,  quam  in 
curvae,    &   duobus  fig.  I  protuli,  esse  nimium  complicatam,  compositam,  &  irregularem,  quae  nimirum  coalescat 
virium  genenbus.     ex  }ngenti  numero  arcuum  jam  attractivorum,  jam  repulsivorum,  qui  inter  se  nullo  pacto 
cohaereant  ;  rem  eo  redire,  ubi  erat  olim,  cum  apud  Peripateticos  pro  singulis  proprietatibus 
corporum  singulae  qualitates  distinctae,  &  pro  diversis  speciebus  diversae  formae  substantiales 
confingebaritur  ad  arbitrium.     Sunt  autem,  qui  &  illud  addant,  repulsionem,  &  attractionem 
esse  virium  genera  inter  se  diversa  ;  satius  esse,  alteram  tantummodo  adhibere,  &  repulsionem 
explicare  tantummodo  per  attractionem  minorem. 

repuisivam  positive  I07-  Inprimis    quod  ad  hoc    postremum  pertinet,  satis  patet,  per  positivam  meae 

demonstrari  prater  Theoriae  probationem  immediate  evinci  repulsionem  ita,  ut  a  minore  attractione  repeti 
omnino  non  possit  ;  nam  duae  materiae  particulae  si  etiam  solae  in  Mundo  essent,  &  ad  se 
invicem  cum  aliqua  velocitatum  inaequalitate  accederent,  deberent  utique  ante  contactum 
ad  sequalitatem  devenire  vi,  quse  a  nulla  attractione  pendere  posset. 


A  THEORY  OF  NATURAL  PHILOSOPHY  97 

untouched,  &  many  other  matters  were  most  felicitously  determined  &  explained  by 
him  ;  &  also  that  which  I  enunciated  in  the  dissertation  De  Lumine,  &  will  repeat  in 
the  third  part  of  this  work.  For  in  other  parts  of  physics  most  of  the  explanations  are 
independent  of,  &  disconnected  from,  one  another,  being  based  on  several  subsidiary 
principles.  Hence  we  may  now  conclude  that  if,  relinquishing  all  idea  of  immediate 
impulses,  we  employ  a  reason  for  the  action  of  Nature  that  is  everywhere  the  same  & 
depends  on  the  distances,  the  remainder  will  be  explained  with  far  greater  ease  &  certainty  ; 
&  indeed  it  is  altogether  successful  in  my  hands,  as  will  be  evident  later,  when  I  come 
to  apply  the  Theory  to  Nature. 

104.  It  is  very  frequently  objected  that,  in  this  Theory  more  especially,  a  sudden  change  There  is  no  sudden 
is  made  in  the  forces,  whilst  the  theory  is  to  be  accepted  for  the  very  purpose  of  avoiding  sitfwPfrom  aiTat- 
such  a  thing.  For  it  is  said  that  the  transition  from  attractions  to  repulsions  is  made  tractive  to  a  repui- 
suddenly,  namely,  when  we  pass  from  the  last  extremely  minute  repulsive  force  to  the 
first  extremely  minute  attractive  force.  But  those  who  raise  these  objections  in  no  wise 
understand  the  nature  of  continuity,  as  it  has  been  explained  above.  The  sudden  change, 
to  avoid  which  the  Theory  has  been  brought  forward,  consists  in  the  fact  that  a  passage 
is  made  from  one  magnitude  to  another  without  going  through  the  intermediate  stages. 
Now  this  kind  of  thing  does  not  take  place  in  the  case  under  consideration.  Take  any 
repulsive  force,  however  small,  &  then  any  attractive  force.  Between  these  two  there 
lie  all  the  repulsive  forces  that  are  less  than  the  former  right  down  to  zero,  in  which  there 
is  the  propensity  for  preserving  the  original  state  of  rest  or  of  uniform  motion  in  a  straight 
line  ;  &  also  all  the  attractive  forces  from  zero  up  to  the  prescribed  attractive  force, 
&  there  will  be  absolutely  no  one  of  all  these  intermediate  states,  which  will  not  be  possessed 
at  some  time  or  other  by  the  points  as  they  pass  from  repulsion  to  attraction.  This  can 
be  readily  understood  from  a  study  of  Fig.  I,  where  indeed  the  passage  is  made  from  the 
repulsive  force  br  to  the  attractive  force  dh  by  the  continuous  motion  of  a  point  from  b  to 
d ;  the  passage  is  made  through  every  intermediate  stage,  &  through  zero  at  E,  without 
any  sudden  change.  For  in  this  motion  there  will  be  obtained  as  ordinates  all  magnitudes, 
less  than  the  first  one  br,  down  to  zero  at  E,  &  after  that  all  magnitudes  of  opposite  sign 
greater  than  zero  as  far  as  the  last  ordinate  dh.  Anyone,  who  will  fix  his  intellectual  vision 
on  this  as  on  a  sort  of  pictorial  illustration  cannot  fail  to  perceive  for  himself  that  all  the 
apparent  difficulty  vanishes  completely. 

i  OS.  Further,  as  regards  what  is  said  in  addition  about  the  last  stage  of  repulsion  &  T^1"6    &  no  !ast 

,       r  .  .  11          11  TI  1111    stage  of  attraction, 

the  first  stage  of  attraction,  it  would  really  not  matter,  even  if  there  were  these  so  called  and  no  first  for  re- 
last  &  first  stages ;  for,  from  one  of  them  to  the  other  the  passage  would  be  made  through  puisjon ;  and  even 

6    ,.'  .  .  ,  i       c°       if    there  were,  the 

the  one  intermediate  stage,  namely  zero  ;  since  it  passes  zero,  &  because  they  are  the  nrst  passage  would  be 

&  last,  therefore  no  intermediate  stage  is  omitted.     Nevertheless   the   omission   of   this  pade  through  ail 

intermediate  alone  would  upset  the   law  of   continuity,  &   introduce  a  sudden  change. 

But,  as  a  matter  of  fact,  there  cannot  possibly  be  a  last  stage  or  a  first ;  just  as  there  cannot 

be  a  last  ordinate  or  a  first  in  the  curve,  that  is  to  say,  a  short  line  that  is  the  least  of 

them  all.     Given  any  short  line,  no  matter  how  short,  there  will  be  others  shorter  than 

it,  less  &  less  in  infinite  succession  without  any  limit  whatever  ;  &  in  this,  as  we  remarked 

also  above,  there  lies  the  nature  of  continuity.     Hence  anyone  who  brings  forward  the 

idea  of  a  first  or  a  last  in  the  case  of  a  line,  or  a  force,  or  a  degree  of  velocity,  or  an 

interval  of  time,  must  be  ignorant  of  continuity ;    this  I  have  mentioned  before  in  this 

work,  &  also  for  this  very  reason  I   explained  it  very  fully  at  the  beginning  of   my 

dissertation  De  Lege  Continuitatis. 

1 06.  It  may  seem  to  some  that  at  least  a  law  of  forces  of  this  nature,  &  the  curve  °gb^cstt10?herappard 
expressing  it,  which  I  gave  in  Fig.  I,  is  very  complicated,  composite  &  irregular,  being  ent  composite  cha- 
indeed  made  up  of  an  immense  number  of  arcs  that  are  alternately  attractive  &  repulsive,  ^te[h°f  t 

&  that  these  are  joined  together  according  to  no  definite  plan  ;  &  that  it  reduces  to  Of  forces, 
the  same  thing  as  obtained  amongst  the  ancients,  since  with  the  Peripatetics  separate 
distinct  qualities  were  invented  for  the  several  properties  of  bodies,  &  different  substantial 
forms  for  different  species.  Moreover  there  are  some  who  add  that  repulsion  &  attraction 
are  kinds  of  forces  that  differ  from  one  another ;  &  that  it  would  be  quite  enough  to 
use  only  the  latter,  &  to  explain  repulsion  merely  as  a  smaller  attraction. 

107.  First  of  all,  as  regards  the  last  objection,  it  is  clear  enough  from  what  has  been  p^ssibie^o    prove 
directly  proved  in  my  Theory  that  the  existence  of  repulsion  has  been  rigorously  demonstrated  directly  the  exist- 
in  such  a  way  that  it  cannot  possibly  be  derived  from  the  idea  of  a  smaller  attraction.     For  f^ce °f apart PUfrom 
two  particles  of  matter,  if  they  were  also  the  only  particles  in  the  universe,  &  approached  attraction. 

one  another  with  some  difference  of  velocity,  would  be  bound  to  attain  to  an  equality  of 
velocity  on  account  of  a  force  which  could  not  possibly  be  derived  from  an  attraction  of 

any  kind, 

H 


PHILOSOPHIC  NATURALIS  THEORIA 


tiva,  &  negativa. 


Hinc  nihu  pbstare,  Iogi  Deinde  vefo  quod  pertinet  ad  duas  diversas  species  attractionis,  &  repulsionis ; 

si  diversi  suit  gene-    .,         .  ,          ,.          .  ,^.          r.r-ii-i  i 

ris;  sed  esse  ejus-  id  quidem  licet  ita  se  haberet,  m-[49j-hil  sane  obesset,  cum  positive  argumento  evmcatur 
dem  uti  sunt  posi-  &  repulsio.  &  attractio,  uti  vidimus;    at  id  ipsum  est  omnino  falsum.     Utraque  vis  ad 

.  f  .  .  .  .       ^  .    .    . 

eandem  pertinet  speciem,  cum  altera  respectu  alterms  negativa  sit,  &  negativa  a  positivis 
specie  non  differant.  Alteram  negativam  esse  respectu  alterius,  patet  inde,  quod  tantum- 
modo  differant  in  directione,  quae  in  altera  est  prorsus  opposita  direction!  alterius ;  in 
altera  enim  habetur  determinatio  ad  accessum,  in  altera  ad  recessum,  &  uti  recessus,  & 
accessus  sunt  positivum,  ac  negativum  ;  ita  sunt  pariter  &  determinationes  ad  ipsos.  Quod 
autem  negativum,  &  positivum  ad  eandem  pertineant  speciem,  id  sane  patet  vel  ex  eo 
principio  :  magis,  W  minus  non  differunt  specie.  Nam  a  positive  per  continuam  subtrac- 
tionem,  nimirum  diminutionem,  habentur  prius  minora  positiva,  turn  zero,  ac  demum 
negativa,  continuando  subtractionem  eandem. 


Probatio  hujus  a 
progressu,  &  re- 
gressu,  in  fluvio. 


109.  Id  facile  patet  exemplis  solitis.  Eat  aliquis  contra  fluvii  directionem  versus  locum 
aliquem  superiori  alveo  proximum,  &  singulis  minutis  perficiat  remis,  vel  vento  too  hexapedas, 
dum  a  cursu  fluvii  retroagitur  per  hexapedas  40  ;  is  habet  progressum  hexapedarum  60 
singulis  minutis.  Crescat  autem  continue  impetus  fluvii  ita,  ut  retroagatur  per  50,  turn  per 
60,  70,  80,  90,  100,  no,  120,  &c.  Is  progredietur  per  50,  40,  30,  20,  10,  nihil ;  turn 
regredietur  per  10,  20,  quae  erunt  negativa  priorum  ;  nam  erat  prius  100 — 50,  100 — 60, 
100—70,100 — 80,100 — 90,  turn  100 — 100=0,100 — no,  =  — 10,  100 — 120  =  —20,  et  ita 
porro.  Continua  imminutione,  sive  subtractione  itum  est  a  positivis  in  negativa,  a 
progressu  ad  regressum,  in  quibus  idcirco  eadem  species  mansit,  non  duae  diversae. 


Probatio  ex  Alge- 
bra, &  Geometria  : 
applicatio  ad  omnes 
quantitates  varia- 
biles. 


An  habeatur  trans- 
itus  e  positivis  in 
negativa  ;  investi- 
gatio  ex  sola  curv- 
arum  natura. 


B 


FHN 


MAC 


FIG.  ii. 


i  to.  Idem  autem  &  algebraicis  formulis,  &  geometricis  lineis  satis  manifeste  ostenditur. 
Sit  formula  10— x,  &  pro  x  ponantur  valores  6,  7,  8,  9,  10,  n,  12,  &c. ;  valor  formulae 
exhibebit  4,  3,  2,  I,  o,  — I,  — 2,  &c.,  quod  eodem  redit,  ubi  erat  superius  in  progressu,  & 
regressu,  qui  exprimerentur  simulper  formulam  10— x.  Eadem  ilia  formula  per  continuam 
mutationem  valoris  x  migrat  e  valore  positive  in  negativum,  qui  aeque  ad  eandem  formulam 
pertinent.  Eodem  pacto  in  Geometria  in  fig. 
u,siduae  lineae  MN,  OP  referantur  invicem 
per  ordinatas  AB,  CD,  &c.  parallelas  inter  se, 
secent  autem  se  in  E ;  continue  motu  ipsius 
ordinatae  a  positive  abitur  in  negativum,  mutata 
directione  AB,  CD,  quae  hie  habentur  pro 
positivis,  in  FG,  HI,  post  evanescentiam  in  E. 
Ad  eandem  lineam  continuam  OEP  aeque 
pertinet  omnis  ea  ordinatarum  series,  nee  est 
altera  linea,  alter  locus  geometricus  OE,  ubi 
ordinatae  sunt  positivae,  ac  EP,  ubi  sunt  nega- 
tivae.  Jam  vero  variabilis  quantitatis  cujusvis 
natura,  &  lex  plerumque  per  formulam  aliquam  analyticam,  semper  per  ordinatas  ad  lineam 
aliquam  exprimi  potest ;  si  [50]  enim  singulis  ejus  statibus  ducatur  perpendicularis 
respondens ;  vertices  omnium  ejusmodi  perpendicularium  erunt  utique  ad  lineam  quandam 
continuam.  Si  ea  linea  nusquam  ad  alteram  abeat  axis  partem,  si  ea  formula  nullum  valorem 
negativum  habeat ;  ilia  etiam  quantitas  semper  positiva  manebit.  Sed  si  mutet  latus  linea, 
vel  formula  valoris  signum  ;  ipsa  ilia  quantitatis  debebit  itidem  ejusmodi  mutationem 
habere.  Ut  autem  a  formulae,  vel  lineae  exprimentis  natura,  &  positione  respectu  axis 
mutatio  pendet ;  ita  mutatio  eadem  a  natura  quantitatis  illius  pendebit ;  &  ut  nee  duas 
formulae,  nee  duae  lineae  speciei  diversae  sunt,  quae  positiva  exhibent,  &  negativa  ;  ita  nee  in 
ea  quantitate  duae  erunt  naturae,  duae  species,  quarum  altera  exhibeat  positiva,  altera 
negativa,  ut  altera  progressus,  altera  regressus ;  altera  accessus,  altera  recessus  ;  &  hie  altera 
attractiones,  altera  repulsiones  exhibeat ;  sed  eadem  erit,  unica,  &  ad  eandem  pertinens 
quantitatis  speciem  tota. 

in.  Quin  immo  hie  locum  habet  argumentum  quoddam,  quo  usus  sum  in  dissertatione 
De  Lege  Continuitatis,  quo  nimirum  Theoria  virium  attractivarum,  &  repulsivarum  pro 
diversis  distantiis,  multo  magis  rationi  consentanea  evincitur,  quam  Theoria  ^  virium 
tantummodo  attractivarum,  vel  tantummodo  repulsivarum.  Fingamus  illud,  nos  ignorare 
penitus,  quodnam  virium  genus  in  Natura  existat,  an  tantummodo  attractivarum,  vel 
repulsivarum  tantummodo,  an  utrumque  simul :  hac  sane  ratiocinatione  ad  earn  perquisi- 
tionem  uti  liceret.  Erit  utique  aliqua  linea  continua,  quae  per  suas  ordinatas  ad  axem 
exprimentem  distantias,  vires  ipsas  determinabit,  &  prout  ipsa  axem  secuerit,  vel  non 


A  THEORY  OF  NATURAL  PHILOSOPHY 


99 


108.  Next,  as    regards  attraction  &  repulsion  being  of  different  species,  even  if   it  Hence  it  does  not 
were  a  fact  that  they  were  so,  it  would  not  matter  in  the  slightest  degree,  since  by  rigorous  Satdifferenthkmds! 
argument  the  existence  of   both  attraction  &  repulsion  is  proved,  as  we  have  seen  ;    but  but  as  a  matter  of 
really  the  supposition  is  untrue.     Both  kinds  of  force  belong  to  the  same  species ;   for  one  same^kmdnusVas 
is  negative  with  regard  to  the  other,  &  a  negative  does  not  differ  in  species  from  positives.  a  positive   and  a 
That  the  one  is  negative  with  regard  to  the  other  is  evident  from  the  fact  that  they  only  negatlve  are  so- 
differ  in  direction,  the  direction  of  one  being  exactly  the  opposite  of  the  direction  of  the 

other  ;  for  in  the  one  there  is  a  propensity  to  approach,  in  the  other  a  propensity  to  recede  ; 
&  just  as  approach  &  recession  are  positive  &  negative,  so  also  are  the  propensities 
for  these  equally  so.  Further,  that  such  a  negative  &  a  positive  belong  to  the  same  species, 
is  quite  evident  from  the  principle  the  greater  &  the  less  are  not  different  in  kind.  For 
from  a  positive  by  continual  subtraction,  or  diminution,  we  first  obtain  less  positives,  then 
zero,  &  finally  negatives,  the  same  subtraction  being  continued  throughout. 

109.  The  matter  is  easily  made   clear  by  the  usual   illustrations.     Suppose    a    man  Demonstration  by 
to  go  against  the  current  of  a  river  to  some  place   on  the    bank  up-stream;    &  suppose  "veTndretrogS 
that  he  succeeds  in  doing,  either  by  rowing  or  sailing,  100  fathoms  a  minute,  whilst  he  motion  on  a  river. 
is  carried  back  by  the  current  of  the  river  through  40  fathoms  ;    then  he  will  get  forward 

a  distance  of  60  fathoms  a  minute.  Now  suppose  that  the  strength  of  the  current  continually 
increases  in  such  a  way  that  he  is  carried  back  first  50,  then  60,  70,  80,  90,  ipo,  no,  120, 
&c.  fathoms  per  minute.  His  forward  motion  will  be  successively  50,  40,  30,  20,  10  fathoms 
per  minute,  then  nothing,  &  then  he  will  be  carried  backward  through  10,  20,  &c.  fathoms 
a  minute  ;  &  these  latter  motions  are  the  negatives  of  the  former.  For  first  of  all  we 
had  100  —  50,  100  —  60,  100  —  70,  100  —  80,  100  —  90,  then  100  —  100  (which  =  o), 
then  100  —  no  (which  =  —  10),  100  —  120  (which  =  —  20),  and  so  on.  By  a  continual 
diminution  or  subtraction  we  have  passed  from  positives  to  negatives,  from  a  progressive 
to  a  retrograde  motion  ;  &  therefore  in  these  there  was  a  continuance  of  the  same  species, 
and  there  were  not  two  different  species. 

no.  Further,  the  same  thing  is  shown  plainly  enough  by  algebraical  formulae,  &  Proof  from  algebra 
by  lines  in  geometry.  Consider  the  formula  10  —  x,  &  for  x  substitute  the  values,  6,  pucatfon™^ Oy :  a  n 
7,  8,  9,  10,  n,  12,  &c.  ;  then  the  value  of  the  formula  will  give  in  succession  4,  3,  2,  variable  quantities. 
I,  o,  —  i,  —  2,  &c.  ;  &  this  comes  to  the  same  thing  as  we  had  above  in  the  case  of  the 
progressive  &  retrograde  motion,  which  may  be  expressed  by  the  formula  10  —  x,  all 
together.  This  same  formula  passes,  by  a  continuous  change  in  the  value  of  x,  from  a 
positive  value  to  a  negative,  which  equally  belong  to  the  same  formula.  In  the  same 
manner  in  geometry,  in  Fig.  1 1,  if  two  lines  MN,  OP  are  referred  to  one  another  by  ordinates 
AB,  CD,  &  also  cut  one  another  in  E  ;  then  by  a  continuous  motion  of  the  ordinate 
itself  it  passes  from  positive  to  negative,  the  direction  of  AB,  CD,  which  are  here  taken 
to  be  positive,  being  changed  to  that  of  FG,  HI,  after  evanescence  at  E.  To  the  same 
continuous  line  OEP  belongs  equally  the  whole  of  this  series  of  ordinates ;  &  OE,  where 
the  ordinates  are  positive,  is  not  a  different  line,  or  geometrical  locus  from  EP,  where  the 
ordinates  are  negative.  Now  the  nature  of  any  variable  quantity,  &  very  frequently 
also  the  law,  can  be  expressed  by  an  algebraical  formula,  &  can  always  be  expressed  by 
some  line  ;  for  if  a  perpendicular  be  drawn  to  correspond  to  each  separate  state  of  the 
quantity,  the  vertices  of  all  perpendiculars  so  drawn  will  undoubtedly  form  some  continuous 
line.  If  the  line  never  passes  over  to  the  other  side  of  the  axis,  if  the  formula  has  no  negative 
value,  then  also  the  quantity  will  always  remain  positive.  But  if  the  line  changes  side, 
or  the  formula  the  sign  of  its  value,  then  the  quantity  itself  must  also  have  a  change  of  the 
same  kind.  Further,  as  the  change  depends  on  the  nature  of  the  formula  &  the  line 
expressing  it,  &  its  position  with  respect  to  the  axis ;  so  also  the  same  change  will  depend 
on  the  nature  of  the  quantity ;  &  just  as  there  are  not  two  formulas,  or  two  lines  of 
different  species  to  represent  the  positives  &  the  negatives,  so  also  there  will  not  be  in  the 
quantity  two  natures,  or  two  species,  of  which  the  one  yields  positives  &  the  other  negatives, 
as  the  one  a  progressive  &  the  other  a  retrograde  motion,  the  one  approach  &  the  other 
recession,  &  in  the  matter  under  consideration  the  one  will  give  attractions  &  the  other 
repulsions.  But  it  will  be  one  &  the  same  nature  &  wholly  belonging  to  the  same 
spec  es  of  quantity. 

in.  Lastly, this  is  the  proper  place  for  me  to  bring  forward  an  argument  that  I  used  whether  there  can 

i        i .  •         T\     T          /-.        -T.r.         ,..,!..  ,  .be       a      transition 

in  the  dissertation  De  Lege  Continmtatis  ;  by  it  indeed  it  is  proved  that  a  theory  of  attractive  {rom    positive    to 
&  repulsive   forces  for  different  distances  is   far  more  reasonable  than  one  of   attractive  negative ;      mves- 
forces  only,  or  of  repulsive  forces  only.     Let  us  imagine  that  we  are  quite  ignorant  of  the  of8the°nature  of  the 
kind  of  forces  that  exist  in  Nature,  whether  they  are  only  attractive  or  only  repulsive,  or  curve  only, 
both  ;    it  would  be  allowable  to  use  the  following  reasoning  to  help  us  to  investigate  the 
matter.     Without  doubt  there  will  be  some  continuous  line  which,  by  means  of  ordinates 
drawn  from  it  to  an  axis  representing  distances,  will  determine  the  forces ;   &  according 


ioo 


PHILOSOPHIC  NATURALIS  THEORIA 


cent.qi 


secuerit  ;  vires  erunt  alibi  attractive,  alibi  repulsivae  ;  vel  ubique  attractive  tantum,  aut 
repulsive  tantum.  Videndum  igitur,  an  sit  ration!  consentaneum  magis,  lineam  ejus 
naturae,  &  positionis  censere,  ut  axem  alicubi  secet,  an  ut  non  secet. 

Transitum    deduci  U2.  Inter  rectas  axem  rectilineum  unica  parallela  ducta  per  quod  vis  datum  punctum 

sint  0>curvse,  Pquas  non  secatj  omnes  alie  numero  infinitae  secant  alicubi.  Curvarum  nulla  est,  quam  infinitae 
recte  secent,  quam  numero  rectae  secare  non  possint  ;  &  licet  aliquae  curvae  ejus  naturae  sint,  ut  eas  aliquae  rectae 
non  secent  ;  tamen  &  eas  ipsas  aliae  infinite  numero  recte  secant,  &  infinite  numero  curve, 
quod  Geometrie  sublimioris  peritis  est  notissimum,  sunt  ejus  nature,  ut  nulla  prorsus  sit 
recta  linea,  a  qua  possint  non  secari.  Hujusmodi  ex.  gr.  est  parabola  ilia,  cujus  ordinate 
sunt  in  ratione  triplicata  abscissarum.  Quare  infinite  numero  curve  sunt,  &  infinite 
numero  rectae,  que  sectionem  necessario  habeant,  pro  quavis  recta,  que  non  habeat,  &  nulla 
est  curva,  que  sectionem  cum  axe  habere  non  possit.  Ergo  inter  casus  possibles  multo 
plures  sunt  ii,  qui  sectionem  admittunt,  quam  qui  ea  careant  ;  adeoque  seclusis  rationibus 
aliis  omnibus,  &  sola  casuum  probabilitate,  &  rei  [51]  natura  abstracte  considerata,  multo 
magis  rationi  consentaneum  est,  censere  lineam  illam,  que  vires  exprimat,  esse  unam  ex  iis, 
que  axem  secant,  quam  ex  iis,  que  non  secant,  adeoque  &  ejusmodi  esse  virium  legem,  ut 
attractiones,  &  repulsiones  exhibeat  simul  pro  diversis  distantiis,  quam  ut  alteras  tantummodo 
referat  ;  usque  adeo  rei  natura  considerata  non  solam  attractionem,  vel  solam  repulsionem, 
sed  utramque  nobis  objicit  simul. 


punctis 

a  recta. 


secabiles 


Ulterior    perqui-  u*    ged  eodem  argumento  licet  ultenus  quoque  progredi,  &  primum  etiam  difficultatis 

sitio:     curvarum  J  ,      °  o    -j    •  •  •  •  •  i  • 

genera  :    quo  aiti-  caput  amovere,  quod  a  sectionum,  &  idcirco  etiam  arcuum  jam  attractivorum,  jam  repulsi- 
ores,  eo  in  piuribus  vorum  multiplicitate  desumitur.     Curvas  lineas  Geometre  in  quasdam  classes  dividunt 

uni  •,  ......... 

°Pe  anaiyseos,  que  earum  naturam  expnmit  per  mas,  quas  Analyste  appellant,  equationes, 
&  que  ad  varies  gradus  ascendunt.  Aequationes  primi  gradus  exprimunt  rectas  ;  equati- 
ones secundi  gradus  curvas  primi  generis  ;  equationes  tertii  gradus  curvas  secundi  generis, 
atque  ita  porro  ;  &  sunt  curve,  que  omnes  gradus  transcendunt  finite  Algebre,  &  que 
idcirco  dicuntur  transcendentes.  Porro  illud  demonstrant  Geometre  in  Analysi  ad 
Geometriam  applicata,  lineas,  que  exprimuntur  per  equationem  primi  gradus,  posse 
secari  a  recta  in  unico  puncto  ;  que  equationem  habent  gradus  secundi,  tertii,  &  ita  porro, 
secari  posse  a  recta  in  punctis  duobus,  tribus,  &  ita  porro  :  unde  fit,  ut  curva  noni,  vel 
nonagesimi  noni  generis  secari  possit  a  recta  in  punctis  decem,  vel  centum. 


itidem 

sum  plures  in  eo- 


Jam  vero  curvae  primi  generis  sunt  tantummodo  tres  conice  sectiones,  ellipis, 
parabola,  hyperbola,  adnumerato  ellipsibus  etiam  circulo,  que  quidem  veteribus  quoque 
Geometris  innotuerunt.  Curvas  secundi  generis  enumeravit  Newtonus  omnium  primus, 
&  sunt  circiter  octoginta  ;  curvarum  generis  tertii  nemo  adhuc  numerum  exhibuit  accura- 
tum,  &  mirum  sane,  quantus  sit  is  ipse  illarum  numerus.  Sed  quo  altius  assurgit  curve 
genus,  eo  plures  in  eo  genere  sunt  curve,  progressione  ita  in  immensum  crescente,  ut  ubi 
aliquanto  altius  ascenderit  genus  ipsum,  numerus  curvarum  omnem  superet  humane 
imaginationis  vim.  Idem  nimirum  ibi  accidit,  quod  in  combinationibus  terminorum,  de 
quibus  supra  mentionem  fecimus,  ubi  diximus  a  24  litterulis  omnes  exhiberi  voces  linguarum 
omnium,  &  que  fuerunt,  aut  sunt,  &  que  esse  possunt. 


Deductio  inde  piu-          jjr    Inde    iam    pronum    est    argumentationem    hujusmodi    instituere.      Numerus 

rimarum     mtersec-   ..  J  ..  .  ,..J.. 

tionum,  axis,  &  linearum,  que  axem  secare  possint  in  punctis  quamplunmis,  est  in  immensum  major  earum 
curvae  exprimentis  numero,  quae  non  possint,  nisi  in  paucis,  vel  unico  :  igitur  ubi  agitur  de  linea  exprimente 
legem  virium,  ei,  qui  nihil  aliunde  sciat,  in  immensum  probabilius  erit,  ejusmodi  lineam 
esse  ex  prio-[52]-rum  genere  unam,  quam  ex  genere  posteriorum,  adeoque  ipsam  virium 
naturam  plurimos  requirere  transitus  ab  attractionibus  ad  repulsiones,  &  vice  versa,  quam 
paucos,  vel  nullum. 


-  Sed  omissa  ista  conjecturali  argumentatione  quadam,  formam  curve  exprimentis 

simpiicem:  in  quo  vires  positive  argumento  a  phenomenis  Nature  deducto  nos  supra  determinavimus  cum 
plurimis  intersectionibus,  que  transitus  ejusmodi  quamplurimos  exhibeant.  Nee  ejusmodi 
curva  debet  esse  e  piuribus  arcubus  temere  compaginata,  &  compacta  :  diximus  enim, 


11        * 


A  THEORY  OF  NATURAL  PHILOSOPHY  101 

as  it  will    cut  the  axis,  or  will  not,  the  forces  will  be  either  partly  attractive  &  partly 
repulsive,  or  everywhere  only  attractive  or  only  repulsive.     Accordingly  it  is  to  be  seen    • 
if  it  is  more  reasonable  to  suppose  that  a  line  of  this  nature  &  position  cuts  the  axis  anywhere, 
or  does  not. 

112.  Amongst  straight  lines  there  is  only  one,  drawn  parallel  to  the  rectilinear  axis,  intersection   is  to 
through  any  given  point  that  does  not  cut  the  axis;  all  the  rest  (infinite  in  number)  will  the  factThat  tfhere 
cut  it  somewhere.     There  is  no  curve  that  an  infinite  number  of  straight  lines  cannot  cut  ;  are  more  lines  that 
&  although  there  are  some  curves  of  such  a  nature  that  some  straight  lines  do  not  cut  them,  thL^es^hat^o 
yet  there  are  an  infinite  number  of  other  straight  lines  that  do  cut  these  curves  ;  &  there  not. 

are  an  infinite  number  of  curves,  as  is  well-known  to  those  versed  in  higher  geometry,  of 
such  a  nature  that  there  is  absolutely  not  a  single  straight  line  by  which  they  cannot  be 
cut.  An  example  of  this  kind  of  curve  is  that  parabola,  in  which  the  ordinates  are  in  the 
triplicate  ratio  of  the  abscissae.  Hence  there  are  an  infinite  number  of  curves  &  an 
infinite  number  of  straight  lines  which  necessarily  have  intersection,  corresponding  to  any 
straight  line  that  has  not  ;  &  there  is  no  curve  that  cannot  have  intersection  with  an 
axis.  Therefore  amongst  the  cases  that  are  possible,  there  are  far  more  curves  that  admit 
intersection  than  those  that  are  free  from  it  ;  hence,  putting  all  other  reasons  on  one  side, 
&  considering  only  the  probability  of  the  cases  &  the  nature  of  the  matter  on  its  own 
merits,  it  is  far  more  reasonable  to  suppose  that  the  line  representing  the  forces  is  one  of 
those,  which  cut  the  axis,  than  one  of  those  that  do  not  cut  it.  Thus  the  law  of  forces 
is  such  that  it  yields  both  attractions  &  repulsions  (for  different  distances),  rather  than 
such  that  it  deals  with  either  alone.  Thus  far  the  nature  of  the  matter  has  been  considered, 
with  the  result  that  it  presents  to  us,  not  attraction  alone,  nor  repulsion  alone,  but  both  of 
these  together. 

113.  But  we  can  also  proceed  still  further  adopting  the  same  line  of  argument,  &  Further  investiga- 
first  of  all  remove  the  chief  point  of  the  difficulty,  that  is  derived  from  the  multiplicity  S^L.^JILhi 

ri*  */i  i  i  p       i  i  i  •  curves  ,    nit,  iijgiicr 

of  the  intersections,  &  consequently  also  of  the  arcs  alternately  attractive  &  repulsive,  their    order,    the 

Geometricians  divide  curves  into  certain  classes  by  the  help  of  analysis,  which  expresses  wWcV^a  ^teaight 

their  nature  by  what  the  analysts  call  equations  ;    these  equations  rise  to  various  degrees,  line  can  cut  them. 

Equations  of  the  first  degree  represent  straight  lines,  equations  of  the  second  degree  represent 

curves  of  the  first  class,  equations  of  the  third  degree  curves  of  the  second  class,  &  so  on. 

There  are  also  curves  which  transcend  all  degrees  of  finite  algebra,  &  on  that  account 

these  are  called  transcendental  curves.     Further,  geometricians  prove,  in  analysis  applied 

to  geometry,  that  lines  that  are  expressed  by  equations  of  the  first  degree  can  be  cut  by  a 

straight  line  in  one  point  only  ;   those  that  have  equations  of  the  second,  third,  &  higher 

degrees  can  be  cut  by  a  straight  line  in  two,  three,  &  more  points  respectively.     Hence 

it  comes  about  that  a  curve  of  the  ninth,  or  the  ninety-ninth  class  can  be  cut  by  a  straight 

line  in  ten,  or  in  a  hundred,  points. 

114.  Now  there  are  only  three  curves  of  the  first  class,  namely  the  conic  sections,  the  As  the  class   gets 
parabola,  the  ellipse  &  the  hyperbola;   the  circle  is  included  under  the  name  of  ellipse;     gh  " 


of  that 
&  these  three  curves  were  known  to  the  ancient   geometricians   also.     Newton  was  the  class  becomes  im- 

first  of  all  persons  to  enumerate  the  curves  of  the  second  class,  &  there  are  about  eighty  mensely  greater. 

of  them.     Nobody  hitherto  has  stated  an  exact  number  for  the  curves  of  the  third  class  ; 

&  it  is  really  wonderful  how  great  is  the  number  of  these  curves.     Moreover,  the  higher 

the  class  of  the  curve  becomes,  the  more  curves  there  are  in  that  class,  according  to  a 

progression  that  increases  in  such  immensity  that,  when  the  class  has  risen  but  a  little  higher, 

the  number  of  curves  will  altogether  surpass  the  fullest  power  of  the  human  imagination. 

Indeed  the  same  thing  happens  in  this  case  as  in  combinations  of  terms  ;   we  mentioned 

the    latter    above,    when    we   said   that   by    means    of  24  little  letters   there  can    be 

expressed  all  the  words  of  all  languages  that  ever  have  been,   or  are,   or  can  be    in 

the  future. 

115.  From  what  has  been  said  above  we  are  led  to  set  up  the  following  line  of  argument.  Hence  we  deduce 
The  number  of  lines  that  can  cut  the  axis  in  very  many  points  is  immensely  greater  than  that  there.  are  ^ 

,    ,  ,  ....  '        ,     '    r.  .      .  '  f>  many  intersections 

the  number  of  those  that  can  cut  it  in  a  few  points  only,  or  in  a  single  point.     Hence,  when  Of  the  axis  and  the 
the  line  representing  the  law  of  forces  is  in  question,  it  will  appear  to  one.  who  otherwise  ?urve  representing 

i  i  •  i  •  i         •     •     •  i  r   i     111  ,  forces. 

knows  nothing  about  its  nature,  that  it  is  immensely  more  probable  that  the  curve  is  of 
the  first  kind  than  that  it  is  of  the  second  kind  ;  &  therefore  that  the  nature  of  the  forces 
must  be  such  as  requires  a  very  large  number  of  transitions  from  attractions  to  repulsions 
&  back  again,  rather  than  a  small  number  or  none  at  all. 

116.  But,  omitting  this  somewhat  conjectural  line  of  reasoning,  we  have  already  it  may  be  that  the 
determined,  by  what  has  been  said  above,  the  form  of  the  curve  representing  forces  by  a  j|£™?  I^SSlJ8 

.  '  rxr  /iii  simple  ,  tnecuarac- 

ngorous  argument  derived  trom  the  phenomena  of  Nature,  &  that  there  are  very  many  teristic  of  simplicity 
intersections  which  represent  just  as  many  of  these  transitions.     Further,  a  curve  of  this  mcurves- 


102 


PHILOSOPHIC  NATURALIS  THEORIA 


notum  esse  Geometris,  infinita  esse  curvarum  genera,  quae  ex  ipsa  natura  sua  debeant  axem 
in  plurimis  secare  punctis,  adeoque  &  circa  ipsum  sinuari ;  sed  praeter  hanc  generalem 
responsionem  desumptam  a  generali  curvarum  natura,  in  dissertatione  De  Lege  Firium  in 
Natura  existentium  ego  quidem  directe  demonstravi,  curvam  illius  ipsius  formae,  cujusmodi 
ea  est,  quam  in  fig.  i  exhibui,  simplicem  esse  posse,  non  ex  arcubus  diversarum  curvarum 
compositam.  Simplicem  autem  ejusmodi  curvam  affirmavi  esse  posse  :  earn  enim  simplicem 
appello,  quae  tota  est  uniformis  naturae,  quae  in  Analysi  exponi  possit  per  aequationem  non 
resolubilem  in  plures,  e  quarum  multiplicatione  eadem  componatur  cujuscunque  demum 
ea  curva  sit  generis,  quotcunque  habeat  flexus,  &  contorsiones.  Nobis  quidem  altiorum 
generum  curvae  videntur  minus  simplices  ;  quh  nimirum  nostrae  humanae  menti,  uti  pluribus 
ostendi  in  dissertatione  De  Maris  Aestu,  &  in  Stayanis  Supplementis,  recta  linea  videtur 
omnium  simplicissima,  cujus  congruentiam  in  superpositione  intuemur  mentis  oculis 
evidentissime,  &  ex  qua  una  omnem  nos  homines  nostram  derivamus  Geometriam  ;  ac 
idcirco,  quae  lineae  a  recta  recedunt  magis,  &  discrepant,  illas  habemus  pro  compositis,  & 
magis  ab  ea  simplicitate,  quam  nobis  confinximus,  recedentibus.  At  vero  lineae  continuae, 
&  uniformis  naturae  omnes  in  se  ipsis  sunt  aeque  simplices  ;  &  aliud  mentium  genus,  quod 
cujuspiam  ex  ipsis  proprietatem  aliquam  aeque  evidenter  intueretur,  ac  nos  intuemur 
congruentiam  rectarum,  illas  maxime  simplices  esse  crederet  curvas  lineas,  ex  ilia  earum 
proprietate  longe  alterius  Geometrise  sibi  elementa  conficeret,  &  ad  illam  ceteras  referret 
lineas,  ut  nos  ad  rectam  referimus ;  quas  quidem  mentes  si  aliquam  ex.  gr.  parabolae  pro- 
prietatem intime  perspicerent,  atque  intuerentur,  non  illud  quaarerent,  quod  nostri 
Geometrae  quaerunt,  ut  parabolam  rectificarent,  sed,  si  ita  loqui  fas  est,  ut  rectam 
parabolarent. 


Problema  continens  1 1 7.  Et  quidem  analyseos  ipsius  profundiorem  cognitionem  requirit  ipsa  investigatio 

naturam  curvaeana-  aequationis,  qua    possit    exprimi    curva  ems   formae,  quae  meam  exhibet  virium  legem. 

lytice  expnmendam.    „/!  j-  •  •  11  ji  -i 

Quamobrem  hie  tantummodo  exponam  conditiones,  quas  ipsa  curva  habere  debet,  &  quibus 
aequatio  ibi  inventa  satis  facere  [53]  debeat.  (c)  Continetur  autem  id  ipsum  num.  75, 
illius  dissertationis,  ubi  habetur  hujusmodi  Problema :  Invenire  naturam  curvce,  cujus 
abscissis  exprimentibus  distantias,  ordinal  exprimant  vires,  mutatis  distantiis  utcunque 
mutatas,  y  in  datis  quotcunque  limitibus  transeuntes  e  repulsivis  in  attractivas,  ac  ex  attractivis 
in  repulsivas,  in  minimis  autem  distantiis  repulsivas,  W  ita  crescentes,  ut  sint  pares  extinguendce 
cuicunque  velocitati  utcunque  magnce.  Proposito  problemate  illud  addo  :  quoniam  posuimus 
mutatis  distantiis  utcunque  mutatas,  complectitur  propositio  etiam  rationem  quee  ad  rationem 
reciprocam  duplicatam  distantiarum  accedat,  quantum  libuerit,  in  quibusdam  satis  magnis 
distantiis. 


Conditiones    ejus 
problematis. 


1 18.  His  propositis  numero  illo  75,  sequenti  numero  propono  sequentes  sex  conditiones, 
quae  requirantur,  &  sufficiant  ad  habendam  curvam,  quse  quaeritur.  Primo  :  ut  sit  regularis, 
ac  simplex,  &  non  composita  ex  aggregate  arcuum  diversarum  curvarum.  Secundo  :  ut  secet 
axem  C'AC  figures  i.  tantum  in  punctis  quibusdam  datis  ad  binas  distantias  AE',  AE ;  AG', 
AG  ;  y  ita  porro  cequales  (d)  bine,  y  inde.  Tertio  :  ut  singulis  abscissis  respondeant  singulcs 
ordinatcf.  (e)  Quarto  :  ut  sumptis  abscissis  cequalibus  hinc,  y  inde  ab  A,  respondeant  ordinal* 


(c)  Qui  velit  ipsam  rei  determinationem  videre,  poterit  hie  in  fine,  ubi  Supphmentorum,  §  3.  exhibebitur  solutio 
problematis,  qua  in  memorata  dissertatione  continetur  a  num.  77.     ad    no.     Sed    W    numerorum   ordo,  &  figurarum 
mutabitur,  ut  cum  reliquis  hujusce  operis  cohtereat. 

Addetur  prieterea  eidem  §.  postremum  scholium  pertinens  ad  qu<sstionem  agitatam  ante  has  aliquot  annos  Parisiis  ; 
an  vis  mutua  inter  materite  particulas  debeat  omnino  exprimi  per  solam  aliquam  distantiee  potenttam,  an  possit  per 
aliquam  ejus  functionem  ;  W  constabit,  posse  utique  per  junctionem,  ut  hie  ego  presto,  qute  uti  superiore  numero  de  curvts 
est  dictum,  est  in  se  eeque  simplex  etiam,  ubi  nobis  potentias  ad  ejus  expressionem  adhibentibus  videatur  admodum 
composita. 

(d)  Id,  ut  y  quarta  conditio,  requiritur,  ut  curva  utrinque  sit  sui  similis,  quod  ipsam  magis  uniformem  reddit  ; 
quanquam  de  illo  crure,  quod  est  citra  asymptotum  AB,  nihil  est,  quod  soliciti  simus  ;   cum  ob  vim  repulsivam  imminutis 
distantiis  ita  in  infinitum  excrescentem,  non  possit  abscissa  distantiam  exprimens  unquam  evadere  zero,   W  abire  in 
negativam. 

(e)  Nam  singulis  distantiis  singulte  vires  respondent. 


A  THEORY -OF  NATURAL  PHILOSOPHY  103 

kind  is  not  bound  to  be  built  up  by  connecting  together  a  number  of  independent  arcs. 
For,  as  I  said,  it  is  well  known  to  Geometricians  that  there  are  an  infinite  number  of  classes 
of  curves  that,  from  their  very  nature,  must  cut  the  axis  in  a  very  large  number  of  points, 
&  therefore  also  wind  themselves  about  it.  Moreover,  in  addition  to  this  general  answer 
to  the  objector,  derived  from  the  general  nature  of  curves,  in  my  dissertation  De  Lege 
Firium  in  Natura  existentium,  I  indeed  proved  in  a  straightforward  manner  that  a  curve, 
of  the  form  that  I  have  given  in  Fig.  i,  might  be  simple  &  not  built  up  of  arcs  of  several 
different  curves.  Further,  I  asserted  that  a  simple  curve  of  this  kind  was  perfectly  feasible  ; 
for  I  call  a  curve  simple,  when  the  whole  of  it  is  of  one  uniform  nature.  In  analysis,  this 
can  be  expressed  by  an  equation  that  is  not  capable  of  being  resolved  into  several  other 
equations,  such  that  the  former  is  formed  from  the  latter  by  multiplication  ;  &  that  too, 
no  matter  of  what  class  the  curve  may  be,  or  how  many  flexures  or  windings  it  may  have. 
It  is  true  that  the  curves  of  higher  classes  seem  to  us  to  be  less  simple  ;  this  is  so  because, 
as  I  have  shown  in  several  places  in  the  dissertation  De  Marts  Aestu,  &  the  supplements 
to  Stay's  Philosophy,  a  straight  line  seems  to  our  human  mind  to  be  the  simplest  of  all 
lines ;  for  we  get  a  real  clear  mental  perception  of  the  congruence  on  superposition  in  the 
case  of  a  straight  line,  &  from  this  we  human  beings  form  the  whole  of  our  geometry. 
On  this  account,  the  more  that  lines  depart  from  straightness  &  the  more  they  differ, 
the  more  we  consider  them  to  be  composite  &  to  depart  from  that  simplicity  that  we  have 
set  up  as  our  standard.  But  really  all  lines  that  are  continuous  &  of  uniform  nature 
are  just  as  simple  as  one  another.  Another  kind  of  mind,  which  might  form  an  equally 
clear  mental  perception  of  some  property  of  any  one  of  these  curves,  as  we  do  the  congruence 
of  straight  lines,  might  believe  these  curves  to  be  the  simplest  of  all  &  from  that  property 
of  these  curves  build  up  the  elements  of  a  far  different  geometry,  referring  all  other  curves 
to  that  one,  just  as  we  compare  them  with  a  straight  line.  Indeed,  these  minds,  if  they 
noticed  &  formed  an  extremely  clear  perception  of  some  property  of,  say,  the  parabola, 
would  not  seek,  as  our  geometricians  do,  to  rectify  the  parabola  ;  they  would  endeavour, 
if  one  may  use  the  words,  to  parabolify  a  straight  line. 

1 17.  The  investigation  of  the  equation,  by  which  a  curve  of  the  form  that  will  represent  pT°bl!Jn  . 

'  i    ~     •  j'  i  i    j          f         i     •    •       ir       1T71.        r          "*™  the  analytical 

my  law  of  forces  can  be  expressed,  requires  a  deeper  knowledge  01  analysis  itselt.     Wnereiore  expression  of    the 

I  will  here  do  no  more  than  set  out  the  necessary  requirements  that  the  curve  must  fulfil  nature  of  the  curve. 

&  those  that  the  equation  thereby  discovered  must  satisfy. (c)     It  is  the  subject  of  Art.  75 

of  the  dissertation  De  Lege  Firium,  where  the  following  problem  is  proposed.     Required 

to  find  the  nature  of  the  curve,  whose  abscissa  represent  distances  &  whose  ordinates  represent 

forces  that  are  changed  as  the  distances  are  changed  in  any  manner,  y  pass  from  attractive 

forces  to  repulsive,  &  from  repulsive  to  attractive,  at  any  given  number  of  limit-points  ;  further, 

the  forces  are  repulsive  at  extremely  small  distances  and  increase  in  such  a  manner  that  they 

are  capable  of  destroying  any  velocity,  however  great  it  may  be.     To  the  problem  as  there 

proposed  I  now  add  the  following  : — As  we  have  used  the  words  are  changed  as  the  distances 

are  changed  in  any  manner,  the  proposition  includes  also  the  ratio  that  approaches  as  nearly 

as  you  please  to  the  reciprocal  ratio  of  the  squares  of  the  distances,  whenever  the  distances  are 

sufficiently  great. 

1 1 8.  In  addition  to  what  is  proposed  in  this  Art.  75,  I  set  forth  in  the  article  that  The  0*  of 
follows  it  the  following  six  conditions ;    these  are  the  necessary  and  sufficient  conditions 

for  determining  the  curve  that  is  required. 

(i)  The  curve  is  regular  &  simple,  &  not  compounded  of  a  number  of  arcs  of  different  curves. 

(ii)  It  shall  cut  the  axis  C'AC  of  Fig.  I,  only  in  certain  given  points,  whose  distances, 
AE',AE,  AG',  AG,  and  so  on,  are  equal  (<t)  in  pairs  on  each  side  of  A  [see  p.  80]. 

(iii)  To  each  abscissa  there  shall  correspond  one  ordinate  y  one  only,  (f) 

(iv)  To  equal  abscisses,  taken  one  on  each  side  of  A,  there  shall  correspond  equal  ordinates. 

(c)  Anyone  who  desires  to  see  the  solution  of  the  -problem  will  be  able  to  do  seat  the  end  of  this  work;  it  will  be 
found  in  §  3  of  the  Supplements  ;  it  is  the  solution  of  the  problem,  as  it  was  given  in  the  dissertation  mentioned  above, 
from  Art.  77  to   no.     But  here  both  the  numbering  of  the  articles  W  of  the  diagrams  have  been  changed,  so  as  to 
agree  with  the  rest  of  the  work.     In  addition,  at  the  end  of  this  section,    there  will  be  found  a  final  note  dealing 
with  a  question  that  was  discussed  some  years  ago  in  Paris.    Namely,  whether  the  mutual  force  between  particles  of  mat- 
ter is  bound  to  be  expressible  by  some  one  power  of  the  distance  only,  or  by  some  function  of  the  distance.     It  will  be 
evident  that  at  any  rate  it  may  be  expressible  by  a  function  as  I  here  assert  ;  y  that  function,  as  has  been  stated  in  the 
article  above,  is  perfectly  simple  in  itself  also  ;  whereas,  if  we  adhere  to  an  expression  by  means  of  powers,  the  curve  will 
seem  to  be  altogether  complex. 

(d)  This,  y  the  fourth  condition  too,  is  required  to  make  the  curve  symmetrical,  thus  giving  it  greater  uniformity  ; 
although  we  are  not  concerned  with  the  branch  on  the  other  side  of  the  asymptote  AB  at  all.     For,  on  account  of  the 
repulsive  force  at  very  small  distances  increasing  indefinitely  in  such  a  manner  as  postulated,  it  is  impossible  that  the 
abscissa  that  represents  the  distance  should  ever  become  zero  y  then  become  negative. 

(e)  For  to  each  distance  one  force,  &  and  only  one,  corresponds. 


104 


PHILOSOPHIC  NATURALIS  THEORIA 


czquales.  Quinto  :  ut  babeant  rectam  AB  pro  asymptoto,  area  asymptotica  BAED  existente  (£) 
infinita.  Sexto  :  ut  arcus  binis  quibuscunque  intersectionibus  terminati  possint  variari,  ut 
libuerit,  fcf?  ad  quascunque  distantias  recedere  ab  axe  C'AC,  ac  accedcre  ad  quoscunque  quarum- 
cunque  curvarum  arcus,  quantum  libuerit,  eos  secanda,  vel  tangendo,  vel  osculando  ubicunque, 
£ff  quomodocunque  libuerit. 

soiutio  IrTUattrac  IS4]  IT9-  Verum  quod  ad  multiplicitatem  virium  pertinet,  quas  diversis  jam  Physici 
tionem  gravitatis  nominibus  appellant,  illud  hie  etiam  notari  potest,  si  quis  singulas  seorsim  considerare 
velit,  licere  illud  etiam,  hanc  curvam  in  se  unicam  per  resolutionem  virium  cogitatione 
nostra,  atque  fictione  quadam,  dividere  in  plures.  Si  ex.  gr.  quis  velit  considerare  in  materia 
gravitatem  generalem  accurate  reciprocam  distantiarum  quadratis  ;  poterit  sane  is  describere 
ex  parte  attractiva  hyperbolam  illam,  quae  habeat  accurate  ordinatas  in  ratione  reciproca 
duplicata  distantiarum,  quse  quidem  erit  quaedam  velut  continuatio  cruris  VTS,  turn  singulis 
ordinatis  ag,  dh  curvae  virium  expressae  in  fig.  I.  adjungere  ordinatas  hujus  novae  hyperbolae 
ad  partes  AB  incipiendo  a  punctis  curvae  g,  b,  &  eo  pacto  orietur  nova  quaedam  curva,  quae 
versus  partes  pV  coincidet  ad  sensum  cum  axe  oC,  in  reliquis  locis  ab  eo  distabit,  &  contor- 
quebitur  etiam  circa  ipsum,  si  vertices  F,  K,  O  distiterint  ab  axe  magis,  quam  distet  ibidem 
hyperbola  ilia.  Turn  poterit  dici,  puncta  omnia  materiae  habere  gravitatem  decrescentem 
accurate  in  ratione  reciproca  duplicata  distantiarum,  &  simul  habere  vim  aliam  expressam 
ab  ilia  nova  curva  :  nam  idem  erit,  concipere  simul  hasce  binas  leges  virium,  ac  illam 
praecedentem  unicam,  &  iidem  effectus  orientur. 


Hujus  posterioris 
vis  resolutio  in  alias 
plures. 


1 20.  Eodem  pacto  haec  nova  curva  potest  dividi  in  alias  duas,  vel  plures,  concipiendo 
aliam  quamcunque  vim,  ut  ut  accurate  servantem  quasdam  determinatas  leges,  sed  simul 
mutando  curvam  jam  genitam,  translatis  ejus  punctis  per  intervalla  aequalia  ordinatis 
respondentibus  novae  legi  ass.umptae.  Hoc  pacto  habebuntur  plures  etiam  vires  diversae, 
quod  aliquando,  ut  in  resolutione  virium  accidere  diximus,  inserviet  ad  faciliorem  deter- 
minationem  effectuum,  &  ea  erit  itidem  vera  virium  resolutio  quaedam  ;  sed  id  omne  erit 
nostrae  mentis  partus  quidam ;  nam  reipsa  unica  lex  virium  habebitur,  quam  in  fig.  I . 
exposui,  &  quae  ex  omnibus  ejusmodi  legibus  componetur. 


Non    obesse    theo- 
r  i  a  m     gravitatis  ; 


distantiis     locum 
non  habet. 


121.  Quoniam  autem  hie  mentio  injecta  est  gravitatis  decrescentis  accurate  in  ratione 
cujusiex1naminimis  reciproca  duplicata  distantiarum  ;  cavendum,  ne  cui  difficultatem  aliquam  pariat  illud, 
'"""m  quod  apud  Physicos,  &  potissimum  apud  Astronomiae  mechanicae  cultores,  habetur  pro 
comperto,  gravitatem  decrescere  in  ratione  reciproca  duplicata  distantiarum  accurate, 
cum  in  hac  mea  Theoria  lex  virium  discedat  plurimum  ab  ipsa  ratione  reciproca  duplicata 
distantiarum.  Inprimis  in  minoribus  distantiis  vis  integra,  quam  in  se  mutuo  exercent 
particulae,  omnino  plurimum  discrepat  a  gravitate,  quae  sit  in  ratione  reciproca  duplicata 
distantiarum.  Nam  &  vapores,  qui  tantam  exercent  vim  ad  se  expandendos,  repulsionem 
habent  utique  in  illis  minimis  distantiis  a  se  invicem,  non  attractionem  ;  &  ipsa  attractio, 
quae  in  cohaesione  se  prodit,  est  ilia  quidem  in  immensum  major,  quam  quae  ex  generali 
gravitate  consequitur  ;  cum  ex  ipsis  Newtoni  compertis  attractio  gravitati  respondens  [55] 
in  globes  homogeneos  diversarum  diametrorum  sit  in  eadem  ratione,  in  qua  sunt  globorum 
diametri,  adeoque  vis  ejusmodi  in  exiguam  particulam  est  eo  minor  gravitate  corporum  in 
Terram,  quo  minor  est  diameter  particulae  diametro  totius  Terrae,  adeoque  penitus  insen- 
sibilis.  Et  idcirco  Newtonus  aliam  admisit  vim  pro  cohaesione,  quae  decrescat  in  ratione 
majore,  quam  sit  reciproca  duplicata  distantiarum  ;  &  multi  ex  Newtonianis  admiserunt 

vim  respondentem  huic  formulae  -3  +  -v   cujus  prior  pars  respectu    posterioris    sit    in 

immensum  minor,  ubi  x  sit  in  immensum  major  unitate  assumpta  ;  sit  vero  major,  ubi  x 
sit  in  immensum  minor,  ut  idcirco  in  satis  magnis  distantiis  evanescente  ad  sensum  prima 
parte,  vis  remaneat  quam  proxime  in  ratione  reciproca  duplicata  distantiarum  x,  in  minimis 
vero  distantiis  sit  quam  proxime  in  ratione  reciproca  triplicata  :  usque  adeo  ne  apud 
Newtonianos  quidem  servatur  omnino  accurate  ratio  duplicata  distantiarum. 


EX 


pianetarum 
™ 


I22.  Demonstravit  quidem  Newtonus,  in  ellipsibus  planetariis,  earn,  quam  Astronomi 
q^ampro™  lineam  apsidum  nominant,  &  est  axis  ellipseos,  habituram  ingentem  motum,  si  ratio  virium 
ime,  non  accurate.     a  reciproca  duplicata  distantiarum  aliquanto  magis  aberret,  cumque  ad  sensum  quiescant 

(f)  Id  requiritur,  quia  in  Mecbanica  demonstrator,  aream  curves,  cujus  abscissa  fxprimant  distantias,  13  ordinatx 
vires,  exprimere  incrementum,  vel  decrementum  quadrati  velocitatis  :  quare  ut  illte  vires  sint  pares  extinguendte  veloci- 
tati  cuivis  utcunque  magna,  debet  ilia  area  esse  omni  finita  major. 


A  THEORY  OF  NATURAL  PHILOSOPHY  105 

(v)  The  straight  line  AB  shall  be  an  asymptote,  and  the  asymptotic  area  BAED  shall  be 
infinite.  (f) 

(vi)  The  arcs  lying  between  any  two  intersections  may  vary  to  any  extent,  may  recede  to  any 
distances  whatever  from  the  axis  C  AC,  and  approximate  to  any  arcs  of  any  curves  to  any  degree 
of  closeness,  cutting  them,  or  touching  them,  or  osculating  them,  at  any  points  and  in  any  manner. 

119.  Now,  as  regards  the  multiplicity  of  forces  which  at  the  present  time  physicists  call  Resolution  of  the 
by  different  names,  it  can  also  here  be  observed  that,  if  anyone  wants  to  consider  one  of  these  £Uj^f  of  N^wtonUn 
separately,  the  curve  though  it  is  of  itself  quite  one-fold  can  yet  be  divided  into  several  attraction     of 
parts  by  a  sort  of   mental  &  fictitious  resolution  of   the  forces.     Thus,  for  instance,  if  f^fother1  force*1  d 
anyone  wishes  to  consider  universal  gravitation  of  matter  exactly  reciprocal  to  the  squares 
of  the  distances  ;    he  can  indeed  describe  on  the  attractive  side  the  hyperbola  which  has 
its  ordinates  accurately  in  the  inverse  ratio  of  the  squares  of  the  distances,  &  this  will  be 
as  it  were  a  continuation  of  the  branch  VTS.     Then  he  can  add  on  to  every  ordinate,  such 
as  ag,  dh,  the  ordinates  of  this  new  hyperbola,  in  the  direction  of  AB,  starting  in  each  case 
from  points  on  the  curve,  as  g,h  ;  &  in  this  way  there  will  be  obtained  a  fresh  curve,  which 
for  the  part  pV  will  approximately  coincide  with  the  axis  0C,  &  for  the  remainder  will 
recede  from  it  &  wind  itself  about  it,  if  the  vertices  F,K,O  are  more  distant  from  the 
axis  than  the  corresponding  point  on  the  hyperbola.     Then  it  can  be  stated  that  all  points 
of  matter  have  gravitation  accurately  decreasing  in  the  inverse  square  of  the  distance, 
together  with  another  force  represented  by  this  new  curve.     For  it  comes  to  the  same 
thing  to  think  of  these  two  laws  of  forces  acting  together  as  of  the  single  law  already 
given  ;    &  the  results  that  arise  will  be  the  same  also. 

1  20.  In  the  same  manner  this  new  curve  can  be  divided  into  two  others,  or  several  The  resolution    of 
others,  by  considering  some  other  force,  in  some  way  or  other  accurately  obeying  certain  ^o    several  other 
fixed  laws,  &  at  the  same  time  altering  the  curve  just  obtained  by  translating  the  points  of  it  forces. 
through  intervals  equal  to  the  ordinates  corresponding  to  the  new  law  that  has  been  taken. 
In  this  manner  several  different  forces  will  be  obtained  ;    &  this  will  be  sometimes  useful, 
as  we  mentioned  that  it  would  be  in  resolution  of  forces,  for  determining  their  effects  more 
readily  ;   &  will  be  a  sort  of  true  resolution  of  forces.     But  all  this  will  be  as  it  were  only 
a  conception  of  our  mind  ;   for,  in  reality,  there  is  a  single  law  of  forces,  &  that  is  the  one 
which  I  gave  in  Fig.  i,  &  it  will  be  the  compounded  resultant  of  all  such  forces  as  the  above. 

121.  Moreover,  since  I  here  make  mention  of  gravitation  decreasing  accurately  in  the  The..  t.heor.y    °* 

,  r11.  ..  111  1111       gravitation     is  not 

inverse  ratio  ot  the  squares  ot  the  distances,  it  is  to  be  remarked  that  no  one  should  make  in  opposition  ;  this 

any  difficulty  over  the  fact  that,  amongst  physicists  &  more  especially  those  who  deal  with  l!J^0'Hd°tesv  not  holn 

celestial  mechanics,  it  is  considered  as  an  established  fact  that  gravitation  decreases  accurately  distances. 

in  the  inverse  ratio  of  the  squares  of  the  distances,  whilst  in  my  Theory  the  law  of  forces 

is  very  different  from  this  ratio.     Especially,  in  the  case  of  extremely  small  distances,  the 

whole  force,  which  the  particles  exert  upon  one  another,  will  differ  very  much  in  every 

case  from  the  force  of  gravity,  if  that  is   supposed  to  be  inversely  proportional    to    the 

squares   of  these  distances.     For,  in  the  case  of  gases,  which    exercise  such  a    mighty 

force  of  self-expansion,  there  is  certainly  repulsion  at  those  very  small  distances  from  one 

another,  &    not  attraction  ;    again,  the  attraction  that  arises  in  cohesion  is  immensely 

greater  than  it  ought  to  be  according  to  the  law  of  universal  gravitation.     Now,  from  the 

results  obtained  by  Newton,  the  attraction  corresponding  to  gravitation  in  homogeneous 

spheres  of   different  diameters  varies  as  the  diameters  of  the  spheres  ;    &  therefore  this 

kind  of  force  for  the  case  of  a  tiny  particle  is  as  small  in  proportion  to  the  gravitation  of 

bodies  to  the  Earth  as  the  diameter  of  the  particle  is  small  in  proportion  to  the  diameter 

of  the  whole  Earth  ;   &  is  thus  insensible  altogether.     Hence  Newton  admitted  another 

force  in  the  case  of  cohesion,  decreasing  in  a  greater  ratio  than  the  inverse  square  of  the 

distances  ;    also  many  of  the  followers  of  Newton  have  admitted  a  force  corresponding  to 

the  formula,  a'x3  +  b'x2  ;  in  this  the  first  term  is  immensely  less  than  the  second,  when  x 

is  immensely  greater  than  some  distance  assumed  as  unit  distance  ;  &  immensely  greater, 

when  x  is  immensely  less.     By  this  means,  at  sufficiently  great  distances  the  first  part 

practically  vanishes  &  the  force  remains  very  approximately  in  the  inverse  ratio  of  the  squares 

of  the  distances  x  ;    whilst,  at  very  small  distances,  it  is  very  nearly  in  the  inverse  ratio 

of  the  cubes  of  the  distances.    Thus  indeed,  not  even  amongst  the  followers  of  Newton  has 

the  inverse  ratio  of  the  squares  of  the  distances  been  altogether  rigidly  adhered  to. 

122.  Now  Newton  proved,  in  the  case  of  planetary  elliptic  orbits,  that  that  which  The     law    follows 
Astronomers  call  the  apsidal  line,  i.e.,  the  axis  of  the  ellipse,  would  have  a  very  great  motion,  not7 


, 

if  the  ratio  of  the  forces  varied  to  any  great  extent  from  the  inverse  ratio  of  the  squares  from  the  apheiia  of 
of  the  distances  ;   &  since  as  far  as  could  be  observed  the  lines  of  apses  were  stationary 

(f)  This  is  required  because  in  Mechanics  it  is  shown  that  the  area  of  a  curve,  whose  abscissa  r'present  distances 
y  ordinates  forces,  represents  the  increase  or  decrease  of  the  square  of  the  velocity.  Hence  in  order  that  the  forces 
should  be  capable  of  destroying  any  velocity  however  great,  this  area  must  be  greater  than  any  finite  area. 


io6  PHILOSOPHIC  NATURALIS  THEORIA 

in  earum  orbitis  apsidum  linese,  intulit,  earn  rationem  observari  omnino  in  gravitate.  At 
id  nequaquam  evincit,  accurate  servari  illam  legem,  sed  solum  proxime,  neque  inde  ullum 
efficax  argumentum  contra  meam  Theoriam  deduci  potest.  Nam  inprimis  nee  omnino 
quiescunt  illae  apsidum  lineae,  sive,  quod  idem  est,  aphelia  planetarum,  sed  motu  exiguo 
quidem,  at  non  insensibili  prorsus,  moventur  etiam  respectu  fixarum,  adeoque  motu  non 
tantummodo  apparente,  sed  vero.  Tribuitur  is  motus  perturbationi  virium  ortae  ex  mutua 
planetarum  actione  in  se  invicem  ;  at  illud  utique  hue  usque  nondum  demonstratum  est, 
ilium  motum  accurate  respondere  actionibus  reliquorum  planetarum  agentium  in  ratione 
reciproca  duplicata  distantiarum  ;  neque  enim  adhuc  sine  contemptibus  pluribus,  & 
approximationibus  a  perfectione,  &  exactitudine  admodum  remotis  solutum  est  problema, 
quod  appellant,  trium  corporum,  quo  quasratur  motus  trium  corporum  in  se  mutuo 
agentium  in  ratione  reciproca  duplicata  distantiarum,  &  utcunque  projectorum,  ac  illae 
ipsae  adhuc  admodum  imperfectae  solutiones,  quae  prolatae  hue  usque  sunt,  inserviunt 
tantummodo  particularibus  quibusdam  casibus,  ut  ubi  unum  corpus  sit  maximum,  & 
remotissimum,  quemadmodum  Sol,  reliqua  duo  admodum  minora  &  inter  se  proxima,  ut 
est  Luna,  ac  Terra,  vel  remota  admodum  a  majore,  &  inter  se,  ut  est  Jupiter,  &  Saturnus. 
Hinc  nemo  hucusque  accuratum  instituit,  aut  etiam  instituere  potuit  calculum  pro  actione 
perturbativa  omnium  planetarum,  quibus  si  accedat  actio  perturbativa  cometarum,  qui, 
nee  scitur,  quam  multi  sint,  nee  quam  longe  abeant ;  multo  adhuc  magis  evidenter  patebit, 
nullum  inde  confici  posse  argumentum  [56]  pro  ipsa  penitus  accurata  ratione  reciproca 
duplicata  distantiarum. 


I23-  Clairautius  quidem  in  schediasmate  ante  aliquot  annos  impresso,  crediderat,  ex 
autem  hanc  legem  ipsis  motibus  Kneje  apsidum  Lunae  colligi  sensibilem  recessum  a  ratione  reciproca  duplicata 
auantum  iftmerit[m  distantiae,  &  Eulerus  in  dissertatione  De  Aberrationibus  Jovis,  W  Saturni,  quas  premium 
retulit  ab  Academia  Parisiensi  an.  1748,  censuit,  in  ipso  Jove,  &  Saturno  haberi  recessum 
admodum  sensibilem  ab  ilia  ratione  ;  sed  id  quidem  ex  calculi  defectu  non  satis  product! 
sibi  accidisse  Clairautius  ipse  agnovit,  ac  edidit ;  &  Eulero  aliquid  simile  fortasse  accidit  : 
nee  ullum  habetur  positivum  argumentum  pro  ingenti  recessu  gravitatis  generalis  a  ratione 
duplicata  distantiarum  in  distantia  Lunae,  &  multo  magis  in  distantia  planetarum.  Vero 
nee  ullum  habetur  argumentum  positivum  pro  ratione  ita  penitus  accurata,  ut  discrimen 
sensum  omnem  prorsus  effugiat.  At  &  si  id  haberetur  ;  nihil  tamen  pati  posset  inde 
Theoria  mea  ;  cum  arcus  ille  meae  curvae  postremus  VT  possit  accedere,  quantum  libuerit, 
ad  arcum  illius  hyperbolae,  quae  exhibet  legem  gravitatis  reciprocam  quadratorum  dis- 
tantiae, ipsam  tangendo,  vel  osculando  in  punctis  quotcunque,  &  quibuscunque  ;  adeoque 
ita  possit  accedere,  ut  discrimen  in  iis  majoribus  distantiis  sensum  omnem  effugiat,  & 
effectus  nullum  habeat  sensibile  discrimen  ab  effectu,  qui  responderet  ipsi  legi  gravitatis ; 
si  ea  accurate  servaret  proportionem  cum  quadratis  distantiarum  reciproce  sumptis. 


Difficuitas  a  Mau-  124.  Nee  vero  quidquam  ipsi  meae  virium  Theorias  obsunt  meditationes  Maupertuisii, 

tionemaxfma^Nlw-  ingeniosae  illae  quidem,  sed  meo  judicio  nequaquam  satis  conformes  Natune  legibus  circa 

tonianae  legis.          legem  virium  decrescentium  in  ratione  reciproca  duplicata  distantiarum,  cujus  ille  perfec- 

tiones  quasdam  persequitur,  ut  illam,  quod  in  hac  una  integri  globi  habeant  eandem  virium 

legem,  quam  singulae  particulae.     Demonstravit  enim  Newtonus,  globos,  quorum  singuli 

paribus  a  centre  distantiis  homogenei  sint,  &  quorum  particulae  minimae  se  attrahant  in 

ratione  reciproca  duplicata  distantiarum,  se  itidem  attrahere  in  eadem  ratione  distantiarum 

reciproca  duplicata.     Ob  hasce  perfectiones  hujus  Theoriae  virium  ipse  censuit  hanc  legem 

reciprocam  duplicatam  distantiarum  ab  Auctore  Naturae  selectam  fuisse,  quam  in  Natura 

esse  vellet. 

Prima    responsio :  125.  At  mihi  quidem  inprimis  nee  unquam  placuit,  nee    placebit   sane  unquam  in 

n!^Jf  8Ts^rwt8  investieatione  Naturae  causarum  fmalium  usus,  quas  tantummodo  ad  meditationem  quandam, 

onmcs,    *x    jjcricui~  o  f  i   •  i  •  t  XT  1*1*  *  "VT 

iones,  ac  seiigi  et-  contemplationemque,  usui  esse  posse  abitror,  ubi  leges  JNaturse  aliunde  innotuennt.     JNam 
^J$£L*£5*    nee  perfectiones  omnes  innotescere  nobis  possunt,  qui  intimas  rerum  naturas  nequaquam 

III    grH.ilclIU     pcrlcC~  *•        f  .  *  a       C 

tionum.  inspicimus,  sed  externas  tantummodo  propnetates  quasdam  agnoscimus,  &  lines  omnes, 

quos  Naturae  Auctor  sibi  potuit  [57]  proponere,  ac  proposuit,  dum  Mundum  conderet, 


A  THEORY  OF  NATURAL  PHILOSOPHY  107 

in  the  orbits  of  each,  he  deduced  that  the  ratio  of  the  inverse  square  of  the  distances  was 
exactly  followed  in  the  case  of  gravitation.  But  he  only  really  proved  that  that  law  was 
very  approximately  followed,  &  not  that  it  was  accurately  so  ;  nor  from  this  can  any 
valid  argument  against  my  Theory  be  brought  forward.  For,  in  the  first  place  these  lines 
of  apses,  or  what  comes  to  the  same  thing,  the  aphelia  of  the  planets  are  not  quite  stationary  ; 
but  they  have  some  motion,  slight  indeed  but  not  quite  insensible,  with  respect  to  the  fixed 
stars,  &  therefore  move  not  only  apparently  but  really.  This  motion  is  attributed  to 
the  perturbation  of  forces  which  arises  from  the  mutual  action  of  the  planets  upon  one 
another.  But  the  fact  remains  that  it  has  never  up  till  now  been  proved  that  this  motion 
exactly  corresponds  with  the  actions  of  the  rest  of  the  planets,  where  this  is  in  accordance 
with  the  inverse  ratio  of  the  squares  of  the  distances.  For  as  yet  the  problem  of  three  bodies, 
as  they  call  it,  has  not  been  solved  except  by  much  omission  of  small  quantities  &  by 
adopting  approximations  that  are  very  far  from  truth  and  accuracy ;  in  this  problem  is 
investigated  the  motion  of  three  bodies  acting  mutually  upon  one  another  in  the  inverse 
ratio  of  the  squares  of  the  distances,  &  projected  in  any  manner.  Moreover,  even  these 
still  only  imperfect  solutions,  such  as  up  till  now  have  been  published,  hold  good  only 
in  certain  particular  cases ;  such  as  the  case  in  which  one  of  the  bodies  is  very  large  &  at 
a  very  great  distance,  the  Sun  for  instance,  whilst  the  other  two  are  quite  small  in  comparison 
&  very  near  one  another,  as  are  the  Earth  and  the  Moon,  or  at  a  large  distance  from  the 
greater  &  from  one  another  as  well,  as  Jupiter  &  Saturn.  Hence  nobody  has  hitherto 
made,  nor  indeed  could  anybody  make,  an  accurate  calculation  of  the  disturbing  influence 
of  all  the  other  planets  combined.  If  to  this  is  added  the  disturbing  influence  of  the  comets, 
of  which  we  neither  know  the  number,  nor  how  far  off  they  are  ;  it  will  be  still  more  evident 
that  from  this  no  argument  can  be  built  up  in  favour  of  a  perfectly  exact  observance  of 
the  inverse  ratio  of  the  squares  of  the  distances. 

123.  Clairaut  indeed,  in  a  pamphlet  printed  several  years  ago,  asserted  his  belief  that  The  same  thing  is 
he  had  obtained  from  the  motions  of  the  line  of  apses  for  the  Moon  a  sensible  discrepancy  J?  ^  ^duced  from 

,  ,       .  r     i         i  •  AIT--I          •      i  •      i  •  •          r>^r  •       •  T         the  rest    of  astro- 

from  the  inverse  square  of  the  distance.     Also  Euler,  in  his  dissertation  De  Aberratiombus  nomy  ;    moreover 
Jovis,  y  Saturni,  which  carried  off  the  prize  given  by  the  Paris  Academy,  considered  that  thls  Iaw  of  .mi1e 

•/,  ,.    T       .  „      <-,  ,         r  °    .  '          ..  ,       ,.  *',  can       approximate 

in  the  case  of  Jupiter  &  Saturn  there  was  quite  a  sensible  discrepancy  from  that  ratio,  to   the   other    as 

But  Clairaut  found  out,  &  proclaimed  the  fact,  that  his  result  was  indeed  due  to  a  defect  nearly as  is  desired. 

in  his  calculation  which  had  not  been  carried  far  enough  ;   &  perhaps  something  similar 

happened  in  Euler's  case.     Moreover,  there  is  no  positive  argument  in  favour  of  a  large 

discrepancy  from  the  inverse  ratio  of  the  squares  of  the  distances  for  universal  gravitation 

in  the  case  of  the  distance  of  the  Moon,  &  still  more  in  the  case  of  the  distances  of  the  planets. 

Neither  is  there  any  rigorous  argument  in  favour  of  the  ratio  being  so  accurately  observed 

that  the  difference  altogether  eludes  all  observation.     But  even  if  this  were  the  case,  my 

Theory  would  not  suffer  in  the  least  because  of  it.     For  the  last  arc  VT  of  my  curve  can 

be  made  to  approximate  as  nearly  as  is  desired  to  the  arc  of  the  hyperbola  that  represents 

the  law  of  gravitation  according  to  the  inverse  squares  of  the  distances,  touching  the  latter, 

or  osculating  it  in  any  number  of  points  in  any  positions  whatever  ;  &  thus  the  approximation 

can  be  made  so  close  that  at  these  relatively  great  distances  the  difference  will  be  altogether 

unnoticeable,  &    the  effect  will   not  be   sensibly  different  from  the  effect    that  would 

correspond  to  the  law  of  gravitation,  even  if  that  exactly  conformed  to  the  inverse  ratio 

of  the  squares  of  the  distances. 

124.  Further,  there  is  nothing  really  to  be  objected  to  my  Theory  on  account  of  the  Objection     arising 
meditations  of  Maupertuis ;    these  are  certainly  most  ingenious,  but  in  my  opinion  in  no  p°r™ction  fccord* 
way  sufficiently  in  agreement  with  the  laws  of  Nature.     Those  meditations  of  his,  I  mean,  ing  to  Maupertuis, 
with  regard  to  the  law  of  forces  decreasing  in  the  inverse  ratio  of  the  squares  of  the  distances ;  j^fw»the  Newtoman 
for  which  law  he  strives  to  adduce  certain  perfections  as  this,  that  in  this  one  law  alone 

complete  spheres  have  the  same  law  of  forces  as  the  separate  particles  of  which  they  are 
formed.  For  Newton  proved  that  spheres,  each  of  which  have  equal  densities  at  equal 
distances  from  the  centre,  &  of  which  the  smallest  particles  attract  one  another  in  the 
inverse  ratio  of  the  squares  of  the  distances,  themselves  also  attract  one  another  in  the  same 
ratio  of  the  inverse  squares  of  the  distances.  On  account  of  such  perfections  as  these  in 
this  Theory  of  forces,  Maupertuis  thought  that  this  law  of  the  inverse  squares  of  the  distances 
had  been  selected  by  the  Author  of  Nature  as  the  one  He  willed  should  exist  in  Nature. 

125.  Now,  in  the  first  place  I  was  never  satisfied,  nor  really  shall  I  ever  be  satisfied,  First  reply  to  this ; 
with  the  use  of  final  causes  in  the  investigation  of  Nature  ;  these  I  think  can  only  be  employed  perfections™^ 'not 
for  a  kind  of  study  &  contemplation,  in  such  cases  as  those  in  which  the  laws  of  Nature  known ;  and  even 
have  already  been  ascertained  from  other  methods.     For  we  cannot  possibly  be  acquainted  ^sdlcted^fo^'fhe 
with  all  perfections ;  for  in  no  wise  do  we  observe  the  inmost  nature  of  things,  but  all  we  sake  of  greater  per- 
know  are  certain  external  properties.     Nor  is  it  at  all  possible  for  us  to  see  &  know  all  fl 

the  intentions  which  the  Author  of  Nature  could  and  did  set  before  Himself  when  He  founded 


io8  PHILOSOPHIC  NATURALIS  THEORIA 

videre,  &  nosse  omnino  non  possumus.  Quin  immo  cum  juxta  ipsos  Leibnitianos  inprimis, 
aliosque  omnes  defensores  acerrimos  principii  rationis  sufficients,  &  Mundi  perfectissimi, 
qui  inde  consequitur,  multa  quidem  in  ipso  Mundo  sint  mala,  sed  Mundus  ipse  idcirco 
sit  optimus,  quod  ratio  boni  ad  malum  in  hoc,  qui  electus  est,  omnium  est  maxima  ;  fieri 
utique  poterit,  ut  in  ea  ipsius  Mundi  parte,  quam  hie,  &  nunc  contemplamur,  id,  quod 
electum  fuit,  debuerit  esse  non  illud  bonum,  in  cujus  gratiam  tolerantur  alia  mala,  sed 
illud  malum,  quod  in  aliorum  bonorum  gratiam  toleratur.  Quamobrem  si  ratio  reciproca 
duplicata  distantiarum  esset  omnium  perfectissima  pro  viribus  mutuis  particularum,  non 
inde  utique  sequeretur,  earn  pro  Natura  fuisse  electam,  &  constitutam. 

Eandem  legem  nee  I26.  At  nee  revera  perfectissima  est,  quin  immo  meo  quidem  judicio  est  omnino 

pcrfcctam  esse,  nee    •  r  0  •  v          1      •  i  •  •  ... 

in  corporibus,  non  imperfecta,  &  tarn  ipsa,  quam  aliae  plunmse  leges,  quas  requirunt  attractionem  immmutis 
utique  accurate  distantiis  crcscentcm  in  ratione  reciproca  duplicata  distantiarum,  ad  absurda  deducunt 
'  plurima,  vel  saltern  ad  inextricabiles  difficultates,  quod  ego  quidem  turn  alibi  etiam,  turn 
inprimis  demonstravi  in  dissertatione  De  Lege  Firium  in  Natura  existentium  a  num.  59.  (g) 
Accedit  autem  illud,  quod  ilia,  qua;  videtur  ipsi  esse  perfectio  maxima,  quod  nimirum 
eandem  sequantur  legem  globi  integri,  quam  particulae  minimae,  nulli  fere  usui  est  in 
Natura  ;  si  res  accurate  ad  exactitudinem  absolutam  exigatur ;  cum  nulli  in  Natura  sint 
accurate  perfecti  globi  paribus  a  centre  distantiis  homogenei,  nam  praeter  non  exiguam 
inaequalitatem  interioris  textus,  &  irregularitatem,  quam  ego  quidem  in  Tellure  nostra 
demonstravi  in  Opere,  quod  de  Litteraria  Expeditione  per  Pontificiam  ditionem  inscripsi, 
in  reliquis  autem  planetis,  &  cometis  suspicari  possumus  ex  ipsa  saltern  analogia,  prater 
scabritiem  superficiei,  quaj  utique  est  aliqua,  satis  patet,  ipsa  rotatione  circa  proprium 
axem  induci  in  omnibus  compressionem  aliquam,  quae  ut  ut  exigua,  exactam  globositatem 
impedit,  adeoque  illam  assumptam  perfectionem  maximam  corrumpit.  Accedit  autem 
&  illud,  quod  Newtoniana  determinatio  rationis  reciprocal  duplicatae  distantiarum  locum 
habet  tantummodo  in  globis  materia  continua  constantibus  sine  ullis  vacuolis,  qui  globi 
in  Natura  non  existunt,  &  multo  minus  a  me  admitti  possunt,  qui  non  vacuum  tantummodo 
disseminatum  in  materia,  ut  Philosophi  jam  sane  passim,  sed  materiam  in  immenso  vacuo 
innatantem,  &  punctula  a  se  invicem  remota,  ex  quibus,  qui  apparentes  globi  fiant,  illam 
habere  proprietatem  non  possunt  rationis  reciprocal  duplicatae  distantiarum,  adeoque  nee 
illius  perfectionis  creditas  maxime  perfectam,  absolutamque  applicationem. 


o   ex  prae-  \<:$\  \2j.   Demum  &    illud     nonnullis    difficultatem    parit    summam  in    hac    Theoria 

juuiv-.w   pro   impul-    £~    *         '  .  .  .  .  .  .  f  i.        •     i    i  .. 

sione,  &  ex  testi-  Virium,  quod  censeant,  phaenomena  omnia  per  impulsionem  explicari  debere,  &  immedi- 
monio     sensuum :  atum  contactum,  quern  ipsum  credant  evidenti  sensuum  testimonio  evinci ;  hinc  huiusmodi 

responsio    ad  hanc  .  •  r  n  «  „    XT  i  • 

posteriorem.  nostras  vires  immechamcas  appellant,  &  eas,  ut  &  Newtomanorum  generalem  gravitatem, 

vel  idcirco  rejiciunt,  quod  mechanicae  non  sint,  &  mechanismum,  quem  Newtoniana 
labefactare  coeperat,  penitus  evertant.  Addunt  autem  etiam  per  jocum  ex  serio  argumento 
petito  a  sensibus,  baculo  utendum  esse  ad  persuadendum  neganti  contactum.  Quod  ad 
sensuum  testimonium  pertinet,  exponam  uberius  infra,  ubi  de  extensione  agam,  quae  eo 
in  genere  habeamus  praejudicia,  &  unde :  cum  nimirum  ipsis  sensibus  tribuamus  id, 
quod  nostrae  ratiocinationis,  atque  illationis  vitio  est  tribuendum.  Satis  erit  hie  monere 
illud,  ubi  corpus  ad  nostra  organa  satis  accedat,  vim  repulsivam,  saltern  illam  ultimam, 
debere  in  organorum  ipsorum  fibris  excitare  motus  illos  ipsos,  qui  excitantur  in 
communi  sententia  ab  impenetrabilitate,  &  contactu,  adeoque  eundem  tremorem  ad 
cerebrum  propagari,  &  eandem  excitari  debere  in  anima  perceptionem,  quae  in 
communi  sententia  excitaretur ;  quam  ob  rem  ab  iis  sensationibus,  quae  in  hac  ipsa 
Theoria  Virium  haberentur,  nullum  utique  argumentum  desumi  potest  contra  ipsam, 
quod  ullam  vim  habeant  utcunque  tenuem. 

Felicius     explicari  128.  Quod  pertinet  ad  explicationem  phaenomenorum  per  impulsionem  immediatam, 

sione*-  "eam^nus-  rnonui  sane  superius,  quanto  felicius,  ea  prorsus  omissa,  Newtonus  explicarit  Astronomiam, 

quam  positive  pro-  &  Opticam  ;    &  patebit  inferius,  quanto  felicius   phaenomena  quaeque  praecipua  sine  ulla 

immediata  impulsione  explicentur.     Cum  iis  exemplis,  turn  aliis,  commendatur  abunde 

ea  ratio  explicandi  phsenomena,  quae  adhibet  vires  agentes  in  aliqua  distantia.     Ostendant 

(g)  Qute  hue  pertinent,  (J  continentur  novem  numeris  ejus  Dissertations  incipiendo  a  59,  habentur  in  fine  Supplem. 
§4- 


A  THEORY  OF  NATURAL  PHILOSOPHY  109 

the  Universe.  Nay  indeed,  since  in  the  doctrine  of  the  followers  of  Leibniz  more  especially, 
and  of  all  the  rest  of  the  keenest  defenders  of  the  principle  of  sufficient  reason,  and  a  most 
perfect  Universe  which  is  a  direct  consequence  of  that  idea,  there  may  be  many  evils  in  the 
Universe,  and  yet  the  Universe  may  be  the  best  possible,  just  because  the  ratio  of 
good  to  evil,  in  this  that  has  been  chosen,  is  the  greatest  possible.  It  might  certainly  happen 
that  in  this  part  of  the  Universe,  which  here  &  now  we  are  considering,  that  which  was 
chosen  would  necessarily  be  not  that  goodness  in  virtue  of  which  other  things  that  are 
evil  are  tolerated,  but  that  evil  which  is  tolerated  because  of  the  other  things  that  are  good. 
Hence,  even  if  the  inverse  ratio  of  the  squares  of  the  distances  were  the  most  perfect  of  all 
for  the  mutual  forces  between  particles,  it  certainly  would  not  follow  from  that  fact  that 
it  was  chosen  and  established  for  Nature. 

126.  But  this  law  as  a  matter  of  fact  is  not  the  most  perfect  of  all;    nay  rather,  in  This  law  is  neither 
my  opinion,  it   is    altogether    imperfect.     Both   it,  &   several   other   laws,  that    require  £0^ec^Tori0Dodt- 
attraction  at  very  small  distances  increasing  in  the  inverse  ratio  of  the  squares  of  the  distances  ies   that   are  not 
lead  to  very  many  absurdities ;    or  at  least,  to  insuperable  difficulties,  as  I  showed  in  the  exactly  spherical, 
dissertation  De  Lege  Virium  in  Natura  existentium  in  particular,  as  well  as  in  other  places. (g) 

In  addition  there  is  the  point  that  the  thing,  which  to  him  seems  to  be  the  greatest 
perfection,  namely,  the  fact  that  complete  spheres  obey  the  same  law  as  the  smallest 
particles  composing  them,  is  of  no  use  at  all  in  Nature  ;  for  there  are  in  Nature  no  exactly 
perfect  spheres  having  equal  densities  at  equal  distances  from  the  centre.  Besides  the 
not  insignificant  inequality  &  irregularity  of  internal  composition,  of  which  I  proved  the 
existence  in  the  Earth,  in  a  work  which  I  wrote  under  the  title  of  De  Litteraria  Ex-peditione 
per  Pontificiam  ditionem,  we  can  assume  also  in  the  remaining  planets  &  the  comets  (at 
least  by  analogy),  in  addition  to  roughness  of  surface  (of  which  it  is  sufficiently  evident  that 
at  any  rate  there  is  some),  that  there  is  some  compression  induced  in  all  of  them  by  the 
rotation  about  their  axes.  This  compression,  although  it  is  indeed  but  slight,  prevents 
true  sphericity,  &  therefore  nullifies  that  idea  of  the  greatest  perfection.  There  is  too 
the  further  point  that  the  Newtonian  determination  of  the  inverse  ratio  of  the  squares 
of  the  distances  holds  good  only  in  spheres  made  up  of  continuous  matter  that  is  free  from 
small  empty  spaces ;  &  such  spheres  do  not  exist  in  Nature.  Much  less  can  I  admit 
such  spheres ;  for  I  do  not  so  much  as  admit  a  vacuum  disseminated  throughout  matter, 
as  philosophers  of  all  lands  do  at  the  present  time,  but  I  consider  that  matter  as  it  were 
swims  in  an  immense  vacuum,  &  consists  of  little  points  separated  from  one  another. 
These  apparent  spheres,  being  composed  of  these  points,  cannot  have  the  property  of  the 
inverse  ratio  of  the  squares  of  the  distances ;  &  thus  also  they  cannot  bear  the  true  & 
absolute  application  of  that  perfection  that  is  credited  so  highly. 

127.  Finally,  some  persons  raise  the  greatest  objections  to  this  Theory  of  mine,  because  Objection  founded 
they  consider  that  all  the  phenomena  must  be  explained  by  impulse  and  immediate  contact ;  "mpui^and'on  the 
this  they  believe  to  be  proved  by  the  clear  testimony  of  the  senses.     So  they  call  forces  testimony   of   the 
like  those  I  propose  non-mechanical,  and  reject  them,  just  as  they  also  reject  the  universal  th?s  latter.  rep'y  t0 
gravitation  of  Newton,  for  the  alleged  reason  that  they  are  not  mechanical,  and  overthrow 

altogether  the  idea  of  mechanism  which  the  Newtonian  theory  had  already  begun  to 
undermine.  Moreover,  they  also  add,  by  way  of  a  joke  in  the  midst  of  a  serious  argument 
derived  from  the  senses,  that  a  stick  would  be  useful  for  persuading  anyone  who  denies 
contact.  Now  as  far  as  the  evidence  of  the  senses  is  concerned,  I  will  set  forth  below, 
when  I  discuss  extension,  the  prejudices  that  we  may  form  in. such  cases,  and  the  origin 
of  these  prejudices.  Thus,  for  instance,  we  may  attribute  to  the  senses  what  really  ought 
to  be  attributed  to  the  imperfection  of  our  reasoning  and  inference.  It  will  be  enough 
just  for  the  present  to  mention  that,  when  a  body  approaches  close  enough  to  our  organs, 
my  repulsive  force  (at  any  rate  it  is  that  finally),  is  bound  to  excite  in  the  nerves  of  those 
organs  the  motions  which,  according  to  the  usual  idea,  are  excited  by  impenetrability  and 
contact ;  &  that  thus  the  same  vibrations  are  sent  to  the  brain,  and  these  are  bound  to 
excite  the  same  perception  in  the  mind  as  would  be  excited  in  accordance  with  the  usual 
idea.  Hence,  from  these  sensations,  which  are  also  obtained  in  my  Theory  of  Forces,  no 
argument  can  be  adduced  against  the  theory,  which  will  have  even  the  slightest  validity. 

128.  As  regards  the  explanation  of  phenomena  by  means  of  immediate  contact  I,  hsaver^thineg  is^°^ 
indeed,  mentioned  above  how  much  more  happily  Newton  had  explained  Astronomy  and  without  the  idea  of 
Optics  by  omitting  it  altogether  ;   and  it  will  be  evident,  in  what  follows,  how  much  more  |^^lse^  nowhere 
happily  every  one  of  the  important  phenomena  is  explained  without  any  idea  of  immediate  rigorously    proved 
contact.   -  Both  by  these  instances,  and  by  many  others,  this  method  of  explaining  phenomena,  to  exist- 

by  employing  forces  acting  at  a  distance,  is  strongly  recommended.      Let  objectors  bring 


(g)  That  which  refers  to  this  point,  &  which  is  contained  in  nine  articles  of  the  dissertation  commencing  with  Art.  59, 
is  to  bf  found  at  the  end  of  this  work  as  Supplement  IV, 


no  PHILOSOPHISE  NATURALIS  THEORIA 

isti  vel  unicum  exemplum,  in  quo  positive  probare  possint,  per  immediatam  impulsionem 
communicari  motum  in  Natura.  Id  sane  ii  praestabunt  nunquam  ;  cum  oculorum  testi- 
monium  ad  excludendas  distantias  illas  minimas,  ad  quas  primum  crus  repulsivum  pertinet, 
&  contorsiones  curvae  circa  axem,  quae  oculos  necessario  fugiunt,  adhibere  non  possint  ;  cum 
e  contrario  ego  positive  argumento  superius  excluserim  immediatum  contactum  omnem, 
&  positive  probaverim,  ipsum,  quern  ii  ubique  volunt,  haberi  nusquam. 

Vires  hujus  Theo-  I2g    j)e  nominibus  quidem  non  esset,  cur  solicitudinem  haberem  ullam  ;    sed  ut  & 

rise  pertineread  ve-    ...,•*,..,  .      ~t    .  .  ,  '  .     .  ,.  ... 

rum,  nee  occuitum  in  nsdem  aliquid  prasjudicio  cmdam,  quod  ex  communi  loquendi  usu  provenit,  mud 
mechanismum.  notandum  duco,  Mechanicam  non  utique  ad  solam  impulsionem  immediatam  fuisse 
restrictam  unquam  ab  iis,  qui  de  ipsa  tractarunt,  sed  ad  liberos  inprimis  adhibitam  contem- 
plandos  motus,  qui  independenter  ab  omni  impulsione  habeantur.  Quae  Archimedes  de 
aequilibrio  tradidit,  quse  Galilaeus  de  li-[59]-bero  gravium  descensu,  ac  de  projectis,  quae 
de  centralibus  in  circulo  viribus,  &  oscillationis  centre  Hugenius,  quae  Newtonus  generaliter 
de  motibus  in  trajectoriis  quibuscunque,  utique  ad  Mechanicam  pertinent,  &  Wolfiana 
&  Euleriana,  &  aliorum  Scriptorum  Mechanica  passim  utique  ejusmodi  vires,  &  motus  inde 
ortos  contemplatur,  qui  fiant  impulsione  vel  exclusa  penitus,  vel  saltern  mente  seclusa. 
Ubicunque  vires  agant,  quae  motum  materiae  gignant,  vel  immutent,  &  leges  expandantur, 
secundum  quas  velocitas  oriatur,  mutetur  motus,  ac  motus  ipse  determinetur  ;  id  omne 
inprimis  ad  Mechanicam  pertinet  in  admodum  propria  significatione  acceptam.  Quam- 
obrem  ii  maxime  ea  ipsa  propria  vocum  significatione  abutuntur,  qui  impulsionem  unicam 
ad  Mechanismum  pertinere  arbitrantur,  ad  quern  haec  virium  genera  pertinent  multo  magis, 
qu33  idcirco  appellari  jure  possunt  vires  Mechanic*?,  &  quidquid  per  illas  fit,  jure  affirmari 
potest  fieri  per  Mechanismum,  nee  vero  incognitum,  &  occuitum,  sed  uti  supra  demonstra- 
vimus,  admodum  patentem,  a  manifestum. 

Discrimen     inter  j  -m    Eodem  etiam  pacto  in  omnino  propria  significatione  usurpare  licebit  vocem  con- 

contactum    mathe-  J    .  .  ..       *  T  i  i  • 

maticum,  &  physi-  tactus  ;  licet  intervallum  semper  remaneat  aliquod  ;  quanquam  ego  ad  aequivocationes  evi- 
cum  :    hunc    did  tandas  soleo  distinguere  inter  contactum  Mathematicum,  in  quo  distantia  sit  prorsus  nulla, 

proprie  contactum.  ni      •  •  j-  •  a.     •  o       •  1  • 

&  contactum  Physicum,  in  quo  distantia  sensus  effugit  omnes,  &  vis  repulsiva  satis  magna 
ulteriorem  accessum  per  nostras  vires  inducendum  impedit.  Voces  ab  hominibus  institutae 
sunt  ad  significandas  res  corporeas,  &  corporum  proprietates,  prout  nostris  sensibus  subsunt, 
iis,  quae  continentur  infra  ipsos,  nihil  omnino  curatis.  Sic  planum,  sic  laeve  proprie  dicitur 
id,  in  quo  nihil,  quod  sensu  percipi  possit,  sinuetur,  nihil  promineat  ;  quanquam  in  communi 
etiam  sententia  nihil  sit  in  Natura  mathematice  planum,  vel  laeve.  Eodem  pacto  &  nomen 
contactus  ab  hominibus  institutum  est,  ad  exprimendum  physicum  ilium  contactum  tantum- 
modo,  sine  ulla  cura  contactus  mathematics,  de  quo  nostri  sensus  sententiam  ferre  non 
possunt.  Atque  hoc  quidem  pacto  si  adhibeantur  voces  in  propria  significatione  ilia,  quae 
ipsarum  institutioni  respondeat  ;  ne  a  vocibus  quidem  ipsis  huic  Theoriae  virium  invidiam 
creare  poterunt  ii,  quibus  ipsa  non  placet. 


extensionis  sit  orta. 


Transitus   ab   ob-  j^j.  Atque  haec  de  iis,  quae  contra  ipsam  virium  legem  a  me  propositam  vel  objecta 

Theoriam     virium  sunt  hactenus,  vel  objici  possent,  sint  satis,  ne  res  in  infinitum  excrescat.     Nunc  ad  ilia 
ad  objections  con-  transibimus,  quae  contra  constitutionem  elementorum  materiae  inde  deductam  se  menti 

tra  puncta.  .•*....  i  .  j. 

oiferunt,  in  quibus  itidem,  quae  maxime  notatu  digna  sunt,  persequar. 

Objectio    ab    idea  132.  Inprimis   quod   pertinet   ad   hanc   constitutionem   elementorum   materise,   sunt 

puncti     inextensi,  multi,  qui  nullo  pacto  in  animum  sibi  possint  inducere,  ut  admittant  puncta  prorsus 

qua    caremus  :   re-  ,r  i         n       T  11  •  j  A      -J 

sponsio  :  unde  idea  mdi-[6o]-visibiha,  &  mextensa,  quod  nullam  se  dicant  habere  posse  eorum  ideam.  At  id 
a-  hominum  genus  praejudiciis  quibusdam  tribuit  multo  plus  aequo.  Ideas  omnes,  saltern 
eas,  quae  ad  materiam  pertinent,  per  sensus  hausimus.  Porro  sensus  nostri  nunquam 
potuerunt  percipere  singula  elementa,  quae  nimirum  vires  exerunt  nimis  tenues  ad  movendas 
fibras,  &  propagandum  motum  ad  cerebrum  :  massis  indiguerunt,  sive  elementorum 
aggregatis,  quae  ipsas  impellerent  collata  vi.  Haec  omnia  aggregata  constabant  partibus, 
quarum  partium  extremae  sumptae  hinc,  &  inde,  debebant  a  se  invicem  distare  per  aliquod 
intervallum,  nee  ita  exiguum.  Hinc  factum  est,  ut  nullam  unquam  per  sensus  acquirere 
potuerimus  ideam  pertinentem  ad  materiam,  quae  simul  &  extensionem,  &  partes,  ac 
divisibilitatem  non  involverit.  Atque  idcirco  quotiescunque  punctum  nobis  animo  sistimus, 
nisi  reflexione  utamur,  habemus  ideam  globuli  cujusdam  perquam  exigui,  sed  tamen  globuli 
rotundi,  habentis  binas  superficies  oppositas  distinctis. 


A  THEORY  OF  NATURAL  PHILOSOPHY  in 

forward  but  a  single  instance  in  which  they  can  positively  prove  that  motion  in  Nature 
is  communicated  by  immediate  impulse.  Of  a  truth  they  will  never  produce  one  ;  for 
they  cannot  use  the  testimony  of  the  eyes  to  exclude  those  very  small  distances  to  which 
the  first  repulsive  branch  of  my  curve  refers  &  the  windings  about  the  axis ;  for  these 
necessarily  evade  ocular  observation.  Whilst  I,  on  the  other  hand,  by  the  rigorous  argument 
given  above,  have  excluded  all  idea  of  immediate  contact ;  &  I  have  positively  proved 
that  the  thing,  which  they  wish  to  exist  everywhere,  as  a  matter  of  fact  exists  nowhere. 

129.  There  is  no  reason  why  I  should  trouble  myself  about  nomenclature  ;    but,  as  The  forces  in  this 
in  that  too  there  is  something  that,  from  the  customary  manner  of  speaking,  gives  rise  to  ^j^/^ot  to  an 
a  kind  of    prejudice,  I  think  it  should  be  observed  that  Mechanics  was  certainly  never  occult  mechanism, 
restricted  to  immediate  impulse  alone  by  those  who  have  dealt  with  it ;    but  that  in  the 

first  place  it  was  employed  for  the  consideration  of  free  motions,  such  as  exist  quite 
independently  of  any  impulse.  The  work  of  Archimedes  on  equilibrium,  that  of  Galileo 
on  the  free  descent  of  heavy  bodies  &  on  projectiles,  that  of  Huygens  on  central  forces 
in  a  circular  orbit  &  on  the  centre  of  oscillation,  what  Newton  proved  in  general  for 
motion  on  all  sorts  of  trajectories  ;  all  these  certainly  belong  to  the  science  of  Mechanics. 
The  Mechanics  of  Wolf,  Euler  &  other  writers  in  different  lands  certainly  treats  of  such 
forces  as  these  &  the  motions  that  arise  from  them,  &  these  matters  have  been  accomplished 
with  the  idea  of  impulse  excluded  altogether,  or  at  least  put  out  of  mind.  Whenever 
forces  act,  &  there  is  an  investigation  of  the  laws  in  accordance  with  which  velocity  is 
produced,  motion  is  changed,  or  the  motion  itself  is  determined  ;  the  whole  of  this  belongs 
especially  to  Mechanics  in  a  truly  proper  signification  of  the  term.  Hence,  they  greatly 
abuse  the  proper  signification  of  terms,  who  think  that  impulse  alone  belongs  to  the  science 
of  Mechanics ;  to  which  these  kinds  of  forces  belong  to  a  far  greater  extent.  Therefore 
these  forces  may  justly  be  called  Mechanical ;  &  whatever  comes  about  through  their 
action  can  be  justly  asserted  to  have  come  about  through  a  mechanism  ;  &  one  too  that 
is  not  unknown  or  mysterious,  but,  as  we  proved  above,  perfectly  plain  &  evident. 

130.  Also  in  the  same  way  we  may  employ  the  term  contact  in  an  altogether  special  Distinction  be- 
sense  ;    the  interval  may  always  remain  something  definite.     Although,  in  order  to  avoid  ticainandmph^eskai 
ambiguity,  I  usually  distinguish  between  mathematical  contact,  in  which  the  distance  is  contact ;  the  latter 
absolutely  nothing,  &  -physical  contact,  in  which  the  distance  is  too  small  to  affect  our  Caned™orftact>.per  y 
senses,  and  the  repulsive  force  is  great  enough  to  prevent  closer  approach  being  induced 

by  the  forces  we  are  considering.  Words  are  formed  by  men  to  signify  corporeal  things 
&  the  properties  of  such,  as  far  as  they  come  within  the  scope  of  the  senses ;  &  those 
that  fall  beneath  this  scope  are  absolutely  not  heeded  at  all.  Thus,  we  properly  call  a 
thing  plane  or  smooth,  which  has  no  bend  or  projection  in  it  that  can  be  perceived  by  the 
senses ;  although,  in  the  general  opinion,  there  is  nothing  in  Nature  that  is  mathematically 
plane  or  smooth.  In  the  same  way  also,  the  term  contact  was  invented  by  men  to  express 
•physical  contact  only,  without  any  thought  of  mathematical  contact,  of  which  our  senses 
can  form  no  idea.  In  this  way,  indeed,  if  words  are  used  in  their  correct  sense,  namely, 
that  which  corresponds  to  their  original  formation,  those  who  do  not  care  for  my  Theory 
of  forces  cannot  from  those  words  derive  any  objection  against  it. 

131.  I  have  now  said  sufficient  about  those  objections  that  either  up  till  now  have  Passing    on    from 
been  raised,  or  might  be  raised,  against  the  law  of  forces  that  I  have  proposed  ;   otherwise  ^y^Theorf  ""of 
the  matter  would  grow  beyond  all  bounds.     Now  we  will  pass  on  to  objections  against  forces  to  objections 
the  constitution  of  the  elements  of  matter  derived  from  it,  which  present  themselves  to  the  agamst  P°mts- 
mind  ;  &  in  these  also  I  will  investigate  those  that  more  especially  seem  worthy  of  remark. 

132.  First  of  all,  as  regards  the  constitution  of  the  elements  of  matter,  there  are  indeed  Potion   to^the 
many  persons  who  cannot  in  any  way  bring  themselves  into  that  frame  of  mind  to  admit  tended     points, 
the  existence  of  points  that  are  perfectly  indivisible  and  non-extended  ;    for  they  say  that  which    we    postu- 

*•  <        '    •  T*  i  r  1_  13. tc  ,      reply  ,      tiic 

they  cannot  form  any  idea  of  such  points.     But  that  type  of  men  pays  more  heed  than  origin  of  the  idea 

is  right  to  certain  prejudices.     We  derive  all  our  ideas,  at  any  rate  those  that  relate  to  of  extension. 

matter,  from  the  evidences  of  our  senses.     Further,  our  senses  never  could  perceive  single 

elements,  which  indeed  give  forth  forces  that  are  too  slight  to  affect  the  nerves  &  thus 

propagate  motion  to  the  brain.     The  senses  would  need  masses,  or  aggregates  of  the  elements, 

which  would  affect  them  as  a  result  of  their  combined  force.     Now  all  these  aggregates  are 

made  up  of  parts ;   &  of  these  parts  the  two  extremes  on  the  one  side  and  on  the^  other 

must   be   separated   from  one  another  by  a  certain  interval,  &  that  not  an  insignificant 

one.     Hence  it  comes  about  that  we  could  never  obtain  through  the  senses  any  idea  relating 

to  matter,  which  did  not  involve  at  the   same  time   extension,  parts  &  divisibility.     So, 

as  often  as  we  thought  of  a  point,  unless  we  used  our  reflective  powers,  we  should  get  the 

idea  of  a  sort  of  ball,  exceedingly  small  indeed,  but  still  a  round  ball,  having  two  distinct 

and  opposite  faces. 


"2  PHILOSOPHISE  NATURALIS   THEORIA 

idea  m    puncti  133.  Quamobrem  ad  concipiendum  punctum  indivisibile,  &  inextensum  ;  non  debemus 

refl^xionemT'quo-  consulere  ideas>  quas  immediate  per  sensus  hausimus  ;    sed  earn  nobis  debemus  efformare 

modo    ejus    idea  per  reflexionem.     Reflexione  adhibita  non  ita  difficulter  efformabimus  nobis  ideam  ejusmodi. 

negativa    acqmra-  Nam  inprimi  s  ubi  &  extensionem,  &  partium  compositionem  conceperimus  ;    si  utranque 

negemus  ;   jam  inextensi,  &  indivisibilis  ideam  quandam  nobis  comparabimus   per  negati- 

onem  illam  ipsam  eorum,  quorum  habemus  ideam  ;    uti  foraminis  ideam  habemus  utique 

negando  existentiam  illius  materias,  quas  deest  in  loco  foraminis. 

Quomodo  ejus  idea  134.  Verum  &  positivam  quandam  indivisibilis,  &  inextensi  puncti  ideam  poterimus 

posfit^per  itmlte"  comParare  n°bis  ope  Geometrias,  &  ope  illius  ipsius  ideas  extensi  continui,  quam  per  sensus 
&  limitum  inter-  hausimus,  &  quam  inferius  ostendemus,  fallacem  esse,  ac  fontem  ipsum  fallacies  ejusmodi 
aperiemus,  quas  tamen  ipsa  ad  indivisibilium,  &  inextensorum  ideam  nos  ducet  admodum 
claram.  Concipiamus  planum  quoddam  prorsus  continuum,  ut  mensam,  longum  ex.  gr. 
pedes  duos  ;  atque  id  ipsum  planum  concipiamus  secari  transversum  secundum  longitudinem 
ita,  ut  tamen  iterum  post  sectionem  conjungantur  partes,  &  se  contingant.  Sectio  ilia 
erit  utique  limes  inter  partem  dexteram  &  sinistram,  longus  quidem  pedes  duos,  quanta 
erat  plani  longitude,  at  latitudinis  omnino  expers  :  nam  ab  altera  parte  immediate  motu 
continue  transitur  ad  alteram,  quse,  si  ilia  sectio  crassitudinem  haberet  aliquam,  non  esset 
priori  contigua.  Ilia  sectio  est  limes  secundum  crassitudinem  inextensus,  &  indivisibilis, 
cui  si  occurrat  altera  sectio  transversa  eodem  pacto  indivisibilis,  &  inextensa  ;  oportebit 
utique,  intersectio  utriusque  in  superficie  plani  concepti  nullam  omnino  habeat  extensionem 
in  partem  quamcumque.  Id  erit  punctum  peni-[6i]-tus  indivisibile,  &  inextensum,  quod 
quidem  punctum,  translate  piano,  movebitur,  &  motu  suo  lineam  describet,  longam  quidem, 
sed  latitudinis  expertem. 


Natura    inextensi,  j^c.  Quo  autem  melius  ipsius    indivisibilis  natura  concipi    possit  ;    quasrat  a  nobis 

quod     non    potest         .     /"  r  ,     .  £         ,  .  ^.      .    .  '     ". 

esse  inextenso  con-  quispiam,  ut  aliam  faciamus  ejus  planae  massas  sectionem,  quas  priori  ita  sit  proxima,  ut 
tiguum  in  Uneis.  nihil  prorsus  inter  utramque  intersit.  Respondebimus  sane,  id  fieri  non  posse  :  vel  enim 
inter  novam  sectionem,  &  veteram  intercedet  aliquid  ejus  materias,  ex  qua  planum  con- 
tinuum constare  concipimus,  vel  nova  sectio  congruet  penitus  cum  praecedente.  En 
quomodo  ideam  acquiremus  etiam  ejus  naturas  indivisibilis  illius,  &  inextensi,  ut  aliud 
indivisibile,  &  inextensum  ipsi  proximum  sine  medio  intervallo  non  admittat,  sed  vel  cum 
eo  congruat,  vel  aliquod  intervallum  relinquat  inter  se,  &  ipsum.  Atque  hinc  patebit 
etiam  illud,  non  posse  promoveri  planum  ipsum  ita,  ut  ilia  sectio  promoveatur  tantummodo 
per  spatium  latitudinis  sibi  asqualis.  Utcunque  exiguus  fuerit  motus,  jam  ille  novus 
sectionis  locus  distabit  a  praecedente  per  aliquod  intervallum,  cum  sectio  sectioni  contigua 
esse  non  possit. 

Eademin  punctis  :  136.  Hasc  si  ad  concursum  sectionum  transferamus,  habebimus  utique  non  solum  ideam 

idea    puncti      eeo-  ..,...,.,.„.  .          ,      .  ,.  ...  v     j  -i  • 

metricf  transiata  puncti  indivisibilis,  &  inextensi,  sed  ejusmodi  naturae  puncti  ipsius,  ut  aliud  punctum  sibi 
ad  physicum,  &  contiguum  habere  non  possit,  sed  vel  congruant,  vel  aliquo  a  se  invicem  intervallo  distent. 
Et  hoc  pacto  sibi  &  Geometrae  ideam  sui  puncti  indivisibilis,  &  inextensi,  facile  efformare 
possunt,  quam  quidem  etiam  efformant  sibi  ita,  ut  prima  Euclidis  definitio  jam  inde  incipiat  : 
•punctum  est,  cujus  nulla  •pars  est.  Post  hujusmodi  ideam  acquisitam  illud  unum  intererit 
inter  geometricum  punctum,  &  punctum  physicum  materiae,  quod  hoc  secundum  habebit 
proprietates  reales  vis  inertias,  &  virium  illarum  activarum,  quas  cogent  duo  puncta  ad  se 
invicem  accedere,  vel  a  se  invicem  recedere,  unde  net,  ut  ubi  satis  accesserint  ad  organa 
nostrorum  sensuum,  possint  in  iis  excitare  motus,  qui  propagati  ad  cerebrum,  perceptiones 
ibi  eliciant  in  anima,  quo  pacto  sensibilia  erunt,  adeoque  materialia,  &  realia,  non  pure 
imaginaria. 

Punctorum    exist-  j--    gn  jgjtur  per  reffexionem  acquisitam  ideam  punctorum  realium,  materialium, 

entiam     aliunde    .     ..    ,J.(  °.  .  .  ,  .     .    r     r. 

demonstrari  :     per  indivisibilium,  inextensorum,  quam  inter  ideas  ab  infantia  acquisitas  per  sensus  mcassum 
ideam     acquisitam  quaerimus.     Idea  ejusmodi  non  evincit  eorum  existentiam.     Ipsam  quam  nobis  exhibent 

ea  tantum  concipi.    ^     .  .  J  .        ,  ,       .      .  ..*-."«... 

positiva  argumenta  superms  facta,  quod  mmirum,  ne  admittatur  in  colhsione  corporum 
saltus,  quern  &  inductio,  &  impossibilitas  binarum  velocitatum  diversarum  habendarum 
omnino  ipso  momento,  quo  saltus  fieret,  excludunt,  oportet  admittere  in  materia  vires, 
quas  repulsivae  sint  in  minimis  distantiis,  &  iis  in  infinitum  imminutis  augeantur  in  infinitum  ; 


A  THEORY  OF  NATURAL  PHILOSOPHY  113 

133.  Hence  for  the  purpose  of  forming  an  idea  of  a  point  that  is  indivisible  &  non-  The  idea  of  a  point 
extended,  we  cannot  consider  the  ideas  that  we  derive  directly  from  the  senses ;    but  we  ^"refleTti  obtaihrxed 
must  form  our  own  idea  of  it  by  reflection.     If  we  reflect  upon  it,  we  shall  form  an  idea  a  negative>nidea°of 
of  this  sort  for  ourselves  without  much  difficulty.     For,  in  the  first  place,  when  we  have  con-  rt  may  ^  ac(iuired- 
ceived  the  idea  of  extension  and  composition  by  parts,  if  we  deny  the  existence  of  both,  then 

we  shall  get  a  sort  of  idea  of  non-extension  &  indivisibility  by  that  very  negation  of  the 
existence  of  those  things  of  which  we  already  have  formed  an  idea.  For  instance,  we  have 
the  idea  of  a  hole  by  denying  the  existence  of  matter,  namely,  that  which  is  absent  from 
the  position  in  which  the  hole  lies. 

134.  But  we  can  also  get  an  idea  of  a  point  that  is  indivisible  &  non-extended,  by  HOW  a  positive  idea 
the  aid  of  geometry,  and  by  the  help  of  that  idea  of  an  extended  continuum  that  we  derive  ^^  ^of^bourf 
from  the  senses ;  this  we  will  show  below  to  be  a  fallacy,  &  also  we  will  open  up  the  very  daries,  and  inter- 
source  of  this  kind  of  fallacy,  which  nevertheless  will  lead  us  to  a  perfectly  clear  idea  of  ^g°ns  of   boun" 
indivisible   &   non-extended  points.     Imagine  some    thing   that   is    perfectly  plane    and 
continuous,  like  a  table-top,  two  feet  in  length  ;   &  suppose  that  this  plane  is  cut  across 

along  its  length  ;  &  let  the  parts  after  section  be  once  more  joined  together,  so  that  they 
touch  one  another.  The  section  will  be  the  boundary  between  the  left  part  and  the  right 
part ;  it  will  be  two  feet  in  length  (that  being  the  length  of  the  plane  before  section),  & 
altogether  devoid  of  breadth.  For  we  can  pass  straightaway  by  a  continuous  motion 
from  one  part  to  the  other  part,  which  would  not  be  contiguous  to  the  first  part  if  the  section 
had  any  thickness.  The  section  is  a  boundary  which,  as  regards  breadth,  is  non-extended 
&  indivisible  ;  if  another  transverse  section  which  in  the  same  way  is  also  indivisible  & 
non-extended  fell  across  the  first,  then  it  must  come  about  that  the  intersection  of  the 
two  in  the  surface  of  the  assumed  plane  has  no  extension  at  all  in  any  direction.  It  will 
be  a  point  that  is  altogether  indivisible  and  non-extended  ;  &  this  point,  if  the  plane 
be  moved,  will  also  move  and  by  its  motion  will  describe  a  line,  which  has  length  indeed 
but  is  devoid  of  breadth. 

135.  The  nature  of  an  indivisible  itself  can  be  better  conceived  in  the  following  way.   The  nature   of  a 
Suppose  someone  should  ask  us  to  make  another  section  of  the  plane  mass,  which  shall  lie  °h°in~gextwIhich 
so  near  to  the  former  section  that  there  is  absolutely  no  distance  between  them.     We  cannot  he  next  to 
should  indeed  reply  that  it  could  not  be  done.     For  either  between  the  new  section  &  ^  S 

the  old  there  would  intervene  some  part  of  the  matter  of  which  the  continuous  plane  was  concerned, 
composed  ;  or  the  new  section  would  completely  coincide  with  the  first.  Now  see  how 
we  acquire  an  idea  also  of  the  nature  of  that  indivisible  and  non-extended  thing,  which 
is  such  that  it  does  not  allow  another  indivisible  and  non-extended  thing  to  lie  next  to  it 
without  some  intervening  interval ;  but  either  coincides  with  it  or  leaves  some  definite 
interval  between  itself  &  the  other.  Hence  also  it  will  be  clear  that  it  is  not  possible 
so  to  move  the  plane,  that  the  section  will  be  moved  only  through  a  space  equal  to  its  own 
breadth.  However  slight  the  motion  is  supposed  to  be,  the  new  position  of  the  section 
would  be  at  a  distance  from  the  former  position  by  some  definite  interval ;  for  a  section 
cannot  be  contiguous  to  another  section. 

136.  If  now  we  transfer  these  arguments  to  the  intersection  of  sections,  we  shall  truly  Th.e  same  thing  for 
have  not  only  the  idea  of  an  indivisible  &  non-extended  point,  but  also  an  idea  of  the  ^geometrical  point 
nature  of  a  point  of  this  sort ;  which  is  such  that  it  cannot  have  another  point  contiguous  transferred  to  a 
to  it,  but  the  two  either  coincide  or  else  they  are  separated  from  one  another  by  some  interval.  riafpoLt^ 

In  this  way  also  geometricians  can  easily  form  an  idea  of  their  own  kind  of  indivisible  & 
non-extended  points ;  &  indeed  they  do  so  form  their  idea  of  them,  for  the  first  defi- 
nition of  Euclid  begins  : — A  -point  is  that  which  has  no  parts.  After  an  idea  of  this  sort  has 
been  acquired,  there  is  but  one  difference  between  a  geometrical  point  &  a  physical  point 
of  matter  ;  this  lies  in  the  fact  that  the  latter  possesses  the  real  properties  of  a  force  of 
inertia  and  of  the  active  forces  that  urge  the  two  points  to  approach  towards,  or  recede 
from,  one  another ;  whereby  it  comes  about  that  when  they  have  approached  sufficiently 
near  to  the  organs  of  our  senses,  they  can  excite  motions  in  them  which,  when  propagated 
to  the  brain,  induce  sensations  in  the  mind,  and  in  this  way  become  sensible,  &  thus 
material  and  real,  &  not  imaginary. 

137.  See  then  how  by  reflection  the  idea  of  real,  material,  indivisible,  non-extended  The    existence    of 
points  can  be  acquired  ;    whilst  we  seek  for  it  in  vain  amongst  those  ideas  that  we  have  o^herwise^demon- 
acquired  since  infancy  by  means  of  the  senses.     But  an  idea  of  this  sort  about  things  does  strated ;  they  can 
not  prove  that  these  things  exist.     That  is  just  what  the  rigorous  arguments  given  above  through  ^cquir- 
point  out  to  us ;    that  is  to  say,  because,  in  order  that  in  the  collision  of  solids  a  sudden  ing  an  idea  of  them, 
change  should  not  be  admitted  (which    change   both   induction  &   the   impossibility  of 

there  being  two  different  velocities  at  the  same  instant  in  which  the  change  should  take 
place),  it  had  to  be  admitted  that  in  matter  there  were  forces  which  are  repulsive  ^at  very 
small  distances,  &  that  these  increased  indefinitely  as  the  distances  were  diminished. 

I 


ii4  PHILOSOPHIC  NATURALIS  THEORIA 

unde  fit,  ut  duse  particulae  materiae  sibi  [62]  invicem  contiguae  esse  non  possint  :  nam  illico 
vi  ilia  repulsiva  resilient  a  se  invicem,  ac  particula  iis  constans  statim  disrumpetur,  adeoque 
prima  materiae  elementa  non  constant  contiguis  partibus,  sed  indivisibilia  sunt  prorsus, 
atque  simplicia,  &  vero  etiam  ob  inductionem  separabilitatis,  ac  distinctionis  eorum,  quae 
occupant  spatii  divisibilis  partes  diversas,  etiam  penitus  inextensa.  Ilia  idea  acquisita  per 
reflexionem  illud  praestat  tantummodo,  ut  distincte  concipiamus  id,  quod  ejusmodi  rationes 
ostendunt  existere  in  Natura,  &  quod  sine  reflexione,  &  ope  illius  supellectilis  tantummodo, 
quam  per  sensus  nobis  comparavimus  ab  ipsa  infantia,  concipere  omnino  non  liceret. 


Ceterum  simplicium,  &  inextensorum  notionem  non  ego  primus  in  Physicam 
aiiis   quoque    ad-  induco.     Eorum  ideam  habuerunt  veteres  post  Zenonem,  &  Leibnitiani  monades  suas  & 
"rastare  "hanc  simP^ces  utique  volunt,  &  inextensas ;  ego  cum  ipsorum  punctorum  contiguitatem  auferam, 
eorum  theoriam.       &  distantias  velim  inter  duo  quaelibet  materiae  puncta,  maximum  evito  scopulum,  in  quern 
utrique  incurrunt,  dum  ex  ejusmodi  indivisibilibus,  &    inextensis  continuum    extensum 
componunt.     Atque  ibi  quidem  in  eo  videntur  mini  peccare  utrique,  quod  cum  simplicitate, 
&  inextensione,  quam  iis  elementis  tribuunt,  commiscent  ideam  illam  imperfectam,  quam 
sibi  compararunt  per  sensus,  globuli  cujusdam  rotundi,  qui  binas  habeat  superficies  a  se 
distinctas,  utcumque  interrogati,  an  id  ipsum  faciant,  omnino  sint  negaturi.     Neque  enim 
aliter  possent  ejusmodi  simplicibus  inextensis  implere  spatium,  nisi  concipiendo  unum 
elementum  in  medio  duorum  ab  altero  contactum  ad  dexteram,  ab  altero  ad  laevam,  quin 
ea  extrema  se  contingant;    in  quo,  praeter  contiguitatem  indivisibilium,  &  inextensorum 
impossibilem,  uti  supra  demonstravimus,  quam   tamen  coguntur  admittere,  si  rem  altius 
perpenderint ;    videbunt  sane,  se  ibi  illam  ipsam  globuli  inter  duos  globules  inter jacentis 
ideam  admiscere. 


impugnatur     con-  139.  Nee    ad    indivisibilitatem,    &   inextensionem    elementorum    conjungendas    cum 

formats  ^b^inex-  continua  extensione  massarum  ab  iis  compositarum  prosunt  ea,  quae  nonnulli  ex  Leibniti- 


tensis  petita  ab  anorum  familia  proferunt,  de  quibus  egi  in  una  adnotatiuncula  adjecta  num.  13.  dissertationis 
impenetrabiiitate.  j)g  Mater  its  Divisibilitate,  £?  Principiis  Corporum,  ex  qua,  quae  eo  pertinent,  hue  libet 
transferre.  Sic  autem  habet  :  Qui  dicunt,  monades  non  compenetrari,  quia  natura  sua 
impenetrabiles  sunt,  ii  difficultatem  nequaquam  amovenf  ;  nam  si  e?  natura  sua  impenetrable  s 
sunt,  y  continuum  debent  componere,  adeoque  contigua  esse  ;  compenetrabuntur  simul,  W  non 
compenetrabuntur,  quod  ad  absurdum  deducit,  W  ejusmodi  entium  impossibilitatem  evincit. 
Ex  omnimodfs  inextensionis,  &  contiguitatis  notione  evincitur,  compenetrari  debere  argumento 
contra  Zenonistas  institute  per  tot  stecula,  £if  cui  nunquam  satis  responsum  est.  Ex  natura, 
qua  in  [63]  iis  supponitur,  ipsa  compenetratio  excluditur,  adeoque  habetur  contradictio,  & 
absurdum. 

inductionem  a  140'.  Sunt  alii,  quibus  videri  poterit,  contra  haec  ipsa  puncta  indivisibilia,  &  inextensa 

sensibihbus     com-       ,1  .,  T.  .     ,    ^  .      .          .      .  r.  .     r.      K      . 

positis,  &  extensis  adniberi  posse  mductionis  prmcipmm,  a    quo  contmuitatis    legem,  &    alias    propnetates 
haud  vaiere  contra  derivavimus  supra,  quae  nos  ad  haec  indivisibilia,  &  inextensa  puncta  deduxerunt.     Videmus 

puncta  simplicia,  &  t*          *.  .  .       ...    ...  ,.    .  ..  ... 

inextensa.  enim  in  matena  omni,  quae  se  uspiam  nostns  objiciat  sensibus,  extensionem,  divisibihtatem, 

partes  ;  quamobrem  hanc  ipsam  proprietatem  debemus  transferre  ad  elementa  etiam  per 
inductionis  principium.  Ita  ii  :  at  hanc  difficultatem  jam  superius  praeoccupavimus,  ubi 
egimus  de  inductionis  principio.  Pendet  ea  proprietas  a  ratione  sensibilis,  &  aggregati,  cum 
nimirum  sub  sensus  nostros  ne  composita  quidem,  quorum  moles  nimis  exigua  sit,  cadere 
possint.  Hinc  divisibilitatis,  &  extensionis  proprietas  ejusmodi  est  ;  ut  ejus  defectus,  si 
habeatur  alicubi  is  casus,  ex  ipsa  earum  natura,  &  sensuum  nostrorum  constitutione  non 
possit  cadere  sub  sensus  ipsos,  atque  idcirco  ad  ejusmodi  proprietates  argumentum  desumptum 
ab  inductione  nequaquam  pertingit,  ut  nee  ad  sensibilitatem  extenditur. 


Per   ipsam   etiam          141.  Sed  etiam  si  extenderetur,  esset  adhuc  nostrae  Theoriae  causa  multo  melior  in  eo, 

tensT^Hn^uctioms  q110^  circa,  extensionem,  &  compositionem  partium  negativa  sit.     Nam  eo    ipso,  quod 

habitam  ipsum  ex-  continuitate  admissa,  continuitas  elementorum  legitima  ratiocinatione  excludatur,  excludi 

omnino  debet    absolute  ;    ubi  quidem  illud  accidit,  quod  a  Metaphysicis,  &  Geometris 

nonnullis  animadversum  est  jam  diu,  licere  aliquando    demonstrare    propositionem  ex 


A  THEORY  OF  NATURAL  PHILOSOPHY  115 

From  this  it  comes  about  that  two  particles  of  matter  cannot  be  contiguous ;  for  thereupon 
they  would  recoil  from  one  another  owing  to  that  repulsive  force,  &  a  particle  composed 
of  them  would  at  once  be  broken  up.  Thus,  the  primary  elements  of  matter  cannot  be 
composed  of  contiguous  parts,  but  must  be  perfectly  indivisible  &  simple  ;  and  also  on 
account  of  the  induction  from  separability  &  the  distinction  between  those  that  occupy 
different  divisible  parts  of  space,  they  must  be  perfectly  non-extended  as  well.  The  idea 
acquired  by  reflection  only  yields  the  one  result,  namely,  that  through  it  we  may  form 
a  clear  conception  of  that  which  reasoning  of  this  kind  proves  to  be  existent  in  Nature  ; 
of  which,  without  reflection,  using  only  the  equipment  that  we  have  got  together  for 
ourselves  by  means  of  the  senses  from  our  infancy,  we  could  not  have  formed  any 
conception. 

138.  Besides,  I  was  not  the  first  to  introduce  the  notion  of  simple  non-extended  points  Simple  and 
into  physics.  The  ancients  from  the  time  of  Zeno  had  an  idea  of  them,  &  the  followers  are^admitt 
of  Leibniz  indeed  suppose  that  their  monads  are  simple  &  non-extended.  I,  since  I  do  others  as  well ;  but 
not  admit  the  contiguity  of  the  points  themselves,  but  suppose  that  any  two  points  of  ^m  is  "the7 best." 
matter  are  separated  from  one  another,  avoid  a  mighty  rock,  upon  which  both  these  others 
come  to  grief,  whilst  they  build  up  an  extended  continuum  from  indivisible  &  non-extended 
things  of  this  sort.  Both  seem  to  me  to  have  erred  in  doing  so,  because  they  have  mixed 
up  with  the  simplicity  &  non-extension  that  they  attribute  to  the  elements  that  imperfect 
idea  of  a  sort  of  round  globule  having  two  surfaces  distinct  from  one  another,  an  idea  they 
have  acquired  through  the  senses ;  although,  if  they  were  asked  if  they  had  made  this 
supposition,  they  would  deny  that  they  had  done  so.  For  in  no  other  way  can  they  fill  up 
space  with  indivisible  and  non-extended  things  of  this  sort,  unless  by  imagining  that  one 
element  between  two  others  is  touched  by  one  of  them  on  the  right  &  by  the  other  on 
the  left.  If  such  is  their  idea,  in  addition  to  contiguity  of  indivisible  &  non-extended 
things  (which  is  impossible,  as  I  proved  above,  but  which  they  are  forced  to  admit  if  they 
consider  the  matter  more  carefully) ;  in  addition  to  this,  I  say,  they  will  surely  see  that  they 
have  introduced  into  their  reasoning  that  very  idea  of  the  two  little  spheres  lying  between 
two  others. 

I3Q.  Those  arguments  that  some  of  the  Leibnitian  circle  put  forward  are  of  no  use  The  deduction  from 

,        i      ~  r  •         •     T    •  -i  •!•        o  •          r    i        i  -i.  •  impenetrability    of 

for  the  purpose  of  connecting  indivisibility  &  non-extension  of  the  elements  with  continuous  a    conciliation    of 

extension  of  the  masses  formed  from  them.     I  discussed  the  arguments  in  question  in  extension  ^j1  ^ 

a   short   note   appended   to   Art.    13    of  the   dissertation  De  Materies  Divisibilitate  and  extendeTthings. 

Principiis  Corporum  ;  &  I  may  here  quote  from  that  dissertation  those  things  that  concern 

us  now.     These  are  the  words  : — Those,  who  say  that  monads  cannot  be  corn-penetrated,  because 

they  are  by  nature  impenetrable,  by  no  means  remove  the  difficulty.     For,  if  they  are  both  by 

nature  impenetrable,  &  also  at  the  same  time  have  to  make  up  a  continuum,  i.e.,  have  to  be 

contiguous,  then  at  one  &  the  same  time  they  are  compenetrated  &  they  are  not  compenetrated  ; 

y  this  leads  to  an  absurdity  \3  proves  the  impossibility  of  entities  of  this  sort.     For,  from  the 

idea  of  non-extension  of  any  sort,  &  of  contiguity,  it  is  proved  by  an  argument  instituted 

against  the  Zenonists  many  centuries  ago  that  there  is  bound  to  be  compenetration  ;    &  -this 

argument  has  never  been  satisfactorily  answered.     From  the  nature  that  is  ascribed  to  them, 

this  compenetration  is  excluded.     Thus  there  is  a  contradiction  13  an  absurdity. 

140.  There  are  others,  who  will  think  that  it  is  possible  to  employ,  for  the  purpose  induction    derived 
of  opposing  the  idea  of  these  indivisible  &  non-extended  points,  the  principle  of  induction,  ^T'senslSf3  <£m*- 
by  which  we  derived  the  Law  of  Continuity  &  other  properties,  which  have  led  us  to  pound,    and    ex- 
these  indivisible  &  non-extended  points.     For  we  perceive  (so   they  say)  in  all  matter,  avauedforrthefpur° 
that  falls  under  our  notice  in  any  way,  extension,  divisibility  &  parts.     Hence  we  must  pose   of   opposing 
transfer  this  property  to  the  elements  also  by  the  principle  of  induction.     Such  is  their 

argument.  But  we  have  already  discussed  this  difficulty,  when  we  dealt  with  the  principle 
of  induction.  The  property  in  question  depends  on  a  reasoning  concerned  with  a  sensible 
body,  &  one  that  is  an  aggregate  ;  for,  in  fact,  not  even  a.  composite  body  can  come  within 
the  scope  of  our  senses,  if  its  mass  is  over-small.  Hence  the  property  of  divisibility  & 
extension  is  such  that  the  absence  of  this  property  (if  this  case  ever  comes  about),  from 
the  very  nature  of  divisibility  &  extension,  &  from  the  constitution  of  our  senses,  cannot 
fall  within  the  scope  of  those  senses.  Therefore  an  argument  derived  from  induction  will 
not  apply  to  properties  of  this  kind  in  any  way,  inasmuch  as  the  extension  does  not  reach 
the  point  necessary  for  sensibility. 

141.  But  even  if  this  point  is  reached,  there  would  only  be  all  the  more  reason  for  our  Extension 
Theory  from  the  fact  that  it  denies  extension  and  composition  by  parts.    For,  from  the  very  exclusion  of 
fact  that,  if  continuity  be  admitted,  continuity  of  the  elements  is  excluded  by  legitimate  exte^seio 
argument,  it  follows  that  continuity  ought  to  be  absolutely  excluded  in  all  cases.     For  in  duCtion. 
that  case  we  get  an  instance  of  the  argument  that  has  been  observed  by  metaphysicists 

and  some  geometers  for  a  very  long  time,  namely,  that  a  proposition  may  sometimes  be 


n6  PHILOSOPHIC  NATURALIS  THEORIA 

assumpta  veritate  contradictoriae  propositionis ;  cum  enim  ambae  simul  verae  esse  non 
possint,  si  ab  altera  inferatur  altera,  hanc  posteriorem  veram  esse  necesse  est.  Sic  nimirum, 
quoniam  a  continuitate  generaliter  assumpta  defectus  continuitatis  consequitur  in  materiae 
elementis,  &  in  extensione,  defectum  hunc  haberi  vel  inde  eruitur  :  nee  oberit 
quidquam  principium  inductionis  physicae,  quod  utique  non  est  demonstrativum,  nee  vim 
habet,  nisi  ubi  aliunde  non  demonstretur,  casum  ilium,  quern  inde  colligere  possumus, 
improbabilem  esse  tantummodo,  adhuc  tamen  haberi,  uti  aliquando  sunt  &  falsa  veris 
probabiliora. 

Cujusmodi      con-  142.  Atque  hie  quidem,  ubi  de  continuitate  seipsam  excludente  mentio  injecta  est, 

TheoiSadrnittatur  n°tandum  &  illud,  continuitatis  legem  a  me  admitti,  &  probari  pro  quantitatibus,  quae 
quid  sit  spatium,  magnitudinem  mutent,  quas  nimirum  ab  una  magnitudine  ad  aliam  censeo  abire  non  posse, 
&  tempus.  njg-  transeant  per  intermedias,  quod  elementorum  materiae,  quse  magnitudinem  nee  mutant, 

nee  ullam  habent  variabilem,  continuitatem  non  inducit,  sed  argumento  superius  facto 
penitus  summovet.  Quin  etiam  ego  quidem  continuum  nullum  agnosco  coexistens,  uti  & 
supra  monui ;  nam  nee  spatium  reale  mihi  est  ullum  continuum,  sed  [64]  imaginarium 
tantummodo,  de  quo,  uti  &  de  tempore,  quae  in  hac  mea  Theoria  sentiam,  satis  luculenter 
exposui  in  Supplementis  ad  librum  i.  Stayanae  Philosophise  (*).  Censeo  nimirum  quodvis 
materiae  punctum,  habere  binos  reales  existendi  modos,  alterum  localem,  alterum  tem- 
porarium,  qui  num  appellari  debeant  res,  an  tantummodo  modi  rei,  ejusmodi  litem,  quam 
arbitror  esse  tantum  de  nomine,  nihil  omnino  euro.  Illos  modos  debere  admitti,  ibi  ego 
quidem  positive  demonstro  :  eos  natura  sua  immobiles  esse,  censeo  ita,  ut  idcirco  ejusmodi 
existendi  modi  per  se  inducant  relationes  prioris,  &  posterioris  in  tempore,  ulterioris,  vel 
citerioris  in  loco,  ac  distantiae  cujusdam  deter minatae,  &  in  spatio  determinatae  positionis 
etiam,  qui  modi,  vel  eorum  alter,  necessario  mutari  debeant,  si  distantia,  vel  etiam  in  spatio 
sola  mutetur  positio.  Pro  quovis  autem  modo  pertinente  ad  quodvis  punctum,  penes 
omnes  infinites  modos  possibiles  pertinentes  ad  quodvis  aliud,  mihi  est  unus,  qui  cum  eo 
inducat  in  tempore  relationem  coexistentiae  ita,  ut  existentiam  habere  uterque  non  possit, 
quin  simul  habeant,  &  coexistant ;  in  spatio  vero,  si  existunt  simul,  inducant  relationem 
compenetrationis,  reliquis  omnibus  inducentibus  relationem  distantiae  temporarise,  vel 
localis,  ut  &  positionis  cujusdam  localis  determinatae.  Quoniam  autem  puncta  materiae 
existentia  habent  semper  aliquam  a  se  invicem  distantiam,  &  numero  finita  sunt ;  finitus  est 
semper  etiam  localium  modorum  coexistentium  numerus,  nee  ullum  reale  continuum 
efformat.  Spatium  vero  imaginarium  est  mihi  possibilitas  omnium  modorum  localium 
confuse  cognita,  quos  simul  per  cognitionem  praecisivam  concipimus,  licet  simul  omnes 
existere  non  possint,  ubi  cum  nulli  sint  modi  ita  sibi  proximi,  vel  remoti,  ut  alii  viciniores, 
vel  remotiores  haberi  non  possint,  nulla  distantia  inter  possibiles  habetur,  sive  minima 
omnium,  sive  maxima.  Dum  animum  abstrahimus  ab  actuali  existentia,  &  in  possibilium 
serie  finitis  in  infinitum  constante  terminis  mente  secludimus  tarn  minimae,  quam  maximae 
distantiae  limitem,  ideam  nobis  efformamus  continuitatis,  &  infinitatis  in  spatio,  in  quo 
idem  spatii  punctum  appello  possibilitatem  omnium  modorum  localium,  sive,  quod  idem 
est,  realium  localium  punctorum  pertinentium  ad  omnia  materiae  puncta,  quae  si  existerent, 
compenetrationis  relationem  inducerent,  ut  eodem  pacto  idem  nomino  momentum  tem- 
poris  temporarios  modos  omnes,  qui  relationem  inducunt  coexistentiae.  Sed  de  utroque 
plura  in  illis  dissertatiunculis,  in  quibus  &  analogiam  persequor  spatii,  ac  temporis 
multiplicem. 


Ubi  habeat  con-  [65]  143.  Continuitatem  igitur  agnosco  in  motu  tantummodo,  quod  est  successivum 
uibilitaffee1ctetNatUra  ^u^'  non  coexistens,  &  in  eo  itidem  solo,  vel  ex  eo  solo  in  corporeis  saltern  entibus  legem 
continuitatis  admitto.  Atque  hinc  patebit  clarius  illud  etiam,  quod  superius  innui, 
Naturam  ubique  continuitatis  legem  vel  accurate  observare,  vel  affectare  saltern.  ^  Servat  in 
motibus,  &  distantiis,  affectat  in  aliis  casibus  multis,  quibus  continuity,  uti  etiam  supra 
definivimus,  nequaquam  convenit,  &  in  aliis  quibusdam,  in  quibus  haberi  omnino  non  pptest 
continuitas,  quae  primo  aspectu  sese  nobis  objicit  res  non  aliquanto  intimius  inspectantibus, 
ac  perpendentibus  :  ex.  gr.  quando  Sol  oritur  supra  horizontem,  si  concipiamus  Solis  discum 

(h)  Binte  dissertatiunculis,  qua  hue  pertinent,  inde  excerptte  habentur  hie  Supplementorum  §  I,  13  2,  quarum  mentio 
facta  est  etiam  superius  num.  66,  W  86. 


con- 


A  THEORY  OF  NATURAL  PHILOSOPHY  117 

proved  by  assuming  the  truth  of  the  contradictory  proposition.  For  since  both  propositions 
cannot  be  true  at  the  same  time,  if  from  one  of  them  the  other  can  be  inferred,  then  the  latter 
of  necessity  must  be  the  true  one.  Thus,  for  instance,  because  it  follows,  from  the 
assumption  of  continuity  in  general,  that  there  is  an  absence  of  continuity  in  the  elements 
of  matter,  &  also  in  the  case  of  extension,  we  come  to  the  conclusion  that  there  is  this 
absence.  Nor  will  any  principle  of  physical  induction  be  prejudicial  to  the  argument, 
where  the  induction  is  not  one  that  can  be  proved  in  every  case  ;  neither  will  it  have  any 
validity,  except  in  the  case  where  it  cannot  be  proved  in  other  ways  that  the  conclusion 
that  we  can  come  to  from  the  argument  is  highly  improbable  but  yet  is  to  be  held  as 
true  ;  for  indeed  sometimes  things  that  are  false  are  more  plausible  than  the  true  facts. 

142.  Now,  in  this  connection,  whilst  incidental  mention  has  been  made  of  the  exclusion  xhe   sort  of 

of  continuity,  it  should  be  observed  that  the  Law  of  Continuity  is  admitted  by  me,  &  tinuum  that  is 
proved  for  those  quantities  that  change  their  magnitude,  but  which  indeed  I  consider  Th^r^fthe^ature 
cannot  pass  from  one  magnitude  to  another  without  going  through  intermediate  stages ;  of  sPace  and  time, 
but  that  this  does  not  lead  to  continuity  in  the  case  of  the  elements  of  matter,  which  neither 
change  their  magnitude  nor  have  anything  variable  about  them  ;  on  the  contrary  it  proves 
quite  the  opposite,  as  the  argument  given  above  shows.  Moreover,  I  recognize  no  co- 
existing continuum,  as  I  have  already  mentioned  ;  for,  in  my  opinion,  space  is  not  any 
real  continuum,  but  only  an  imaginary  one  ;  &  what  I  think  about  this,  and  about  time 
as  well,  as  far  as  this  Theory  is  concerned,  has  been  expounded  clearly  enough  in  the 
supplements  to  the  first  book  of  Stay's  Philosophy.  (A)  For  instance,  I  consider  that  any 
point  of  matter  has  two  modes  of  existence,  the  one  local  and  the  other  temporal ;  I  do 
not  take  the  trouble  to  argue  the  point  as  to  whether  these  ought  to  be  called  things,  or 
merely  modes  pertaining  to  a  thing,  as  I  consider  that  this  is  merely  a  question  of  terminology. 
That  it  is  necessary  that  these  modes  be  admitted,  I  prove  rigorously  in  the  supplements 
mentioned  above.  I  consider  also  that  they  are  by  their  very  nature  incapable  of  being 
displaced  ;  so  that,  of  themselves,  such  modes  of  existence  lead  to  the  relations  of  before 
&  after  as  regards  time,  far  &  near  as  regards  space,  &  also  of  a  given  distance  & 
a  given  position  in  space.  These  modes,  or  one  of  them,  must  of  necessity  be  changed, 
if  the  distance,  or  even  if  only  the  position  in  space  is  altered.  Moreover,  for  any  one 
mode  belonging  to  any  point,  taken  in  conjunction  with  all  the  infinite  number  of  possible 
modes  pertaining  to  any  other  point,  there  is  in  my  opinion  one  which,  taken  in  conjunction 
with  the  first  mode,  leads  as  far  as  time  is  concerned  to  a  relation  of  coexistence  ;  so  that 
both  cannot  have  existence  unless  they  have  it  simultaneously,  i.e.,  they  coexist.  But, 
as  far  as  space  is  concerned,  if  they  exist  simultaneously,  the  conjunction  leads  to  a  relation 
of  compenetration.  All  the  others  lead  to  a  relation  of  temporal  or  of  local  distance,  as 
also  of  a  given  local  position.  Now  since  existent  points  of  matter  always  have  some  distance 
between  them,  &  are  finite  in  number,  the  number  of  local  modes  of  existence  is  also 
always  finite  ;  &  from  this  finite  number  we  cannot  form  any  sort  of  real  continuum. 
But  I  have  an  ill-defined  idea  of  an  imaginary  space  as  a  possibility  of  all  local  modes,  which 
are  precisely  conceived  as  existing  simultaneously,  although  they  cannot  all  exist  simul- 
taneously. In  this  space,  since  there  are  not  modes  so  near  to  one  another  that  there 
cannot  be  others  nearer,  or  so  far  separated  that  there  cannot  be  others  more  so,  there 
cannot  therefore  be  a  distance  that  is  either  the  greatest  or  the  least  of  all,  amongst  those 
that  are  possible.  So  long  as  we  keep  the  mind  free  from  the  idea  of  actual  existence  &,  in 
a  series  of  possibles  consisting  of  an  indefinite  number  of  finite  terms,  we  mentally  exclude 
the  limit  both  of  least  &  greatest  distance,  we  form  for  ourselves  a  conception  of  continuity 
&  infinity  in  space.  In  this,  I  define  the  same  point  of  space  to  be  the  possibility  of  all 
local  modes,  or  what  comes  to  the  same  thing,  of  real  local  points  pertaining  to  all  points 
of  matter,  which,  if  they  existed,  would  lead  to  a  relation  of  compenetration  ;  just  as  I 
define  the  same  instant  of  time  as  all  temporal  modes,  which  lead  to  a  relation  of  coexistence. 
But  there  is  a  fuller  treatment  of  both  these  subjects  in  the  notes  referred  to  ;  &  in  them 
I  investigate  further  the  manifold  analogy  between  space  &  time. 

143.  Hence  I  acknowledge  continuity  in  motion  only,  which  is  something  successive  where  there  is  con- 

i      TJ  .  .       .°  .       ,  '  ,  f   .      V  °.  .  tmuity  in  Nature ; 

and  not  co-existent ;  &  also  in  it  alone,  or  because  or  it  alone,  in  corporeal  entities  at  any  Where  Nature  does 
rate,  lies  my  reason  for  admitting  the  Law  of  Continuity.     From  this  it  will  be  all  the  no  more  than  at- 
more  clear  that,  as  I  remarked  above,  Nature  accurately  observes  the  Law  of  Continuity,  jteml 
or  at  least  tries  to  do  so.     Nature  observes  it  in   motions  &  in  distance,  &  tries  to  in  many 
other  cases,  with  which  continuity,  as  we  have  defined  it  above,  is  in  no  wise  in  agree- 
ment ;  also  in  certain  other  cases,  in  which  continuity  cannot  be  completely  obtained.   This 
continuity  does  not  present  itself  to  us  at  first  sight,  unless  we  consider  the  subjects  somewhat 
more  deeply  &  study  them  closely.     For  instance,  when  the  sun  rises  above  the  horizon, 

(h)  The  two  notes,  which  refer  to  this  matter,  have  been  quoted  in  this  work  as  supplements  IS-  II :   these  have 
been  already  referred  to  in  Arts.  66  &  86  above. 


n8  PHILOSOPHISE  NATURALIS  THEORIA 

ut  continuum,  &  horizontem  ut  planum  quoddam  ;  ascensus  Solis  fit  per  omnes  magnitudines 
ita,  ut  a  primo  ad  postremum  punctum  &  segmenta  Solaris  disci,  &  chordae  segmentorum 
crescant  transeundo  per  omnes  intermedias  magnitudines.  At  Sol  quidem  in  mea  Theoria 
non  est  aliquid  continuum,  sed  est  aggregatum  punctorum  a  se  invicem  distantium,  quorum 
alia  supra  illud  imaginarium  planum  ascendunt  post  alia,  intervallo  aliquo  temporis  inter- 
posito  semper.  Hinc  accurata  ilia  continuitas  huic  casui  non  convenit,  &  habetur  tantummodo 
in  distantiis  punctorum  singulorum  componentium  earn  massam  ab  illo  imaginario  piano. 
Natura  tamen  etiam  hie  continuitatem  quandam  affectat,  cum  nimirum  ilia  punctula  ita 
sibi  sint  invicem  proxima,  &  ita  ubique  dispersa,  ac  disposita,  ut  apparens  quaedam  ibi  etiam 
continuitas  habeatur,  ac  in  ipsa  distributione,  a  qua  densitas  pendet,  ingentes  repentini 
saltus  non  riant. 

Exempla    continu-  144.  Innumera  ejus  rei  exempla  liceret  proferre,  in  quibus  eodem  pacto  res  pergit. 

it  at  is  apparent  gjc  jn  fluviorum  alveis,  in  frondium  flexibus,  in  ipsis  salium,  &  crystallorum,  ac  aliorum 

tantum  :    unde  ea  .....  .  ,.,  .    •  •••*.«, 

ortum  ducat.  corporum  angulis,  in  ipsis  cuspidibus  unguium,  quae  acutissimae  in  quibusdam  ammalibus 
apparent  nudo  oculo  ;  si  microscopio  adhibito  inspiciantur  ;  nusquam  cuspis  abrupta 
prorsus,  nusquam  omnino  cuspidatus  apparet  angulus,  sed  ubique  flexus  quidam,  qui 
curvaturam  habeat  aliquam,  &  ad  continuitatem  videatur  accedere.  In  omnibus  tamen  iis 
casibus  vera  continuitas  in  mea  Theoria  habetur  nusquam  ;  cum  omnia  ejusmodi  corpora 
constent  indivisibilibus,  &  a  se  distantibus  punctis,  quse  continuam  superficiem  non  efformant, 
&  in  quibus,  si  quaevis  tria  puncta  per  rectas  lineas  conjuncta  intelligantur  ;  triangulum 
habebitur  utique  cum  angulis  cuspidatis.  Sed  a  motuum,  &  virium  continuitate  accurata 
etiam  ejusmodi  proximam  continuitatem  massarum  oriri  censeo,  &  a  casuum  possibilium 
multitudine  inter  se  collata,  quod  ipsum  innuisse  sit  satis. 

Motuum    omnium  145-  Atque  hinc  fiet    manifestum,  quid  respondendum   ad    casus    quosdam,  qui    eo 

continuitas  in  -pertinent,  &  in  quibus  violari  quis  crederet  F661  continuitatis  legem.     Quando  piano  aliquo 

line  is      continuis    r  .*.  fr      .  n        •  •          n      •  • 

nusquam  inter-  speculo  lux  excipitur,  pars  relrmgitur,  pars  renectitur  :  in  renexione,  &  retractione,  uti  earn 
ruptis,  aut  mutatis.  olim  creditum  est  fieri,  &  etiamnum  a  nonnullis  creditur,  per  impulsionem  nimirum,  & 
incursum  immediatum,  fieret  violatio  quaedam  continui  motus  mutata  linea  recta  in  aliam  ; 
sed  jam  hoc  Newtonus  advertit,  &  ejusmodi  saltum  abstulit,  explicando  ea  phenomena  per 
vires  in  aliqua  distantia  agentes,  quibus  fit,  ut  quaevis  particula  luminis  motum  incurvet 
paullatim  in  accessu  ad  superficiem  re  flectentem,  vel  refringentem  ;  unde  accessuum,  & 
recessuum  lex,  velocitas,  directionum  flexus,  omnia  juxta  continuitatis  legem  mutantur. 
Quin  in  mea  Theoria  non  in  aliqua  vicinia  tantum  incipit  flexus  ille,  sed  quodvis  materiae 
punctum  a  Mundi  initio  unicam  quandam  continuam  descripsit  orbitam,  pendentem  a 
continua  ilia  virium  lege,  quam  exprimit  figura  I ,  quae  ad  distantias  quascunque  protenditur  ; 
quam  quidem  lineae  continuitatem  nee  liberae  turbant  animarum  vires,  quas  itidem  non  nisi 
juxta  continuitatis  legem  exerceri  a  nobis  arbitror  ;  unde  fit,  ut  quemadmodum  omnem 
accuratam  quietem,  ita  omnem  accurate  rectilineum  motum,  omnem  accurate  circularem, 
ellipticum,  parabolicum  excludam  ;  quod  tamen  aliis  quoque  sententiis  omnibus  commune 
esse  debet ;  cum  admodum  facile  sit  demonstrare,  ubique  esse  perturbationem  quandam, 
&  mutationum  causas,  quae  non  permittant  ejusmodi  linearum  nobis  ita  simplicium  accuratas 
orbitas  in  motibus. 

Apparens  saltus  in  146.  Et  quidem  ut  in  iis  omnibus,  &  aliis  ejusmodi  Natura  semper  in  mea  Theoria 

diffusione     reflexi,  accuratissimam  continuitatem  observat,  ita  &  hie  in  reflexionibus,  ac  refractionibus  luminis. 

ac  refracti  luminis.     .  ,.     ,  .  ..'..,.  ,         ,     ,      '.     .  , 

At  est  ahud  ea  in  re,  in  quo  continuitatis  violatio  quaedam  haben  videatur,  quam,  qui  rem 
altius  perpendat,  credet  primo  quidem,  servari  itidem  accurate  a  Natura,  turn  ulterius 
progressus,  inveniet  affectari  tantummodo,  non  servari.  Id  autem  est  ipsa  luminis  diffusio, 
atque  densitas.  Videtur  prima  fronte  discindi  radius  in  duos,  qui  hiatu  quodam  intermedio 
a  se  invicem  divellantur  velut  per  saltum,  alia  parte  reflexa,  ali  refracta,  sine  ullo  intermedio 
flexu  cujuspiam.  Alius  itidem  videtur  admitti  ibidem  saltus  quidam  :  si  enim  radius 
integer  excipiatur  prismate  ita,  ut  una  pars  reflectatur,  alia  transmittatur,  &  prodeat  etiam 
e  secunda  superficie,  turn  ipsum  prisma  sensim  convertatur  ;  ubi  ad  certum  devenitur  in 
conversione  angulum,  lux,  quae  datam  habet  refrangibilitatem,  jam  non  egreditur,  sed 
reflectitur  in  totum  ;  ubi  itidem  videtur  fieri  transitus  a  prioribus  angulis  cum  superficie 
semper  minoribus,  sed  jacentibus  ultra  ipsam,  ad  angulum  reflexionis  aequalem  angulo 


A  THEORY  OF  NATURAL  PHILOSOPHY  119 

if  we  think  of  the  Sun's  disk  as  being  continuous,  &  the  horizon  as  a  certain  plane  ;  then 
the  rising  of  the  Sun  is  made  through  all  magnitudes  in  such  a  way  that,  from  the  first  to 
the  last  point,  both  the  segments  of  the  solar  disk  &  the  chords  of  the  segments  increase  by 
passing  through  all  intermediate  magnitudes.  But,  in  my  Theory,  the  Sun  is  not  something 
continuous,  but  is  an  aggregate  of  points  separate  from  one  another,  which  rise,  one  after 
the  other,  above  that  imaginary  plane,  with  some  interval  of  time  between  them  in  all 
cases.  Hence  accurate  continuity  does  not  fit  this  case,  &  it  is  only  observed  in  the  case 
of  the  distances  from  the  imaginary  plane  of  the  single  points  that  compose  the  mass  of  the 
Sun.  Yet  Nature,  even  here,  tries  to  maintain  a  sort  of  continuity ;  for  instance,  the 
little  points  are  so  very  near  to  one  another,  &  so  evenly  spread  &  placed  that,  even  in 
this  case,  we  have  a  certain  apparent  continuity,  and  even  in  this  distribution,  on  which 
the  density  depends,  there  do  not  occur  any  very  great  sudden  changes. 

144.  Innumerable  examples  of  this  apparent  continuity  could  be  brought  forward,  in  Examples  of  con- 
which  the  matter  comes  about  in  the  same  manner.     Thus,  in  the  channels  of  rivers,  the  ^"reiy    apparent'3 
bends  in  foliage,  the  angles  in  salts,  crystals  and  other  bodies,  in  the  tips  of  the  claws  that  its  origin, 
appear  to  the  naked  eye  to  be  very  sharp  in  the  case  of  certain  animals ;   if  a  microscope 

were  used  to  examine  them,  in  no  case  would  the  point  appear  to  be  quite  abrupt,  or  the 
angle  altogether  sharp,  but  in  every  case  somewhat  rounded,  &  so  possessing  a  definite 
curvature  &  apparently  approximating  to  continuity.  Nevertheless  in  all  these  cases 
there  is  nowhere  true  continuity  according  to  my  Theory ;  for  all  bodies  of  this  kind  are 
composed  of  points  that  are  indivisible  &  separated  from  one  another  ;  &  these  cannot 
form  a  continuous  surface  ;  &  with  them,  if  any  three  points  are  supposed  to  be  joined 
by  straight  lines,  then  a  triangle  will  result  that  in  every  case  has  three  sharp  angles.  But 
I  consider  that  from  the  accurate  continuity  of  motions  &  forces  a  very  close  approximation 
of  this  kind  arises  also  in  the  case  of  masses ;  &,  if  the  great  number  of  possible  cases  are 
compared  with  one  another,  it  is  sufficient  for  me  to  have  just  pointed  it  out. 

145.  Hence  it  becomes  evident  how  we  are  to  refute  certain  cases,  relating  to  this  The.  continuity  of 
matter,  in  which  it  might  be  considered  that  the  Law  of  Continuity  was  violated.     When  ™uous    lines  ""is 
light  falls  upon  a  plane  mirror,   part  is  refracted  &  part  is  reflected.     In   reflection  &  nowhere    inter- 
refraction,  according  to  the  idea  held  in  olden  times,  &  even  now  credited  by  some  people,  rup  e 
namely,  that  it  took  place  by  means  of  impulse  &  immediate  collision,  there  would  be 

a  breach  of  continuous  motion  through  one  straight  line  being  suddenly  changed  for 
another.  But  already  Newton  has  discussed  this  point,  &  has  removed  any  sudden  change 
of  this  sort,  by  explaining  the  phenomena  by  means  of  forces  acting  at  a  distance ;  with 
these  it  comes  about  that  any  particle  of  light  will  have  its  path  bent  little  by  little  as  it 
approaches  a  reflecting  or  refracting  surface.  Hence,  the  law  of  approach  and  recession, 
the  velocity,  the  alteration  of  direction,  all  change  in  accordance  with  the  Law  of  Continuity. 
Nay  indeed,  in  my  Theory,  this  alteration  of  direction  does  not  only  begin  in  the  immediate 
neighbourhood,  but  any  point  of  matter  from  the  time  that  the  world  began  has  described 
a  single  continuous  orbit,  depending  on  the  continuous  law  of  forces,  represented  in  Fig.  i, 
a  law  that  extends  to  all  distances  whatever.  I  also  consider  that  this  continuity  of  path 
is  undisturbed  by  any  voluntary  mental  forces,  which  also  cannot  be  exerted  by  us  except 
in  accordance  with  the  Law  of  Continuity.  Hence  it  comes  about  that,  just  as  I  exclude 
all  idea  of  absolute  rest,  so  I  exclude  all  accurately  rectilinear,  circular,  elliptic,  or  parabolic 
motions.  This  too  ought  to  be  the  general  opinion  of  all  others ;  for  it  is  quite  easy  to  show 
that  there  is  everywhere  some  perturbation,  &  reasons  for  alteration,  which  do  not  allow 
us  to  have  accurate  paths  along  such  simple  lines  for  our  motions. 

146.  Just  as  in  all  the  cases  I  have  mentioned,  &  in  others  like  them,  Nature  always  Apparent    discon- 

'.-,,,  J  ..  i       •       i  •      i  i  •        i  tinuity  in  diffusion 

in  my  Theory  observes  the  most  accurate  continuity,  so  also  is  this  done  here  in  the  case  Of  renected  and  re- 

of  the  reflection  and  refraction  of  light.     But  there  is  another  thing  in  this  connection,  fracted  light. 

in  which  there  seems  to  be  a  breach  of  continuity ;   &  anyone  who  considers  the  matter 

fairly  deeply,  will  think  at  first  that  Nature  has  observed  accurate  continuity,  but  on  further 

consideration   will  find  that  Nature  has  only  endeavoured  to  do  so,  &  has  not  actually 

observed  it ;  that  is  to  say,  in  the  diffusion  of  light,  &  its  density.     At  first  sight  the  ray 

seems  to  be  divided  into  two  parts,  which  leave  a  gap  between  them  &  diverge  from  one 

another  as  it  were  suddenly,  the  one  part  being    reflected  &  the  other   part    refracted 

without  any  intermediate  bending  of  the  path.     It  also  seems  that  another  sudden  change 

must  be  admitted  ;   for  suppose  that  a  beam  of  light  falls  upon  a  prism,  &  part  of  it  is 

reflected  &  the  rest  is  transmitted  &  issues    from  the  second   surface,  and  that  then  the 

prism   is  gradually  rotated ;  when  a  certain  angle  of    rotation  is   reached,  light,  having 

a  given  refrangibility,  is  no  longer  transmitted,  but  is  totally  reflected.     Here  also  it 

seems  that  there  is  a  sudden  transition  from  the  first  case  in  which  the  angles  made^with 

the  surface  by  the  issuing  rays  are  always  less  than   the  angle  of  incidence,  &  lie  on 

the  far  side  of  the  surface,  to  the  latter  case  in  which  the  angles  of  reflection  are  equal  to 


120  PHILOSOPHIC  NATURALIS  THEORIA 

incidentiae,  &  jacentem  citra,  sine  ulla  reflexione  in  angulis  intermediis  minoribus  ab  ipsa 
superficie  ad  ejusmodi  finitum  angulum. 

Apparens    concili-  14.7.  Huic  cuidam  velut  laesioni  continuitatis  videtur  responderi  posse  per  illam  lucem 

Unuitafe  pel  radios  qua3  reflectitur,  vel  refrin-[67]-gitur  irregulariter  in  quibusvis  angulis.  Jam  olim  enim 
irregulariter  disper-  observatum  est  illud,  ubi  lucis  radius  reflectitur,  non  reflecti  totum  ita,  ut  angulus 
reflexionis  aequetur  angulo  incidentiae,  sed  partem  dispergi  quaquaversus  ;  quam  ob  causam 
si  Solis  radius  in  partem  quandam  speculi  incurrat,  quicunque  est  in  conclavi,  videt,  qui  sit 
ille  locus,  in  quern  incurrit  radius,  quod  utique  non  fieret,  nisi  e  solaribus  illis  directis  radiis 
etiam  ad  oculum  ipsius  radii  devenirent,  egressi  in  omnibus  iis  directionibus,  quae  ad  omnes 
oculi  positiones  tendunt ;  licet  ibi  quidem  satis  intensum  lumen  non  appareat,  nisi  in 
directione  faciente  angulum  reflexionis  aequalem  incidentiae,  in  qua  resilit  maxima  luminis 
pars.  Et  quidem  hisce  radiis  redeuntibus  in  angulis  hisce  inaequalibus  egregie  utitur 
Newtonus  in  fine  Opticae  ad  explicandos  colores  laminarum  crassarum  :  &  eadem  irregularis 
dispersio  in  omnes  plagas  ad  sensum  habetur  in  tenui  parte,  sed  tamen  in  aliqua,  radii 
refracti.  Hinc  inter  vividum  ilium  reflexum  radium,  &  refractum,  habetur  intermedia 
omnis  ejusmodi  radiorum  series  in  omnibus  iis  intermediis  angulis  prodeuntium,  &  sic  etiam 
ubi  transitur  a  refractione  ad  reflexionem  in  totum,  videtur  per  hosce  intermedios  angulos 
res  posse  fieri  citissimo  transitu  per  ipsos,  atque  idcirco  illaesa  perseverare  continuitas. 

Cur   ea    apparens  148.  Verum  si  adhuc  altius  perpendatur  res  ;  patebit  in  ilia  intermedia  serie  non  haberi 

dSitio1  pe^contiii-  accuratam  continuitatem,  sed  apparentem  quandam,  quam  Natura  affectat,  non  accurate 

ujtatem  yiae  cujus-  servat  illaesam.      Nam  lumen  in  mea  Theoria  non  est  corpus  quoddam  continuum,  quod 

vis  puncti  diffundatur  continue  per  illud  omne  spatium,  sed  est  aggregatum  punctorum  a  se  invicem 

disjunctorum,  atque  distantium,   quorum  quodlibet  suam  percurrit  viam  disjunctam  a 

proximi  via  per  aliquod  intervallum.     Continuitas  servatur  accuratissime    in  singulorum 

punctorum  viis,  non  in  diffusione  substantiae  non  continuae,  &  quo  pacto  ea  in  omnibus  iis 

motibus  servetur,  &  mutetur,  mutata  inclinatione  incidentiae,  via  a  singulis  punctis  descripta 

sine  saltu,  satis  luculenter  exposui  in  secunda  parte  meae  dissertationis  De  Lumine  a  num.  98. 

Sed  haec  ad  applicationem  jam  pertinent  Theoriae  ad  Physicam. 


QUO  pacto  servetur  149.  Haud  multum  absimiles  sunt  alii  quidam  casus,  in  quibus  singula  continuitatem 

bu^dam^casibusTui  observant,  non  aggregatum  utique  non  continuum,  sed  partibus  disjunctis  constans. 
quibus  videtur  tedi.  Hujusmodi  est  ex.  gr.  altitude  cujusdam  domus,  quae  aedificatur  de  novo,  cui  cum  series 
nova  adjungitur  lapidum  determinatae  cujusdam  altitudinis,  per  illam  additionem  repente 
videtur  crescere  altitude  domus,  sine  transitu  per  altitudines  intermedias  :  &  si  dicatur  id 
non  esse  Naturae  opus,  sed  artis ;  potest  difficultas  transferri  facile  ad  Naturae  opera,  ut  ubi 
diversa  inducuntur  glaciei  strata,  vel  in  aliis  incrustationibus,  ac  in  iis  omnibus  casibus,  in 
quibus  incrementum  fit  per  externam  applicationem  partium,  ubi  accessiones  finitae  videntur 
acquiri  simul  totae  sine  [68]  transitu  per  intermedias  magnitudines.  In  iis  casibus 
continuitas  servatur  in  motu  singularum  partium,  quae  accedunt.  Illae  per  lineam  quandam 
continuam,  &  continua  velocitatis  mutatione  accedunt  ad  locum  sibi  deditum  :  quin  immo 
etiam  posteaquam  eo  advenerunt,  pergunt  adhuc  moveri,  &  nunquam  habent  quietem  nee 
absolutam,  nee  respectivam  respectu  aliarum  partium,  licet  jam  in  respectiva  positione 
sensibilem  mutationem  non  subeant  :  parent  nimirum  adhuc  viribus  omnibus,  quae 
respondent  omnibus  materiae  punctis  utcunque  distantibus,  &  actio  proximarum  partium, 
quae  novam  adhaesionem  parit,  est  continuatio  actionis,  quam  multo  minorem  exercebant, 
cum  essent  procul.  Hoc  autem,  quod  pertineant  ad  illam  domum,  vel  massam,  est  aliquid 
non  in  se  determinatum,  quod  momento  quodam  determinato  fiat,  in  quo  saltus  habeatur, 
sed  ab  aestimatione  quadam  pendet  nostrorum  sensuum  satis  crassa  ;  ut  licet  perpetuo 
accedant  illae  partes,  &  pergant  perpetuo  mutare  positionem  respectu  ipsius  massae  ;  turn 
incipiant  censeri  ut  pertinentes  ad  illam  domum,  vel  massam  :  cum  desinit  respectiva 
mutatio  esse  sensibilis,  quae  sensibilitatis  cessatio  fit  ipsa  etiam  quodammodo  per  gradus 
omnes,  &  continue  aliquo  tempore,  non  vero  per  saltum. 


Generate  responsio  ISO-  Hinc  distinctius  ibi  licebit  difHcultatem  omnem  amovere  dicendo,  non  servari 

de  emta.3  similes  m"  mutationem  continuam  in    magnitudinibus    earum    rerum,  quae    continuae    non    sunt,  & 

magnitudinem  non  habent  continuam,  sed  sunt  aggregata  rerum  disjunctarum  ;   vel  in  iis 

rebus,  quae  a  nobis  ita  censentur  aliquod  totum  constituere,  ut  magnitudinem  aggregati  non 


A  THEORY  OF  NATURAL  PHILOSOPHY  121 

the  angles  of  incidence  &  lie  on  the  near  side  of  the  surface,  without  any  reflection  for 
rays  at  intermediate  angles  with  the  surface  less  than  a  certain  definite  angle. 

147.  It  seems  that  an  explanation  of  this  apparent  breach  of  continuity  can  be  given  Apparent  recontiii- 
by  means  of  light  that  is  reflected  or  refracted  irregularly  at  all  sorts  of  angles.     For  long  ago  of  Continuity6  effect 
it  was  observed  that,  when  a  ray  of  light  is  reflected,  it  is  not  reflected  entirely  in  such  a  *ed  fay  means    of 
manner  that  the  angle  of  reflection  is  equal  to  the  angle  of  incidence,  but  that  a  part  of  it 

is  dispersed  in  all  directions.  For  this  reason,  if  a  ray  of  light  from  the  Sun  falls  upon  some 
part  of  a  mirror,  anybody  who  is  in  the  room  sees  where  the  ray  strikes  the  mirror  ;  & 
this  certainly  would  not  be  the  case,  unless  some  of  the  solar  rays  reached  his  eye  directly 
issuing  from  the  mirror  in  all  those  directions  that  reach  to  all  positions  that  the  eye  might 
be  in.  Nevertheless,  in  this  case  the  light  does  not  appear  to  be  of  much  intensity,  unless 
the  eye  is  in  the  position  facing  the  angle  of  reflection  equal  to  the  angle  of  incidence,  along 
which  the  greatest  part  of  the  light  rebounds.  Newton  indeed  employed  in  a  brilliant 
way  these  rays  that  issue  at  irregular  angles  at  the  end  of  his  Optics  to  explain  the  colours 
of  solid  laminae.  The  same  irregular  dispersion  in  all  directions  takes  place  as  far  as  can 
be  observed  in  a  small  part,  but  yet  in  some  part,  of  the  refracted  ray.  Hence,  in  between 
the  intense  reflected  &  refracted  rays,  we  have  a  whole  series  of  intermediate  rays  of  this  sort 
issuing  at  all  intermediate  angles.  Thus,  when  the  transition  is  effected  from  refraction 
to  total  reflection,  it  seems  that  it  can  be  done  through  these  intermediate  angles  by  an 
extremely  rapid  transition  through  them,  &  therefore  continuity  remains  unimpaired. 

148.  But  if  we  inquire  into  the  matter  yet  more  carefully,  it  will  be  evident  that  in  Why  this  is  only  an 
that  intermediate  series  there  is  no  accurate  continuity,  but  only  an  apparent  continuity ;  atimT*  the^true 
&  this   Nature   tries   to   maintain,  but  does  not  accurately  observe  it  unimpaired.     For,  reconciliation  is 
in  my  Theory,  light  is  not  some  continuous  body,  which  is  continuously  diffused  through  t!nurtyhof  ^ath^or 
all  the  space  it  occupies ;   but  it  is  an  aggregate  of  points  unconnected  with  &  separated  any  point  of  light, 
from  one  another  ;   &  of  these  points,  any  one  pursues  its  own  path,  &  this  path  is  separated 

from  the  path  of  the  next  point  to  it  by  a  definite  interval.  Continuity  is  observed  perfectly 
accurately  for  the  paths  of  the  several  points,  not  in  the  diffusion  of  a  substance  that  is 
not  continuous ;  &  the  manner  in  which  continuity  is  preserved  in  all  these  motions, 
&  the  path  described  by  the  several  points  is  altered  without  sudden  change,  when  the  angle 
of  incidence  is  altered,  I  have  set  forth  fairly  clearly  in  the  second  part  of  my  dissertation 
De  Lumine,  Art.  98.  But  in  this  work  these  matters  belong  to  the  application  of  the 
Theory  to  physics. 

140.  There  are  certain  cases,  not  greatly  unlike  those  already  given,  in  which  each  The    manner  in 

/   i     _j«  i_  •  '•  V  j     r   which      continuity 

part  preserves  continuity,  but  not  so  the  whole,  which  is  not  continuous  but  composed  ot  is   maintained    m 

separate  parts.     For  an  instance  of  this  kind,  take  the  height  of  a  new  house  which  is  being  certain     cases    in 

built ;   as  a  fresh  layer  of  stones  of  a  given  height  is  added  to  it,  the  height  of  the  house  ^ 

on  account  of  that  addition  seems  to  increase  suddenly  without  passing  through  intermediate 

heights.     If  it  is  said  that  that  is  not  a  work  of  Nature,  but  of  art ;  then  the  same  difficulty 

can  easily  be  transferred  to  works  of  Nature,  as  when  different  strata  of  ice  are  formed,  or 

in  other  incrustations,  and  in  all  cases  in  which  an  increment  is  caused  by  the  external 

application  of  parts,  where  finite  additions  seem  to  be  acquired  all  at  once  without  any 

passage  through  intermediate  magnitudes.     In  these  cases  the  continuity  is  preserved  in 

the  motions  of  the  separate  parts  that  are  added.     These  reach  the  place  allotted  to  them 

along  some  continuous  line  &  with  a  continuous  change  of  velocity.     Further,  after  they 

have  reached  it,  they  still  continue  to  move,  &  never  have  absolute  rest ;   no,  nor  even 

relative  rest  with  respect  to  the  other  parts,  although  they  do  not  now  suffer  a  sensible 

change  in  their  relative  positions.     Thus,  they  still  submit  to  the  action  of  all  the  forces 

that  correspond  to  all  points  of  matter  at  any  distances  whatever  ;    and  the  action  of  the 

parts  nearest  to  them,  which  produces  a  new  adhesion,  is  the  continuation  of  the  action 

that  they  exert  to  a  far  smaller  extent  when  they  are  some  distance  away.     Moreover,  in 

the  fact  that  they  belong  to  that  house  or  mass,  there  is  something  that  is  not  determinate 

in  itself,  because  it  happens  at  a  determinate  instant  in  which  the  sudden  change  takes 

place  ;   but  it  depends  on  a  somewhat  rough  assessment  by  our  senses.     So  that,  although 

these  parts    are   continually    being    added,  &  continually  go  on  changing  their  position 

with  respect  to  the  mass,  they  both  begin  to  be  thought  of  as  belonging  to  that  house  or 

mass,  &  the  relative  change  ceases  to  be  sensible  ;    also  this  cessation  of  sensibility  itself 

also  takes  place  to  some  extent  through  all  stages,  and  in  some  continuous  interval  of  time, 

&  not  by  a  sudden  jump. 

KO.  From  this  consideration  we  may  here  in  a  clearer  manner  remove  all  difficulty 

-'  *  .        ,    .         -  •jrt,'U*  simuar     cd.b 

by  saying  that  a  continuous  change  is  not  maintained  in  the  magnitudes  ot  those  tmngs,  derived  from  this, 
which  are  not  themselves  continuous,  &  do  not  possess  continuous  magnitude,  but  are 
aggregates  of  separate  things.     That  is  to  say,  in  those  things  that  are  thus  considered  as 
forming  a  certain  whole,  in  such  a  way  that  the  magnitude  of  the  aggregate  is  not  determined 


122 


PHILOSOPHIC  NATURALIS  THEORIA 


determinent  distantias  inter  eadem  extrema,  sed  a  nobis  extrema  ipsa  assumantur  jam  alia, 
jam  alia,  quae  censeantur  incipere  ad  aggregatum  pertinere,  ubi  ad  quasdam  distantias 
devenerint,  quas  ut  ut  in  se  juxta  legem  continuitatis  mutatas,  nos  a  reliquis  divellimus  per 
saltum,  ut  dicamus  pertinere  eas  partes  ad  id  aggregatum.  Id  accidit,  ubi  in  objectis 
casibus  accessiones  partium  novae  fiunt,  atque  ibi  nos  in  usu  vocabuli  saltum  facimus ;  ars, 
&  Natura  saltum  utique  habet  nullum. 

Alii  casus  in  quibus  151.  Non  idem  contingit  etiam,  ubi  plantas,  vel  animantia  crescunt,  succo  se  insinuante 

'uibusUr'  hab'etur  Per  tubulos  fibrarum,  &  procurrente,  ubi  &  magnitude  computata  per  distantias  punctorum 
soium  proxima,  non  maxime  distantium  transit  per  omnes  intermedias ;  cum  nimirum  ipse  procursus  fiat 
accurata  contmm-  pef  omnes  intermedias  distantias.  At  quoniam  &  ibi  mutantur  termini  illi,  qui  distantias 
determinant,  &  nomen  suscipiunt  altitudinis  ipsius  plantas  ;  vera  &  accurata  continuitas  ne 
ibi  quidem  observatur,  nisi  tantummodo  in  motibus,  &  velocitatibus,  ac  distantiis  singularum 
partium  :  quanquam  ibi  minus  recedatur  a  continuitate  accurata,  quam  in  superioribus.  In 
his  autem,  &  in  illis  habetur  ubique  ilia  alia  continuitas  quasdam  apparens,  &  affectata 
tantummodo  a  Natura,  quam  intuemur  etiam  in  progressu  substantiarum,  ut  incipiendo  ab 
inanima-[69]-tis  corporibus  progressu  facto  per  vegetabilia,  turn  per  quasdam  fere 
semianimalia  torpentia,  ac  demum  animalia  perfectiora  magis,  &  perfectiora  usque  ad  simios 
homini  tarn  similes.  Quoniam  &  harum  specierum,  ac  existentium  individuorum  in  quavis 
specie  numerus  est  finitus,  vera  continuitas  haberi  non  potest,  sed  ordinatis  omnibus  in 
seriem  quandam,  inter  binas  quasque  intermedias  species  hiatus  debet  esse  aliquis  necessario, 
qui  continuitatem  abrumpat.  In  omnibus  iis  casibus  habentur  discretas  quasdam  quantitates, 
non  continues  ;  ut  &  in  Arithmetica  series  ex.  gr.  naturalium  numerorum  non  est  continua, 
sed  discreta  ;  &  ut  ibi  series  ad  continuam  reducitur  tantummodo,  si  generaliter  omnes 
intermedias  fractiones  concipiantur  ;  sic  &  in  superiore  exemplo  quasdam  velut  continua 
series  habebitur  tantummodo  ;  si  concipiantur  omnes  intermedias  species  possibiles. 


uitatem. 


Conciusio  pertinens  152.  Hoc  pacto  excurrendo  per    plurimos    justmodi    casus,  in    quibus  accipiuntur 

ad  ea,  quse  veram,  aggregata  rerum  a  se  invicem  certis  intervallis  distantium,  &  unum  aliquid  continuum  non 

(X     CcL,    CJ1.13E    cLttCCtcl"          OO         O  •••iill** 

tam  habent  contin-  constituentium,  nusquam  accurata  occurret  continuitatis  lex,  sed  per  quandam  dispersionem 
quodammodo  affectata,  &  vera  continuitas  habebitur  tantummodo  in  motibus,  &  in  iis,  quas 
a  motibus  pendent,  uti  sunt  distantiae,  &  vires  determinatas  a  distantiis,  &  velocitates  a 
viribus  ortae  ;  quam  ipsam  ob  causam  ubi  supra  num.  39  inductionem  pro  lege  continuitatis 
assumpsimus,  exempla  accepimus  a  motu  potissimum,  &  ab  iis,  quae  cum  ipsis  motibus 
connectuntur,  ac  ab  iis  pendent. 

153.  Sed  jam  ad  aliam  difficultatem  gradum  faciam,  quae  non  nullis  negotium  ingens 
3ito  facessit,  &  obvia  est  etiam,  contra  hanc  indivisibilium,  &  inextensorum  punctorum  Theoriam ; 
'  &  quod  nimirum  ea  nullum  habitura  sint   discrimen  a  spiritibus.     Ajunt   enim,  si   spiritus 
ejusmodi  vires  habeant,  praestituros  eadem  phaenomena,  tolli  nimirum  corpus,  &  omnem 
corporeae  substantiae  notionem  sublata  extensione  continua,  quae    sit    prascipua    materias 
proprietas  ita  pertinens  ad  naturam  ipsius ;    ut  vel  nihil  aliud  materia  sit,  nisi  substantia 
praedita  extensione  continua  ;   vel  saltern  idea  corporis,  &  materiae  haberi  non  ppssit ;  nisi 
in  ea  includatur  idea  extensionis  continuae.     Multa  hie  quidem  congeruntur  simul,  quae 
nexum  aliquem  inter  se  habent,  quae  hie  seorsum  evolvam  singula. 


Difficultates  petitae 
a  discrimine  debito 
inter  materiam 
spiritum. 


DifferrehKcpuncta  154.  Inprimis  falsum  omnino  est,  nullum  esse  horum  punctorum  discrimen  a  spiritibus. 

fm^netrabUttlte^  Discrimen  potissimum  materiae  a   spiritu  situm  est  in   hisce   duobus,  quod  _  materia_  est 

™nSitatem,a  e£-  sensibilis,  &  incapax  cogitationis,  ac  voluntatis,  spiritus  nostros  sensus  non  afficit,  &  cogitare 

capadtatem  cogit-  pOtest)  ac  velle.     Sensibilitas  autem  non  ab  extensione  continua  oritur,  sed  ab  impene- 

trabilitate,  qua  fit,  ut    nostrorum   organorum  fibrae   tendantur  a  corporibus,  quae   ipsis 

sistuntur,  &  motus  ad  cerebrum  pro-[7o]-pagetur.     Nam  si  extensa  quidem  essent  corpora, 

sed  impenetrabilitate  carerent ;    manu  contrectata  fibras  non  sisterent,  nee  motum  ullum 

in  iis  progignerent,  ac  eadem  radios  non  reflecterent,  sed  liberum  intra  se  aditum  luci 

prasberent.     Porro  hoc  discrimen  utrumque  manere  potest  integrum,  &  manet  inter  mea 

indivisibilia  hasc  puncta,  &  spiritus.     Ipsa  impenetrabilitatem  habent,  &  sensus  nostros 

afficiunt,  ob  illud  primum  crus  asymptoticum  exhibens  vim  illam  repulsivam  primam  ; 

spiritus  autem,  quos  impenetrabilitate  carere  credimus,  ejusmodi  viribus  itidem  carent,  & 

sensus  nostros  idcirco  nequaquam  afficiunt,  nee  oculis  inspectantur,  nee  ^manibus  palpari 

possunt.     Deinde  in  meis  hisce  punctis  ego  nihil  admitto  aliud,  nisi  illam  virium  legem  cum 

inertias  vi    conjunctam,  adeoque    ilia    volo   prorsus    incapacia    cogitationis,  &  voluntatis. 


A  THEORY  OF  NATURAL  PHILOSOPHY 


I23 


by  the  distances  between  the  same  extremes  all  the  time,  but  the  extremes  we  take  are 
different,  one  after  another ;  &  these  are  considered  to  begin  to  belong  to  the  aggregate 
when  they  attain  to  certain  distances  from  it ;  &,  although  in  themselves  changed  in 
accordance  with  the  Law  of  Continuity,  we  separate  them  from  the  rest  in  a  discontinuous 
manner,  by  saying  that  these  parts  belong  to  the  aggregate.  This  comes  about,  whenever 
in  the  cases  under  consideration  fresh  additions  of  parts  take  place  ;  &  then  we  make  a 
discontinuity  in  the  use  of  a  term ;  art,  as  well  as  Nature,  has  no  discontinuity. 

151.  It  is  not  the  same  thing  however  in  the  case  of  the  growth  of  plants  or  animals, 
which  is  due  to  a  life-principle  insinuating  itself  into,  &  passing  along  the  fine  tubes  of  the 
fibres  ;  here  the  magnitude,  calculated  by  means  of  the  distance  between  the  points  furthest 
from  one  another,  passes  through  all  intermediate  distances ;  for  the  flow  of  the  life-principle 
takes  place  indeed  through  all  intermediate  distances.     But,  since  here  also  the  extremes 
are  changed,  which  determine  the  distances,  &  denominate  the  altitude  of   the   plant ; 
not  even  in  this  case  is  really  accurate  continuity  observed,  except  only  in  the  motions  & 
velocities  and  distances  of  the  separate  parts ;  however  there  is  here  less  departure  from 
accurate  continuity,  than  there  was  in  the  examples  given  above.     In  both  there  is  indeed 
that  kind  of  apparent  continuity,  which  Nature  does  no  more  than  try  to  maintain  ;   such 
as  we  also  see  in  the  series  of  substantial  things,  which  starting  from  inanimate  bodies, 
continues  through  vegetables,  then  through  certain  sluggish  semianimals,  &  lastly,  through 
animals  more  &  more  perfect,  up  to  apes  that  are  so  like  to  man.     Also,  since  the  number 
of  these  species,  &  the  number  of  existent  individuals  of  any  species,  is  finite,  it  is  impossible 
to  have  true  continuity  ;  but  if  they  are  all  ordered  in  a  series,  between  any  two  intermediate 
species  there  must  necessarily  be  a  gap  ;   &  this  will  break  the  continuity.     In  all  these 
cases  we  have  certain  discrete,  &  not   continuous,  quantities  ;    just  as,  for  instance,  the 
arithmetical  series  of  the  natural  numbers  is  not  continuous,  but  discrete.     Also,  just  as  the 
series  is  reduced  to  continuity  only  by  mentally  introducing  in  general  all  the  intermediate 
fractions ;  so  also,  in  the  example  given  above  a  sort  of  continuous  series  is  obtained,  if 
&  only  if  all  intermediate  possible  species  are  so  included. 

152.  In  the  same  way,  if  we  examine  a  large  number  of  cases  of  the  same  kind,  in  which 
aggregates  of  things  are  taken,  separated  from  one  another  by  certain  definite  intervals, 
&   not   composing  a  single  continuous  whole,  an  accurate  continuity  law  will  never  be 
met  with,  but  only  a  sort  of  counterfeit  depending  on  dispersion.     True  continuity  will 
only  be  obtained  in  motions,  &  in  those  things  that  depend  on  motions,  such  as  distances 
&  forces  determined  by  distances,  &  velocities  derived   from  such   forces.     It   was   for 
this  very  reason  that,  when  we  adopted  induction  for  the  proof  of  the  Law  of  Continuity 
in  Art.  39  above,  we  took  our  examples  mostly  from  motion,  &  from  those  things  which 
are  connected  with  motions  &  depend  upon  them. 

153.  Now  I  will  pass  on  to  another  objection,  which  some  people  have  made  a  great 
to-do  about,  and  which  has  also  been  raised  in  opposition  to  this  Theory  of  indivisible  & 
non-extended   points ;    namely,  that  there  will  be  no  difference   between  my  points  & 
spirits.     For,  they  say  that,  if  spirits  were  endowed  with  such  forces,  they  would  show  the 
same  phenomena   as  bodies,  &  that   bodies  &  all  idea  of  corporeal  substance  would  be 
done  away  with  by  denying  continuous  extension  ;  for  this  is  one  of  the  chief  properties  of 
matter,  so  pertaining  to  Nature  itself ;   so  that  either  matter  is  nothing  else  but  substance 
endowed  with  continuous  extension,  or  the  idea  of  a  body  and  of  matter  cannot  be  obtained 
without  the  inclusion  of  the  idea  of  continuous  extension.     Here  indeed  there  are  many 
matters  all  jumbled  together,  which  have  no  connection  with  one  another ;    these  I  will 
now  separate  &  discuss  individually. 

154.  First  of  all  it  is  altogether  false  that  there  is  no  difference  between  my  points  & 
spirits.     The  most   important   difference   between    matter  &  spirit  lies  in  the  two  facts, 
that  matter  is  sensible  &  incapable  of   thought,  whilst  spirit  does  not  affect  the  senses, 
but  can  think  or  will.     Moreover,  sensibility  does  not  arise  from  continuous  extension, 
but  from  impenetrability,  through  which  it  comes  about  that  the  fibres  of  our  organs  are 
subjected  to  stress  by  bodies  that  are  set  against  them  &  motions  are  thereby  propagated 
to  the  brain.     For  if  indeed  bodies  were  extended,  but  lacked  impenetrability,  they  would 
not  resist  the  fibres  of   the  hand  when  touched,  nor  produce  in  them  any  motion  ;  nor 
would  they  reflect   light,   but   allow  it   an   uninterrupted   passage  through  themselves. 
Further,  it  is  possible  that  each  of  these  distinctions  should  hold  good  independently ; 
&  they   do   so    between    these    indivisible    points    of    mine  &  spirits.     My  points    have 
impenetrability  &  affect  our  senses,  because  of  that  first  asymptotic  branch  representing  that 
first  repulsive  force  ;  but  spirits,  which  we  suppose  to  lack  impenetrability,  lack  also  forces 
of  this  kind,  and  therefore  can  in  no  wise  affect  our  senses,  nor  be  examined  by  the  eyes, 
nor  be  felt  by  the  hands.     Then,  in  these  points  of  mine,  I  admit  nothing  else  but  the 
law  of  forces  conjoined  with  the  force  of  inertia  ;  &  hence  I  intend  them  to  be  incapable 


Cases  in  which 
there  is  a  breach  of 
continuity  ;  others 
in  which  the  con- 
tinuity is  only  very 
nearly,  but  not 
accurately,  ob- 
served. 


Conclusion  as  re- 
gards those  things 
that  possess  true 
continuity,  and 
those  that  have  a 
counterfeit  continu- 
ity. 


Objections  derived 
from  the  distinc- 
tion that  has  to  be 
made  between 
matter  &  spirit. 


These  points  differ 
from  spirits  on 
account  o  f  their 
impenet  rability, 
their  being  sen- 
sible, &  their  inca- 
pacity for  thought. 


I24 


PHILOSOPHIC   NATURALIS  THEORIA 


Si    possibilis    sub- 


earn  nee 
materiam 
spiritum. 


Quamobrem  discrimen  essentiae  illud  utrumque,  quod  inter  corpus,  &  spiritum 
agnoscunt  omnes,  id  &  ego  agnosco,  nee  vero  id  ab  extensione,  &  compositione  continua 
desumitur,  sed  ab  iis,  quae  cum  simplicitate,  &  inextensione  aeque  conjungi  possunt,  & 
cohaerere  cum  ipsis. 

155.  At  si  substantiae  capaces  cogitationis  &  voluntatis  haberent  ejusmodi  virium  legem, 
an  non  eosdem  praestarent  effectus  respectu  nostrorum  sensuum,  quos  ejusmodi  puncta  ? 
capax  cogitationis ;   Respondebo  sane,  me  hie  non  quaerere,  utrum  impenetrabilitas,  &  sensibilitas,  quae  ab  iis 
fnec  viribus  pendent,  conjungi  possint  cum  facultate  cogitandi,  &  volendi,  quae  quidem  quaestio 
eodem  redit,  ac  in  communi  sententia  de  impenetrabilitate  extensorum,  ac  compositorum 
relata    ad    vim   cogitandi,  &    volendi.     Illud  ajo,  notionem,  quam    habemus   partim   ex 
observationibus    tarn   sensuum    respectu    corporurh,    quam   intimae    conscientiae   respectu 
spiritus,  una  cum  reflexione,  partim,  &  vero  etiam  circa  spiritus  potissimum,  ex  principiis 
immediate    revelatis,    vel    connexis    cum    principiis    revelatis,     continere     pro     materia 
impenetrabilitatem,  &  sensibilitatem,   una  cum  incapacitate  cogitationis,  &  pro  spiritu 
incapacitatem  afHcicndi  per  impenetrabilitatem  nostros  sensus,  &  potentiam  cogitandi,  ac 
volendi,  quorum  priores  illas  ego  etiam  in    meis    punctis  admitto,  posteriores    hasce   in 
spiritibus ;    unde  fit,  ut  mea  ipsa    puncta    materialia    sint,  &    eorum    massae    constituant 
corpora  a  spiritibus  longissime  discrepantia.     Si  possibile  sit  illud  substantiae  genus,  quod 
&  hujusmodi  vires  activas  habeat  cum  inertia  conjunctas,  &  simul  cogitare  possit,  ac  velle ; 
id  quidem  nee  corpus  erit,  nee  spiritus,  sed  tertium  quid,  a  corpore  discrepans  per  capacitatem 
cogitationis,  &  voluntatis,  discrepans  autem  a  spiritu  per  inertiam,  &  vires  hasce  nostras, 
quae  impenetrabilitatem  inducunt.     Sed,  ut  ajebam,  ea  quaestio  hue  non  pertinet,  &  aliunde 
resolvi  debet ;  ut  aliunde  utique  debet  resolvi  quaestio,  qua  quaeratur,  an  substantia  extensa, 
&  impenetrabilis  [71]  hasce    proprietates    conjungere    possit    cum    facultate    cogitandi, 
volendique. 


Nihil     amitti,  156.  Nee  vero  illud  reponi  potest,  argumentum  potissimum  ad  evincendum,  materiam 

amisso   argumento  cogitare  non  posse,  deduci  ab  extensione,  &  partium  compositione,  quibus  sublatis,  omne  id 

eorum,  qui  a  com-    r    to  ,  . r, .  •     •  VT  •  i 

positione  partium  lundamentum  prorsus  corruere,  &  ad  materialismum  sterm  viam.  JNam  ego  sane  non  video, 
deducunt  incapaci-  quid  argument!  peti  possit  ab  extensione,  &  partium  compositione  pro  incapacitate  cogitandi, 
&  volendi.  Sensibilitas,  praecipua  corporum,  &  materiae  proprietas,  quae  ipsam  adeo  a 
spiritibus  discriminat,  non  ab  extensione  continua,  &  compositione  partium  pendet,  uti 
vidimus,  sed  ab  impenetrabilitate,  quae  ipsa  proprietas  ab  extensione  continua,  &  compositione 
non  pendet.  Sunt  qui  adhibent  hoc  argumentum  ad  excludendam  capacitatem  cogitandi 
a  materia,  desumptum  a  compositione  partium  :  si  materia  cogitaret ;  singulae  ejus  partes 
deberent  singulas  cogitationis  partes  habere,  adeoque  nulla  pars  objectum  perciperet ;  cum 
nulla  haberet  earn  perceptionis  partem,  quam  habet  altera.  Id  argumentum  in  mea  Theoria 
amittitur  ;  at  id  ipsum,  meo  quidem  judicio,  vim  nullam  habet.  Nam  posset  aliquis 
respondere,  cogitationem  totam  indivisibilem  existere  in  tota  massa  materiae,  quae  certa 
partium  dispositione  sit  praedita,  uti  anima  rationalis  per  tarn  multos  Philosophos,  ut  ut 
indivisibilis,  in  omni  corpore,  vel  saltern  in  parte  corporis  aliqua  divisibili  existit,  &  ad 
ejusmodi  praesentiam  praestandam  certa  indiget  dispositione  partium  ipsius  corporis,  qua 
semel  laesa  per  vulnus,  ipsa  non  potest  ultra  ibi  esse  ;  atque  ut  viventis  corporei,  sive  animalis 
rationalis  natura,  &  determinatio  habetur  per  materiam  divisibilem,  &  certo  modo 
constructam,  una  cum  anima  indivisibili ;  ita  ibi  per  indivisibilem  cogitationem  inhaerentem 
divisibili  materise  natura,  &  determinatio  cogitantis  haberetur.  Unde  aperte  constat  eo 
argumento  amisso,  nihil  omnino  amitti,  quod  jure  dolendum  sit. 


Etiam  si  quidpiam  157.  Sed  quidquid  de  eo  argumento  censeri  debeat,  nihil  refert,  nee  ad  infirmandam 

iam^poStive  ^prob-  Theoriam  positivis,  &  validis  argumentis  comprobatam,  ac  e  solidissimis  principiis  directa 

ari,  &  in  ea  manere  ratiocinatione  deductani,  quidquam  potest  unum,  vel  alterum  argumentum  amissum,  quod 

inte™m™ter1ainme&  a^  probandam  aliquam  veritatem  aliunde  notam,  &  a  revelatis  principiis  aut  directe,  aut 

spiritum.  indirecte  confirmatam,  ab  aliquibus  adhibeatur,  quando  etiam  vim  habeat  aliquam,  quam, 

uti  ostendi,  superius  allatum  argumentum  omnino  non  habet.     Satis  est,  si  ilia  Theoria  cum 

ejusmodi  veritate  conjungi  possit,  uti  haec  nostra  cum   immaterialitate   spirituum   con- 

jungitur  optime,  cum  retineat  pro  materia  inertiam,  impenetrabilitatem,  sensibilitatem, 

incapacitatem  cogitandi,  &  pro  spiritibus  retineat  incapacitatem  afHciendi  sensus  nostros 

per  impenetrabilitatem,  &  facultatem  cogitandi,  ac  volendi.      [72]  Ego  quidem  in  ipsius 


A  THEORY  OF  NATURAL  PHILOSOPHY  125 

of  thought  or  will.  Wherefore  I  also  acknowledge  each  of  those  essential  differences  between 
matter  and  spirit,  which  are  acknowledged  by  everyone  ;  but  by  me  it  is  not  deduced  from 
extension  and  continuous  composition,  but,  just  as  correctly,  from  things  that  can  be 
conjoined  with  simplicity  &  non-extension,  &  can  combine  with  them. 

155.  Now  if  there  were  substances  capable  of  thought  &  will  that  also  had  a  law  of  if  it  were  possible 
forces  of  this  kind,  is  it  possible  that  they  would  produce  the  same  effects  with  respect  to  substanc<Pthatwas 
our  senses,  as  points  of  this  sort  ?     Truly,  I  will  answer  that  I  do  not  seek  to  know  in  this  both  endowed  with 
connection,   whether  impenetrability  &  sensibility,  which  depend    on    these   forces,  can  capsTbieofthoughT 
be  conjoined  with  the  faculty  of  thinking  &  willing  ;   indeed  this  question  comes  to  the  it  would  be  neither 
same  thing  as  the  general  idea  of  the  relations  of  impenetrability  of  extended  &  composite  matter  nor  sP'nt- 
things  to  the  power  of  thinking  &  willing.     I  will  say  but  this,  that  we  form  our  ideas, 

partly  from  observations,  of  the  senses  in  the  case  of  bodies,  &  of  the  inner  consciousness 
in  the  case  of  spirits,  together  with  reflections  upon  them,  partly,  &  indeed  more  especially 
in  the  case  of  spirits,  from  directly  revealed  principles,  or  matters  closely  connected  with 
revealed  principles ;  &  these  ideas  involve  for  matter  impenetrability,  sensibility,  combined 
with  incapacity  for  thought,  &  for  spirit  an  incapacity  for  affecting  our  senses  by  means 
of  impenetrability,  together  with  the  capacity  for  thinking  and  willing.  I  admit  the  former 
of  these  in  the  case  of  my  points,  &  the  latter  for  spirits ;  so  that  these  points  of  mine 
are  material  points,  &  masses  of  them  compose  bodies  that  are  far  different  from  spirits. 
Now  if  it  were  possible  that  there  should  be  some  kind  of  substance,  which  has  both  active 
forces  of  this  kind  together  with  a  force  of  inertia  &  also  at  the  same  time  is  able  to 
think  and  will ;  then  indeed  it  will  neither  be  body  nor  spirit,  but  some  third  thing,  differing 
from  a  body  in  its  capacity  for  thought  &  will,  &  also  from  spirit  by  possessing  inertia 
and  these  forces  of  mine,  which  lead  to  compenetration.  But  as  I  was  saying,  that  question 
does  not  concern  me  now,  &  the  answer  must  be  found  by  other  means.  So  by  other 
means  also  must  the  answer  be  found  to  the  question,  in  which  we  seek  to  know  whether 
a  substance  that  is  extended  &  impenetrable  can  conjoin  these  two  properties  with  the 
faculty  of  thinking  and  willing. 

156.  Now  it  cannot  be  ignored  that  an  argument  of  great  importance  in  proving  that  Nothing     is    lost 
matter  is  incapable  of  thought  is  deduced  from  extension  &  composition  by  parts ;   &  ^n  ^g^^1"1^ 
if  these  are  denied,  the  whole  foundation  breaks  down,  &  the  way  is  laid  open  to  materialism,  those  who  deduce 
But  really  I  do  not  see  what  in  the  way  of  argument  can  be  derived  from  extension  &  i?  °t&%~*yj;°r 

...  ,  *,  i  •     i  •  i        *ii*  n          *i  "v  v         i_  •    f    lutJU5i*^   irom  com- 

composition  by  parts,  to  support  incapacity  for  thinking  and  willing,     bensibmty,  the  cruel  position  by  parts. 

property  of  bodies  &  of  matter,  which  is  so  much  different  from  spirits,  does  not  depend 

on  continuous  extension  &  composition  by  parts,  as  we  have  seen,  but  on  impenetrability  ; 

&  this  latter  property  does  not  depend  on  continuous  extension  &  composition.     There 

are  some,  who  use  the  following  argument,  derived  from  composition  by  parts,  to  exclude 

from  matter  the  capacity  for  thought  : — If  matter  were  to  think,  then  each  of  its  parts 

would  have  a  separate  part  of  the  thought,  &  thus  no  part  would  have  perception  of  the 

object  of  thought ;  for  no  part  can  have  that  part  of  the  perception  that  another  part  has. 

This  argument  is  neglected  in  my  Theory ;   but  the  argument  itself,  at  least  so  I  think,  is 

unsound.     For  one  can  reply  that  the  complete  thought  exists  as  an  indivisible  thing  in 

the  whole  mass  of  matter,  which  is  endowed  with  a  certain  arrangement  of  parts,  in^the 

same  way  as  the  rational  soul  in  the  opinion  of  so  many  philosophers  exists,  although  it  is 

indivisible,  in  the  whole  of  the  body,  or  at  any  rate  in  a  certain  divisible  part  of  the  body ; 

&  to  maintain  a  presence  of  this  kind  there  is  need  for  a  definite  arrangement  of  the  parts  of 

the  body,  which  if  at  any  time  impaired  by  a  wound  would  no  longer  exist  there.     Thus, 

just  as  from  the  nature  of  a  living  body,  or  of  a  rational  animal,  determination  arises  from 

matter  that  is  divisible  &  constructed  on  a  definite  plan,  in  conjunction  with  an  indivisible 

mind  ;  so  also  in  this  case  by  means  of  indivisible  thought  inherent  in  the  nature  of  divisible 

matter,    there   is    a    propensity   for   thought.     From    this   it   is   very  plain  that,   if  this 

argument  is  dismissed,   there  will  be  nothing  neglected   that  we  have   any    reason   to 

regret. 

157.  But  whatever  opinion  we  are  to  form  about  this  argument,  it  makes  no  difference,  Even  a  something 
nor  can  it  weaken  a  Theory  that  has  been  corroborated  by  direct  &  valid  arguments,  &  iheVheorrcln  Cbe 
deduced  from  the  soundest  principles  by  a  straightforward  chain  of  reasoning,  if  we  leave  P^J^.  in&a  dt^t 
out  one  or  other  of  the  arguments,  which  have  been  used  by  some  for  the  purpose  of  ^f^fi  remain  in 
testing  some    truth  that  is  otherwise  known  &  confirmed  by  revealed   principles  either  j^^fj*^ 
directly  or  indirectly  ;   even  when  the  argument  has  some  validity,  which,  as  I  have  shown,  matter  &  spirit. 
that  adduced  above  has  not  in  any  way.     It  is  sufficient  if  that  theory  can  be  conjoined 

with  such  a  truth  ;  just  as  this  Theory  of  mine  can  be  conjoined  in  an  excellent  manner 
with  the  immateriality  of  spirits.  For  it  retains  for  matter  inertia,  impenetrability, 
sensibility,  &  incapacity  for  thinking,  &  for  spirits  it  retains  the  incapacity  for  affecting 
our  senses  by  impenetrability,  &  the  faculty  of  thinking  or  willing.  Indeed  I  assume  the 


126  PHILOSOPHIC  NATURALIS  THEORIA 

materiae,  &  corporeae  substantias  definitione  ipsa  assumo  incapacitatem  cogitandi,  &  volendi, 
&  dico  corpus  massam  compositam  e  punctis  habentibus  vim  inertiae  conjunctam  cum 
viribus  activis  expressis  in  fig.  i,&  cum  incapacitate  cogitandi,  ac  volendi,  qua  definitione 
admissa,  evidens  est,  materiam  cogitare  non  posse  ;  quae  erit  metaphysica  quaedam  conclusio, 
ea  definitione  admissa,  certissima  :  turn  ubi  solae  rationes  physicae  adhibeantur,  dicam,  haec 
corpora,  quae  meos  afficiunt  sensus,  esse  materiam,  quod  &  sensus  afficiant  per  illas  utique 
vires,  &  non  cogitent.  Id  autem  deducam  inde,  quod  nullum  cogitationis  indicium 
praestent ;  quae  erit  conclusio  tantum  physica,  circa  existentiam  illius  materiae  ita  definitae, 
aeque  physice  certa,  ac  est  conclusio,  quae  dicat  lapides  non  habere  levitatem,  quod  nunquam 
earn  prodiderint  ascendendo  sponte,  sed  semper  e  contrario  sibi  relict!  descenderint. 


Sensus  omnino  fain  158.  Quod  autem  pertmet  ad  ipsam  corporum,  &  materiae  ideam,  quae  videtur  exten- 

^nultat^in^xten-  si°nem  continuam,  &  contactum  partium  involvere,  in  eo  videntur  mihi  quidem  Cartesian! 
sionis,  quam  nobis  inprimis,  qui  tantopere  contra  prasjudicia  pugnare  sunt  visi,  praejudiciis  ipsis  ante  omnes 
alios  indulsisse.  Ideam  corporum  habemus  per  sensus ;  sensus  autem  de  continuitate 
accurata  judicare  omnino  non  possunt,  cum  minima  intervalla  sub  sensus  non  cadant.  Et 
quidem  omnino  certo  deprehendimus  illam  continuitatem,  quam  in  plerisque  corporibus 
nobis  objiciunt  sensus  nostri,  nequaquam  haberi.  In  metallis,  in  marmoribus,  in  vitris, 
&  crystallis  continuitas  nostris  sensibus  apparet  ejusmodi,  ut  nulla  percipiamus  in  iis  vacua 
spatiola,  nullos  poros,  in  quo  tamen  hallucinari  sensus  nostros  manifesto  patet,  turn  ex 
diversa  gravitate  specifica,  quae  a  diversa  multitudine  vacuitatum  oritur  utique,  turn  ex 
eo,  quod  per  ilia  insinuentur  substantiae  plures,  ut  per  priora  oleum  diffundatur,  per 
posteriora  liberrime  lux  transeat,  quod  quidem  indicat,  in  posterioribus  hisce  potissi- 
mum  ingentem  pororum  numerum,  qui  nostris  sensibus  delitescunt. 

Fons     prajudici-  159-  Quamobrem  jam  ejusmodi  nostrorum  sensuum  testimonium,  vel  potius  noster 

orum :   haberi  pro  eorum  ratiociniorum  usus,  in  hoc  ipso  genere  suspecta  esse  debent,  in  quo  constat  nos 

nulhs    in    se,   quas     ,  ....  .  •  11-  •   v  •  •      -i_ 

sunt  nuiia  in  nostris  decipi.  Suspican  igitur  licet,  exactam  continuitatem  sine  urns  spatiolis,  ut  in  majonbus 
sensibus  :  eorum  corporibus  ubique  deest,  licet  sensus  nostri  illam  videantur  denotare,  ita  &  in  minimis 
quibusvis  particulis  nusquam  haberi,  sed  esse  illusionem  quandam  sensuum  tantummodo, 
&  quoddam  figmentum  mentis,  reflexione  vel  non  utentis,  vel  abutentis.  Est  enim 
solemne  illud  hominibus,  atque  usitatum,  quod  quidem  est  maximorum  praejudiciorum 
fons,  &  origo  praecipua,  ut  quidquid  in  nostris  sensibus  est  nihil,  habeamus  pro  nihilo 
absolute.  Sic  utique  per  tot  saecula  a  multis  est  creditum,  &  nunc  etiam  a  vulgo  creditur, 
[73]  quietem  Telluris,  &  diurnum  Solis,  ac  fixarum  motum  sensuum  testimonio  evinci, 
cum  apud  Philosophos  jam  constet,  ejusmodi  qusestionem  longe  aliunde  resolvendam  esse, 
quam  per  sensus,  in  quibus  debent  eaedem  prorsus  impressiones  fieri,  sive  stemus  &  nos,  & 
Terra,  ac  moveantur  astra,  sive  moveamur  communi  motu  &  nos,  &  Terra,  ac  astra 
consistant.  Motum  cognoscimus  per  mutationem  positionis,  quam  objecti  imago  habet 
in  oculo,  &  quietem  per  ejusdem  positionis  permanentiam.  Tarn  mutatio,  quam 
permanentia  fieri  possunt  duplici  modo  :  mutatio,  primo  si  nobis  immotis  objectum  movea- 
tur  ;  &  permanentia,  si  id  ipsum  stet  :  secundo,  ilia,  si  objecto  stante  moveamur  nos  ;  haec,  si 
moveamur  simul  motu  communi.  Motum  nostrum  non  sentimus,  nisi  ubi  nos  ipsi  motum 
inducimus,  ut  ubi  caput  circumagimus,  vel  ubi  curru  delati  succutimur.  Idcirco  habemus 
turn  quidem  motum  ipsum  pro  nullo,  nisi  aliunde  admoneamur  de  eodem  motu  per  causas, 
quae  nobis  sint  cognitae,  ut  ubi  provehimur  portu,  quo  casu  vector,  qui  jam  diu  assuevit  idese 
littoris  stantis,  &  navis  promotae  per  remos,  vel  vela,  corrigit  apparentiam  illius,  terrceque 
urbesque  recedunt,  &  sibi,  non  illis,  motum  adjudicat. 


Eorum     correctio  160.  Hinc  Philosophus,  ne  fallatur,  non  debet  primis  hisce  ideis  acquirere,  quas  e 

ubi  deprehendatur,  sensationibus  haurimus,  &  ex  illis  deducere  consectaria  sine  diligent!  perquisitione,  ac  in  ea 

modoalc°umtlsaenn  quae  ab  infantia  deduxit,  debet  diligenter  inquirere.     Si  inveniat,  easdem  illas  sensuum 

suum     apparentia  perceptiones  duplici  modo  aeque  fieri  posse  ;  peccabit  utique  contra  Logicae  etiam  naturalis 

leges,  si  alterum  modum  prze  altero  pergat  eligere,  unice,  quia  alterum  antea  non  viderat, 

&  pro  nullo  habuerat,  &  idcirco  alteri  tantum  assueverat.     Id  vero  accidit  in  casu  nostro  : 


A  THEORY  OF  NATURAL  PHILOSOPHY  127 

incapacity  for  thinking  &  willing  in  the  very  definition  of  matter  itself  &  corporeal 
substance  ;  &  I  say  that  a  body  is  a  mass  composed  of  points  endowed  with  a  force  of 
inertia  together  with  such  active  forces  as  are  represented  in  Fig.  i,  &  an  incapacity  for 
thinking  &  willing.  If  this  definition  is  taken,  it  is  clear  that  matter  cannot  think ;  & 
this  will  be  a  sort  of  metaphysical  conclusion,  which  will  follow  with  absolute  certainty 
from  the  acceptation  of  the  definition.  Again,  where  physical  arguments  are  alone  employed, 
I  say  that  such  bodies  as  affect  our  senses  are  matter,  because  they  affect  the  senses 
by  means  of  the  forces  under  consideration,  &  do  not  think.  I  also  deduce  the  same 
conclusion  from  the  fact  that  they  afford  no  evidence  of  thought.  This  will  be  a  conclusion 
that  is  solely  physical  with  regard  to  the  existence  of  matter  so  defined  ;  &  it  will  be  just 
as  physically  true  as  the  conclusion  that  says  that  stones  do  not  possess  levity,  deduced  from 
the  fact  that  they  never  display  such  a  thing  by  an  act  of  spontaneous  ascent,  but  on  the 
contrary  always  descend  if  left  to  themselves. 

158.  With  regard  to  the  idea  of  bodies  &  matter,  which  seems  to  involve  continuous  The  senses  are 
extension,  it  seems  to  me  indeed  that  in  this  matter  the  Cartesians  in  particular,  who  have  altogether  at  fault 

.  •  r  i  i_  m  the  greatness  of 

appeared  to  impugn  pre judgments  with  so  much  vigour,  have  given  themselves  up  to  these  the   continuity  of 

prejudgments  more  than  anyone  else.     We  obtain  the  idea  of  bodies  through  the  senses ;  f^^'^beiieve5'' 

and  the  senses  cannot  in  any  way  judge  on  a  matter  of  accurate  continuity  ;  for  very  small 

intervals  do  not  fall  within  the  scope  of  the  senses.     Indeed  we  quite  take  it  for  granted 

that  the  continuity,  which  our  senses  meet  with  in  a  large  number  of  bodies,  does  not  really 

exist.     In  metals,  marble,  glass  &  crystals  there  appears  to  our  senses  to  be  continuity, 

of  such  sort  that  we  do  not  perceive  in  them  any  little  empty  spaces,  or  pores ;  but  in  this 

respect  the  senses  have  manifestly  been  deceived.     This  is  clear,  both  from  their  different 

specific  gravities,  which  certainly  arises  from  the  differences  in  the  numbers  of  the  empty 

spaces ;    &  also  from  the  fact  that  several  substances  will  insinuate  themselves  through 

their  substance.     For  instance,  oil  will  diffuse  itself  through  the  former,  &  light  will  pass 

quite  freely  through  the   latter ;    &   this  indeed  indicates,  especially  in  the  case  of  the 

latter,  an  immense  number  of  pores ;   &  these  are  concealed  from  our  senses. 

159.  Hence  such  evidence  of  our  senses,  or  rather  our  employment  of  such  arguments,  The  origin  of  pre- 
must  now  lie  open  to  suspicion  in  that  class,  in  which  it  is  known  that  we  have  been  deceived,  j^fjdered :  as^o- 
We  may  then  suspect  that  accurate  continuity  without  the  presence  of  any  little  empty  thing,    which    are 
spaces — such  as  is  certainly  absent  from  bodies  of  considerable  size,  although  our  senses  SSe1srases0ar^con- 
seem  to  remark  its  presence — is  also  nowhere  existent  in  any  of  their  smallest  particles ;  cemed ;    examples 
but  that  it  is  merely  an  illusion  of  the  senses,  &  a  sort  of  figment  of  the  brain  through  its  ° 

not  using,  or  through  misusing,  reflection.  For  it  is  a  customary  thing  for  men  (&  a 
thing  that  is  frequently  done)  to  consider  as  absolutely  nothing  something  that  is  nothing 
as  far  as  the  senses  are  concerned ;  &  this  indeed  is  the  source  &  principal  origin  of 
the  greatest  prejudices.  Thus  for  many  centuries  it  was  credited  by  many,  &  still  is 
believed  by  the  unenlightened,  that  the  Earth  is  at  rest,  &  that  the  daily  motions  of  the 
Sun  &  the  fixed  stars  is  proved  by  the  evidence  of  the  senses ;  whilst  among  philosophers 
it  is  now  universally  accepted  that  such  a  question  has  to  be  answered  in  a  far  different 
manner  from  that  by  means  of  the  senses.  Exactly  the  same  impressions  are  bound  to  be 
obtained,  whether  we  &  the  Earth  stand  still  &  the  stars  are  moved,  or  we  &  the 
Earth  are  moved  with  a  common  motion  &  the  stars  are  at  rest.  We  recognize  motion 
by  the  change  of  position,  which  the  image  of  an  object  has  in  the  eye  ;  and  rest  by  the 
permanence  of  that  position.  Now  both  the  change  &  the  permanence  can  come  about 
in  two  ways.  Firstly,  if  we  remain  at  rest,  there  is  a  change  of  position  if  the  object  is 
moved,  &  permanence  if  it  too  is  at  rest ;  secondly,  if  we  move,  there  is  a  change  if  the 
object  is  at  rest,  &  permanence  if  we  &  it  move  with  a  motion  common  to  both.  We 
do  not  feel  ourselves  moving,  unless  we  ourselves  induce  the  motion,  as  when  we  turn  the 
head,  or  when  we  are  jolted  as  we  are  borne  in  a  vehicle.  Hence  we  consider  that  the 
motion  is  nothing,  unless  we  are  made  to  notice  in  other  ways  that  there  is  motion  by  causes 
that  are  known  to  us.  Thus,  when  "  we  leave  the  harbour"  a  passenger  who  has  for  some  time 
been  accustomed  to  the  idea  of  a  shore  remaining  still,  &  of  a  ship  being  propelled  by 
oars  or  sails,  corrects  the  apparent  motion  of  the  shore  ;  &,  as  "  the  land  &  buildings  recede" 
he  attributes  the  motion  to  himself  and  not  to  them. 

160.  Hence,  the  philosopher,  to  avoid  being  led  astray,  must  not  seek  to  obtain  from  ^ctionjrf ^ 
these  primary  ideas  that  we  derive  from  the  senses,  or  deduce  from  them,  consequential  known  that  the 
theorems,    without    careful   investigation;    &  he  must  carefully  study  those  things  that  matter^  ^annot^  be 
he  has  deduced  from  infancy.     If  he  find  that  these  very  perceptions  by  the  senses  can  agreement     with 
come  about  in  two  ways,  one  of  which  is  as  probable  as  the  other  ;   then  he  will  certainly  £hattheis  £JV£ e£ 
commit  an  offence  against  the  laws  of  natural  logic,  if  he  should  proceed  to  choose  one  some  other  way. 
method  in  preference  to  the  other,  solely  for  the  reason  that  previously  he  had  not  seen 

the  one  &  took  no  account  of  it,  &  thus  had  become  accustomed  to  the  other.     Now 


128 


PHILOSOPHIC  NATURALIS  THEORIA 


sensationes  habebuntur  eaedem,  sive  materia  constet  punctis  prorsus  inextensis,  &  distantibus 
inter  se  per  intervalla  minima,  quae  sensum  fugiant,  ac  vires  ad  ilia  intervalla  pertinentes 
organorum  nostrorum  fibras  sine  ulla  sensibili  interruptione  afficiant,  sive  continua  sit,  & 
per  immediatum  contactum  agat.  Patebit  autem  in  tertia  hujusce  operis  parte,  quo  pacto 
proprietates  omnes  sensibiles  corporum  generales,  immo  etiam  ipsorum  prsecipua  discrimina, 
cum  punctis  hisce  indivisibilibus  conveniant,  &  quidem  multo  sane  melius,  quam  in  communi 
sententia  de  continua  extensione  materiae.  Quamobrem  errabit  contra  rectae  ratiocinationis 
usum,  qui  ex  praejudicio  ab  hujusce  conciliationis,  &  alterius  hujusce  sensationum  nostrarum 
causae  ignoratione  inducto,  continuam  extensionem  ut  proprietatem  necessariam  corporum 
omnino  credat,  &  multo  magis,  qui  censeat,  materialis  substantive  ideam  in  ea  ipsa  continua 
extensione  debere  consistere. 


Ordo  idearum,  quas 


esse  per  tactum. 


161.  Verum  quo  magis  evidenter  constet  horum  prsejudiciorum  origo,  afferam  hie 
dissertationis  De  Materia  Divisibilita-\j4\-te,  &  Principiis  Corporum,  numeros  tres  inci- 
piendo  a  14,  ubi  sic  :  "  utcunque  demus,  quod  ego  omnino  non  censeo,  aliquas  esse  innatas 
ideas,  &  non  per  sensus  acquisitas  ;  illud  procul  dubio  arbitror  omnino  certum,  ideam 
corporis,  materiae,  rei  corporeae,  rei  materialis,  nos  hausisse  ex  sensibus.  Porro  ideas  prims 
omnium,  quas  circa  corpora  acquisivimus  per  sensus,  fuerunt  omnino  eae,  quas  in  nobis 
tactus  excitavit,  &  easdem  omnium  frequentissimas  hausimus.  Multa  profecto  in  ipso 
materno  utero  se  tactui  perpetuo  offerebant,  antequam  ullam  fortasse  saporum,  aut  odorum, 
aut  sonorum,  aut  colorum  ideam  habere  possemus  per  alios  sensus,  quarum  ipsarum,  ubi  eas 
primum  habere  ccepimus,  multo  minor  sub  initium  frequentia  fuit.  Idese  autem,  quas  per 
tactum  habuimus,  ortae  sunt  ex  phsenomenis  hujusmodi.  Experiebamur  palpando,  vel 
temere  impingendo  resistentiam  vel  a  nostris,  vel  a  maternis  membris  ortam,  quae  cum 
nullam  interruptionem  per  aliquod  sensibile  intervallum  sensui  objiceret,  obtulit  nobis  ideam 
impenetrabilitatis,  &  extensionis  continuae  :  cumque  deinde  cessaret  in  eadem  directione, 
alicubi  resistentia,  &  secundum  aliam  directionem  exerceretur  ;  terminos  ejusdem  quanti- 
tatis  concepimus,  &  figurse  ideam  hausimus." 


Quae  fuerint  turn 
consideranda  :  in- 
fantia  ad  eas  re- 
flexiones,  inepta  :  in 
quo  ea  sita  sit. 


162.  "  Porro  oriebantur  haec  phsenomena  a  corporibus  e  materia  jam  efformatis,  non  a 
singulis  materiae  particulis,  e  quibus  ipsa  corpora  componebantur.  Considerandum 
diligenter  erat,  num  extensio  ejusmodi  esset  ipsius  corporis,  non  spatii  cujusdam,  per  quod 
particulae  corpus  efformantes  diffunderentur  :  num  ea  particulse  ipsae  iisdem  proprietatibus 
essent  praeditae  :  num  resistentia  exerceretur  in  ipso  contactu,  an  in  minimis  distantiis  sub 
sensus  non  cadentibus  vis  aliqua  impedimento  esset,  quae  id  ageret,  &  resistentia  ante  ipsum 
etiam  contactum  sentiretur  :  num  ejusmodi  proprietates  essent  intrinsecae  ipsi  materiae,  ex 
qua  corpora  componuntur,  &  necessariae  :  an  casu  tantum  aliquo  haberentur,  &  ab  extrinseco 
aliquo  determinante.  Haec,  &  alia  sane  multa  considerate  diligentius  oportuisset  :  sed  erat 
id  quidem  tempus  maxime  caliginosum,  &  obscurum,  ac  reflexionibus  minus  obviis  minime 
aptum.  Praster  organorum  debilitatem,  occupabat  animum  rerum  novitas,  phaenomenorum 
paucitas,  &  nullus,  aut  certe  satis  tenuis  usus  in  phaenomenis  ipsis  inter  se  comparandis,  & 
ad  certas  classes  revocandis,  ex  quibus  in  eorum  leges,  &  causas  liceret  inquirere  &  systema 
quoddam  efformare,  quo  de  rebus  extra  nos  positis  possemus  ferre  judicium.  Nam  in  hac 
ipsa  phaenomenorum  inopia,  in  hac  efformandi  systematis  difficultate,  in  hoc  exiguo 
reflexionum  usu,  magis  etiam,  quam  in  organorum  imbecillitate,  arbitror,  sitam  esse 
infantiam." 


inde  [75]   163.  "In  hac  tanta  rerum  caligine  ea  prima  sese  obtulerunt  animo,  quae^  minus 
orta    extensionis     jta  jndagine,  minus  intentis  reflexionibus  indigebant,  eaque  ipsa  ideistoties  repetitis  altius 

continuae  ut  essen-    .  .      .-1 

tiaiis,  odorum,  &c.,  impressa  sunt,  &  tenacius  adhaeserunt,  &  quendam  veluti  campum  nacta  prorsus  vacuum, 
ut  accidentaiium.     &  acjhuc  immunem,  suo  quodammodo  jure  quandam  veluti  possessionem  inierunt.     Inter- 
valla, quae  sub  sensum  nequaquam  cadebant,  pro  nullis  habita  :    ea,  quorum  ideae^ semper 
simul  conjunctae  excitabantur,  habita  sunt  pro  iisdem,  vel  arctissimo,  &  necessario^  nexu 
inter  se  conjunctis.     Hinc  illud    effectum    est,  ut    ideam    extensionis    continuae,  ideam 


A  THEORY  OF  NATURAL  PHILOSOPHY  129 

that  is  just  what  happens  in  the  case  under  consideration.  The  same  sensations  will  be 
experienced,  whether  matter  consists  of  points  that  are  perfectly  non-extended  &  distant 
from  one  another  by  very  small  intervals  that  escape  the  senses,  &  forces  pertaining  to 
those  intervals  affect  the  nerves  of  our  organs  without  any  sensible  interruption  ;  or 
whether  it  is  continuous  and  acts  by  immediate  contact.  Moreover  it  will  be  clearly  shown, 
in  the  third  part  of  this  work,  how  all  the  general  sensible  properties  of  bodies,  nay  even 
the  principal  distinctions  between  them  as  well,  will  fit  in  with  these  indivisible  points ; 
&  that  too,  in  a  much  better  way  than  is  the  case  with  the  common  idea  of  continuous 
extension  of  matter.  Wherefore  he  will  commit  an  offence  against  the  use  of  true  reasoning, 
who,  from  a  prejudgment  derived  from  this  agreement  &  from  ignorance  of  this  alter- 
native cause  for  our  sensations,  persists  in  believing  that  continuous  extension  is  an 
absolutely  necessary  property  of  bodies  ;  and  much  more  so,  one  who  thinks  that 
the  very  idea  of  material  substance  must  depend  upon  this  very  same  continuous 
extension. 

161.  Now  in  order  that  the  source  of  these  prejudices  may  be  the  more  clearly  known,  Order  of  the  ideas 
I  will  here  quote,  from  the  dissertation  De  Materice  Divisibilitate  &  Princi-pii  Corporum,  ^£  b^ies^tte 
three  articles,  commencing  with  Art.  14,  where  we  have  : — "  Even  if  we  allow  (a  thing  quite  first    ideas    come 
opposed  to  my  way  of  thinking)  that  some  ideas  are  innate  &  are  not  acquired  through  o^ifch  the  SenSe 
the  senses,  there  is  no  doubt  in  my  mind  that  it  is  quite  certain  that  we  derive  the  idea 

of  a  body,  of  matter,  of  a  corporeal  thing,  or  a  material  thing,  through  the  senses.  Further, 
the  very  first  ideas,  of  all  those  which  we  have  acquired  about  bodies  through  the  senses, 
would  be  in  every  circumstance  those  which  have  excited  our  sense  of  touch,  &  these 
also  are  the  ideas  that  we  have  derived  on  more  occasions  than  any  other  ideas.  Many 
things  continually  present  themselves  to  the  sense  of  touch  actually  in  the  very  womb  of 
our  mothers,  before  ever  perchance  we  could  have  any  idea  of  taste,  smell,  sound,  or  colour, 
through  the  other  senses ;  &  of  these  latter,  when  first  we  commenced  to  have  them, 
there  were  to  start  with  far  fewer  occasions  for  experiencing  them.  Moreover  the  ideas 
which  we  have  obtained  through  the  sense  of  touch  have  arisen  from  phenomena  of  the 
following  kind.  We  experienced  a  resistance  on  feeling,  or  on  accidental  contact  with,  an 
object ;  &  this  resistance  arose  from  our  own  limbs,  or  from  those  of  our  mothers.  Now, 
since  this  resistance  offered  no  opposition  through  any  interval  that  was  perceptible  to  the 
senses,  it  gave  us  the  idea  of  impenetrability  &  continuous  extension  ;  &  then  when 
it  ceased  in  the  original  direction  at  any  place  &  was  exerted  in  some  other  direction, 
we  conceived  the  boundaries  of  this  quantity,  &  derived  the  idea  of  figure." 

162.  "  Furthermore,  these  phenomena  will  have  arisen  from  bodies  already  formed  from  Such    things    de- 
matter,  not  from  the  single  particles  of  matter  of  which  the  bodies  themselves  were  composed.  S^time  ^tae  tf 
It  would  have  to  be  considered  carefully  whether  such  extension  was  a  property  of  the  tude  of  inf'ancy^for 
body   itself,  &  not  of  some  space  through  which  the  particles   forming  the  body  were  su.fh.  reflection ;  on 

j-rc        JIT  -11  i    r     i        •  i        i  •          wnat  they  maY  be 

diffused  ;  whether  the  particles  themselves  were  endowed  with  the  same  properties ;  founded, 
whether  the  resistance  was  exerted  only  on  actual  contact,  or  whether,  at  very  small 
distances  such  as  did  not  fall  within  the  scope  of  the  senses,  some  force  would  act  as  a 
hindrance  &  produce  the  same  effect,  and  resistance  would  be  felt  even  before  actual 
contact ;  whether  properties  of  this  kind  would  be  intrinsic  in  the  matter  of  which  the 
bodies  are  composed,  &  necessary  to  its  existence ;  or  only  possessed  in  certain  cases, 
being  due  to  some  external  influence.  These,  &  very  many  other  things,  should  have 
been  investigated  most  carefully ;  but  the  period  was  indeed  veiled  in  mist  &  obscurity 
to  a  great  degree,  &  very  little  fitted  for  aught  but  the  most  easy  thought.  In  addition 
to  the  weakness  of  the  organs,  the  mind  was  occupied  with  the  novelty  of  things  &  the 
rareness  of  the  phenomena  ;  &  there  was  no,  or  certainly  very  little,  use  made  of  comparisons 
of  these  phenomena  with  one  another,  to  reduce  them  to  definite  classes,  from  which  it 
would  be  permissible  to  investigate  their  laws  &  causes  &  thus  form  some  sort  of  system, 
through  which  we  could  bring  the  judgment  to  bear  on  matters  situated  outside  our  own 
selves.  Now,  in  this  very  paucity  of  phenomena,  in  this  difficulty  in  the  matter  of  forming 
a  system,  in  this  slight  use  of  the  powers  of  reflection,  to  a  greater  extent  even  than  in  the 
lack  of  development  of  the  organs,  I  consider  that  infancy  consists." 

163.  "  In  this  dense  haze  of  things,  the  first  that  impressed  themselves  on  the  mind  Thence      Pr«J'u<^- 

i  1*1  •        -i  i  11  i  •  •  ••  n        i  •  mcnis>    di  c    uci  i  vcu. 

were  those  which  required  a  less  deep  study  &  less  intent  investigation ;  &  these,  since  that  continuity  of 
the  ideas  were  the  more  often  renewed,  made   the   greater   impression  &  became   fixed  J^^J1^  «S 
the  more  firmly  in  the  mind,  &  as  it  were  took  possession  of,  so  to  speak,  a  land  that  they  continuity  of  odours 
found  quite  empty  &  hitherto  immune,  by  a  sort  of  right  of  discovery.     Intervals,  which  &c-  *  accidental. 
in  no  wise  came  within  the  scope  of  the  senses,  were  considered  to  be  nothing  ;  those  things, 
the  ideas  of  which  were  always   excited   simultaneously  &  conjointly,   were   considered 
as  identical,  or  bound  up  with  one  another  by  an  extremely   close  &  necessary  bond. 
Hence  the  result  is  that  we  have  formed  the  idea  of  continuous  extension,  the  idea  of 


130  PHILOSOPHISE  NATURALIS  THEORIA 

impenetrabilitatis  prohibentis  ulteriorem  motum  in  ipso  tantum  contactu  corporibus 
affinxerimus,  &  ad  omnia,  quae  ad  corpus  pertinent,  ac  ad  materiam,  ex  qua  ipsum  constat, 
temere  transtulerimus  :  quse  ipsa  cum  primum  insedissent  animo,  cum  frequcntissimis,  immo 
perpetuis  phaenomenis,  &  experimentis  confirmarentur ;  ita  tenaciter  sibi  invicem 
adhseserunt,  ita  firmiter  ideae  corporum  immixta  sunt,  &  cum  ea  copulata  ;  ut  ea  ipsa  pro 
primis  corporibus,  &  omnium  corporearum  rerum,  nimirum  etiam  materiae  corpora  compo- 
nentis,  ejusque  partium  proprietatibus  maxime  intrinsecis,  &  ad  naturam,  atque  essentiam 
earundem  pertinentibus,  &  turn  habuerimus,  &  nunc  etiam  habeamus,  nisi  nos  praejudiciis 
ejusmodi  liberemus.  Extensionem  nimirum  continuam,  impenetrabilitatem  ex  contactu, 
compositionem  ex  partibus,  &  figuram,  non  solum  naturae  corporum,  sed  etiam  corporeae 
materiae,  &  singulis  ejusdem  partibus,  tribuimus  tanquam  proprietates  essentiales  :  csetera, 
quae  serius,  &  post  aliquem  reflectendi  usum  deprehendimus,  colorem,  saporem,  odorem 
sonum,  tanquam  accidentales  quasdam,  &  adventitias  proprietates  consideravimus." 

propositiones  164.  Ita  ego  ibi,  ubi  Theoriam  virium  deinde  refero,  quam  supra  hie  exposui,  ac  ad 

Theoriam°continen?  Pr3ecipuas  corporum  proprietates  applico,  quas  ex  ilia  deduco,  quod  hie  praestabo  in  parte 
tis.  tertia.     Ibi  autem  ea  adduxeram  ad  probandam  primam  e  sequentibus  propositionibus, 

quibus  probatis  &  evincitur  Theoria  mea,  &  vindicatur  :  sunt  autem  hujusmodi  :  i.  Nullo 
prorsus  argumento  evincitur  materiam  habere  extensionem  continuam,  W  non  potius  constare  e 
punctis  prorsus  indivisibilibus  a  se  per  aliquod  intervallum  distantibus ;  nee  ulla  ratio  seclusis 
pr&judiciis  suadet  extensionem  ipsam  continuam  potius,  quam  compositionem  e  punctis  prorsus 
indivisibilibus,  inextensis,  y  nullum  continuum  extensum  constituentibus.  2.  Sunt  argumenta, 
y  satis  valida  ilia  quidem,  qua  hanc  compositionem  e  punctis  indivisibilibus  evincant  extensioni 
ipsi  continues  pr&ferri  oportere. 

Quo    pacto    con-  165.  At  quodnam  extensionis  genus  erit  istud,  quod  e  punctis  inextensis,  &  spatio 

coaiescan^lnmassas  imaginario,  sive  puro  nihilo  [76]  constat  ?  Quo  pacto  Geometria  locum  habere  poterit, 
tenaces:  transitus  ubi  nihil  habetur  reale  continue  extensum?  An  non  punctorum  ejusmodi  in  vacuo 
damPartem  secun"  innatantium  congeries  erit,  ut  quaedam  nebula  unico  oris  flatu  dissolubilis  prorsus  sine  ulla 
consistent!  figura,  solidate,  resistentia  ?  Haec  quidem  pertinent  ad  illud  extensionis  ,& 
cohaesionis  genus,  de  quo  agam  in  tertia  parte,  in  qua  Theoriam  applicabo  ad  Physicam,  ubi 
istis  ipsis  difficultatibus  faciam  satis.  Interea  hie  illud  tantummodo  innuo  in  antecessum,  me 
cohaesionem  desumere  a  limitibus  illis,  in  quibus  curva  virium  ita  secat  axem,  ut  a  repulsione 
in  minoribus  distantiis  transitus  fiat  ad  attractionem  in  majoribus.  Si  enim  duo  puncta 
sint  in  distantia  alicujus  limitis  ejus  generis,  &  vires,  quae  immutatis  distantiis  oriuntur,  sint 
satis  magnae,  curva  secante  axem  ad  angulum  fere  rectum,  &  longissime  abeunte  ab  ipso  ; 
ejusmodi  distantiam  ea  puncta  tuebuntur  vi  maxima  ita,  ut  etiam  insensibiliter  compressa 
resistant  ulteriori  compressioni,  ac  distracta  resistant  ulteriori  distractioni ;  quo  pacto  si 
multa  etiam  puncta  cohaereant  inter  se,  tuebuntur  utique  positionem  suam,  &  massam 
constituent  formae  tenacissimam,  ac  eadem  prorsus  phsenomena  exhibentem,  quae  exhiberent 
solidae  massulae  in  communi  sententia.  Sed  de  hac  re  uberius,  uti  monui,  in  parte  tertia  : 
nunc  autem  ad  secundam  faciendus  est  gradus. 


A  THEORY  OF  NATURAL  PHILOSOPHY  131 

impenetrability  preventing  further  motion  only  on  the  absolute  contact  of  bodies ;  & 
then  we  have  heedlessly  transferred  these  ideas  to  all  things  that  pertain  to  a  solid  body, 
and  to  the  matter  from  which  it  is  formed.  Further,  these  ideas,  from  the  time  when  they 
first  entered  the  mind,  would  be  confirmed  by  very  frequent,  not  to  say  continual,  phenomena 
&  experiences.  So  firmly  are  they  mutually  bound  up  with  one  another,  so  closely  are 
they  intermingled  with  the  idea  of  solid  bodies  &  coupled  with  it,  that  we  at  the  time 
considered  these  two  things  as  being  just  the  same  as  primary  bodies,  &  as  peculiarly 
intrinsic  properties  of  all  corporeal  things,  nay  further,  of  the  very  matter  from  which 
bodies  are  composed,  &  of  its  parts ;  indeed  we  shall  still  thus  consider  them,  unless  we 
free  ourselves  from  prejudgments  of  this  nature.  To  sum  up,  we  have  attributed  continuous 
extension,  impenetrability  due  to  actual  contact,  composition  by  parts,  &  shape,  as  if 
they  were  essential  properties,  not  only  to  the  nature  of  bodies,  but  also  to  corporeal  matter 
&  every  separate  part  of  it ;  whilst  others,  which  we  comprehend  more  deeply  &  as 
a  consequence  of  some  considerable  use  of  thought,  such  as  colour,  taste,  smell  &  sound, 
we  have  considered  as  accidental  or  adventitious  properties." 

164.  Such  are  the  words  I  used  ;  &  then  I  stated  the  Theory  of  forces  which  I  have  A  pair  of  proposi- 
expounded  in  the  previous  articles  of  this  work,  and  I  applied  the  theory  to  the  principal  tation0f  containing 
properties  of  bodies,  deducing  them  from  it ;    &  this  I  will  set  forth  in  the  third  part  the  whole  of  ™y 
of  the  present  work.     In  the  dissertation  I  had  brought  forward  the  arguments  quoted  *" 

in  order  to  demonstrate  the  truth  of  the  first  of  the  following  theorems.  If  these  theorems 
are  established,  then  my  Theory  is  proved  &  verified;  they  are  as  follows  : —  i.  There  is 
absolutely  no  argument  that  can  be  brought  forward  to  prove  that  matter  has  continuous  extension, 
y  that  it  is  not  rather  made  up  of  perfectly  indivisible  points  separated  from  one  another  by 
a  definite  interval ;  nor  is  there  any  reason  apart  from  prejudice  in  favour  of  continuous  extension 
in  preference  to  composition  from  points  that  are  perfectly  indivisible,  non-extended,  &  forming 
no  extended  continuum  of  any  sort.  2.  There  are  arguments,  W  fairly  strong  ones  too,  which 
will  prove  that  this  composition  from  indivisible  points  is  preferable  to  continuous  extension. 

165.  Now  what  kind  of  extension  can  that  be  which  is  formed  out  of  non-extended  The  manner   in 

o     •  •  t  i  •          5       TT  /-i  1111    which     groups      of 

points  &  imaginary   space,  i.e.,  out  of  pure  nothing  ?     How  can  Geometry  be   upheld  points  coalesce  into 
if  no  thing  is  considered  to  be  actually  continuously  extended  ?     Will  not  groups  of  points,  tenacious   masses : 

n        •         •  t     i  •  11-1  i        i      T       i    •  •      i      i  i       n     &  then  we  pass  on 

floating  in  an  empty  space  of  this  sort  be  like  a  cloud,  dissolving  at  a  single  breath,  &  to  the  second  part. 

absolutely  without  a  consistent  figure,  or  solidity,  or  resistance  ?     These  matters  pertain 

to  that  kind  of  extension  &  cohesion,  which  I  will  discuss  in  the  third  part,  where  I  apply 

my  Theory  to  physics  &  deal  fully  with  these  very  difficulties.     Meanwhile  I  will  here 

merely  remark  in  anticipation  that  I  derive  cohesion  from  those  limit-points,  in  which  the 

curve  of  forces  cuts  the  axis,  in  such  a  way  that  a  transition  is  made  from  repulsion  at  smaller 

distances  to  attraction  at  greater  distances.     For  if  two  points  are  at  the  distance  that 

corresponds  to  that  of  any  of  the  limit-points  of  this  kind,  &  the  forces  that  arise  when 

the  distances  are  changed  are  great  enough  (the  curve  cutting  the  axis  almost  at  right  angles 

&  passing  to  a  considerable  distance  from  it),  then  the  points  will  maintain  this  distance 

apart  with  a  very  great  force  ;  so  that  when  they  are  insensibly  compressed  they  will  resist 

further   compression,  &  when  pulled  apart  they  resist  further  separation.     In  this  way 

also,  if  a  large  number  of  points  cohere  together,  they  will  in  every  case  maintain  their 

several  positions,  &  thus  form  a  mass  that  is  most  tenacious  as  regards  its  form  ;  &  this 

mass  will  exhibit  exactly  the  same  phenomena  as  little  solid  masses,  as  commonly  understood, 

exhibit.     But  I  will  discuss  this  more  fully,  as  I  have  remarked,  in  the  third  part ;  for  now 

we  must  pass  on  to  the  second  part. 


[77]  PARS  II 
Theories  *Applicato  ad  Mechanicam 

Ante  appHcatipnem  166.  Considerabo  in  hac  secunda  parte  potissimum  generates  quasdam  leges  aequilibrii 
consideratio'curvs!  &  motus  tam  punctorum,  quam  massarum,  quae  ad  Mechanicam  utique  pertinent,  &  ad 
plurima  ex  iis,  quae  in  elementis  Mechanics  passim  traduntur,  ex  unico  principio,  &  adhibito 
constant!  ubique  agendi  modo,  demonstranda  viam  sternunt  pronissimam.  Sed  prius 
praemittam  nonnulla  quae  pertinent  ad  ipsam  virium  curvam,  a  qua  utique  motuum, 
phaenomena  pendent  omnia. 

Quid  in  ea   con-          167.  In  ea  curva  consideranda  sunt  potissimum  tria,  arcus  curvae,  area  comprehensa 
siderandum.  inter  axemj  &  arcum,  quam  general   ordinata  continue  fluxu,  ac  puncta  ilia,  in  quibus 

curva  secat  axem. 

Diversa  arcnum          1 68.  Quod  ad  arcus  pcrtinet,  alii  dici  possunt  repulsivi,  &  alii  attractivi,  prout  nimirum 

asymptotic!  "tiam  Jacent  ac*  partes  cruris  asymptotici  ED,  vel  ad  contrarias,  ac  terminant  ordinatas  exhibentes 

numero  infiniti.       vires  repulsivas,  vel  attractivas.     Primus  arcus  ED  debet  omnino   esse    asymptoticus  ex 

parte  repulsiva,  &  in  infinitum    productus :    ultimus  TV,  si  gravitas  cum   lege    virium 

reciproca  duplicata  distantiarum  protenditur  in  infinitum,  debet  itidem  esse  asymptoticus 

ex  parte  attractiva,  &  itidem  natura  sua  in  infinitum  productus.     Reliquos  figura  I  exprimit 

omnes  finitos.    Verum  curva  Geometrica  etiam  ejus  naturae,  quam  exposuimus,  posset  habere 

alia  itidem  asymptotica  crura,  quot  libuerit,  ut  si  ordinata  mn  in  H  abeat  in  infinitum. 

Sunt  nimirum  curvae  continuae,  &  uniformis  naturae,  quae  asymptotes  habent  plurimas, 

&  habere  possunt  etiam  numero  infinitas.  (') 

Arcus  intermedii.  [78]  169.  Arcus  intermedii,  qui  se  contorquent  circa  axem,  possunt  etiam  alicubi, 
ubi  ad  ipsum  devenerint,  retro  redire,  tangendo  ipsum,  atque  id  ex  utralibet  parte,  & 
possent  itidem  ante  ipsum  contactum  inflecti,  &  redire  retro,  mutando  accessum  in  recessum, 
ut  in  fig.  i.  videre  est  in  arcu  P^R. 

Arcus  prostremus  170.  Si  gravitas  gencralis  legem  vis  proportionalis  inverse  quadrate  distantiae,  quam 
36  non  accurate  servat,  sed  quamproxime,  uti  diximus  in  priore  parte,  retinet  ad  sensum  non 
mutatam  solum  per  totum  planetarium,  &  cometarium  systema,  fieri  utique  poterit,  ut 
curva  virium  non  habeat  illud  postremum  crus  asymptoticum  TV,  habens  pro  asymptoto 
ipsam  rectam  AC,  sed  iterum  secet  axem,  &  se  contorqueat  circa  ipsum.(*)  Turn  vero  inter 

(i)  S»*  ex.  gr.  in  fig.  12.  cyclois  continua  CDEFGH  (3e.,  quam  generet  punctum  peripheries  circuli  continue  revoluti 
supra  rectam  AB,  qute  natura  sua  protenditur  utrinque  in  infinitum,  adeoque  in  infinitis  punctis  C,  E,  G,  I,  &c.  occurrit 
basi  AB.  Si  ubicunque  ducatur  qutevis  ordinata  PQ,  productaturque  in  R  ita,  ut  sit  PR  tertia  post  PQ,  y  datam  quampiam 
rectam  ;  punctum  R  frit  ad  curvam  continuum  constantem  totidem  ramis  MNO,  VXY,  yr.,  quot  erunt  arcus  Cycloidales 
CDE,  EFG,  i3c,,  quorum  ramorum  singuli  habebunt  bina  crura  asymptotica,  cum  ordinata  PQ  in  accessu  ad  omnia  puncta, 
C,  E,  G,  &c.  decrescat  ultra  quoscunque  Unites,  adeoque  ordinata  PR  crescat  ultra  limites  quoscunque.  Erunt  hie  quidem 
omnes  asymptoti  CK,  EL,  GS  &c.  parallels  inter  se,  &  perpendiculares  basi  AB,  quod  in  aliis  curvis  non  est  necessarium, 
cum  etiam  divergentes  utcunque  possint  esse.  Erunt  autem  y  totidem  numero,  quot  puncta.  ilia  C,  E,  G  &c.,  nimirum 
infinite.  Eodem  autem  pacto  curvarum  quarumlibet  singuli  occursus  cum  axe  in  curvis  per  eas  hac  eadem  lege  genitis 
bina  crura  asymptotica  generant,  cruribus  ipsis  jacentibus,  vel,  ut  hie,  ad  eandem  axis  partem,  ubi  curva  genetrix  ab  eo 
regreditur  retro  post  appulsum,  vel  etiam  ad  partes  oppositas,  ubi  curva  genetrix  ipsum  secet,  ac  transiliat :  cumque  possit 
eadem  curva  altiorum  generum  secari  in  punctis  plurimis  a  recta,  vel  contingi  ;  poterunt  utique  haberi  y  rami  asymptotici 
in  curva  eadem  continua,  quo  libuerit  data  numero. 

(k)Nam  ex  ipsa  Geometrica  continuitate,  quam  persecutus  sum  in  dissertatione  De  Lege  Continuitatis,  y  in  dissertatione 
De  Transformatione  Locorum  Geometricorum  adjecta  Sectionibus  Conicis,  exhibui  necessitatem  generalem  secundi 
illius  cruris  asymptotici  redeuntis  ex  infinite.  Quotiescunque  enim  curva  aliqua  saltern  algebraica  habet  asymptoticum 
crus  aliquod,  debet  necessario  habere  y  alterum  ipsi  respondens,  y  habens  pro  asymptoto  eandem  rectam  :  sed  id  habere 

132 


A  THEORY  OF  NATURAL  PHILOSOPHY 


133 


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'34 


PHILOSOPHIC  NATURALIS  THEORIA 


PART  II 
^Application   of  the    Theory   to    Mechanics 

1 66.  I    will   consider  in   this   second   part   more  especially  certain  general  laws  of  Consideration     of 
equilibrium,  &  motions  both  of  points  &  masses ;   these  certainly  belong  to  the  science  of  proceeding    w^t'h 
Mechanics,  &  they  smooth  the  path  that  is  most  favourable  for  proving  very  many  of  those  tne  application  to 
theorems,  that  are  everywhere  expounded  in  the  elements  of  Mechanics,  from  a  single 

principle,  &  in  every  case  by  the  constant  employment  of  a  single  method  of  dealing  with 
them.  But,  before  I  do  that,  I  will  call  attention  to  a  few  points  that  pertain  to  the  curve 
of  forces  itself,  upon  which  indeed  all  the  phenomena  of  motions  depend. 

167.  With  regard  to  the  curve,  there  are  three  points  that  are  especially  to  be  considered  ;  The  points  we  have 
namely,  the  arcs  of  the  curve,  the  area  included  between  the  axis  &  the  curve  swept  out  regard'tolt.1 

by  the  ordinate  by  its  continuous  motion,  &  those  points  in  which  the  curve  cuts  the  axis. 

1 68.  As  regards  the  arcs,  some  may  be  called  repulsive,  &  others  attractive,  according  The  different  kinds 
indeed  as  they  lie  on  the  same  side  of  the  axis  as  the  asymptotic  branch  ED  or  on  the  opposite  totkfarc's  may  even 
side,  &  terminate  ordinates  that  represent  repulsive  or  attractive  forces.     The  first  arc  be  infinite  in  num- 
ED  must  certainly  be  asymptotic  on  the  repulsive  side  of  the  axis,  &  continued  indefinitely.      r' 

The  last  arc  TV,  if  gravity  extends  to  indefinite  distances  according  to  a  law  of  forces  in 
the  inverse  ratio  of  the  squares  of  the  distances,  must  also  be  asymptotic  on  the  attractive 
side  of  the  axis,  &  by  its  nature  also  continued  indefinitely.  All  the  remaining  arcs  are 
represented  in  Fig.  I  as  finite.  But  a  geometrical  curve,  of  the  kind  that  we  have  expounded, 
may  also  have  other  asymptotic  branches,  as  many  in  number  as  one  can  wish  ;  for  instance, 
suppose  the  ordinate  mn  at  H  to  go  away  to  infinity.  There  are  indeed  curves,  that  are 
continuous  &  uniform,  which  have  very  many  asymptotes,  &  such  curves  may  even 
have  an  infinite  number  of  asymptotes. («') 

169.  The  intermediate  arcs,  which  wind  about  the  axis,  can  also,  at  any  point  where  intermediate  arcs, 
they  reach  it,  return  backwards  &  touch  it ;  and  they  can  do  this  on  either  side  of  it ;  they 

may  also  be  reflected  and  recede  before  actual  contact,  the  approach  being  altered  into  a 
recession,  as  is  to  be  seen  in  Fig.  i  with  regard  to  the  arc  P^/yR. 

170.  If  universal  gravity  obeys  the  law  of  a  force  inversely  proportional  to  the  square  of  The   ultimate   arc 

the  distance  (which,  as  I  remarked  in  the  first  part,  it  only  obeys  as  nearly  as  possible,  but  [ tPyeSposs 

not  exactly),  sensibly  unchanged  only  throughout  the  planetary  &  cometary  system,  it  will  asymptotic, 
certainly  be  the  case  that  the  curve  of  forces  will  not  have  the  last  arm  PV  asymptotic  with 
the  straight  line  AC  as  the  asymptote,  but  will  again  cut  the  axis  &  wind  about  it.  (*)     Then 

(i)  Let,  for  example,  in  Fig.  12,  CDEFGH  &c.  be  a.  continuous  cycloid,  generated  by  a  point  on  the  circumference 
of  a  circle  rolling  continuously  along  the  straight  line  AB  ;  this  by  its  nature  extends  on  either  side  to  infinity,  W  thus 
meets  the  base  AB  in  an  infinite  number  of  points  such  as  C,  E,  G,  I,  &c.  //  at  every  point  there  is  drawn  an  ordinate 
such  as  PQ,  and  this  is  produced  to  R,  so  that  PR  is  a  third  proportional  to  PQ  W  some  given  straight  line  ;  then  the  point 
R  will  trace  out  a  continuous  curve  consisting  of  as  many  branches,  MNO,  VXY,  &c.,  as  there  are  cycloidal  arcs,  CDE, 
EFG,  &c. ;  each  of  these  branches  will  have  a  pair  of  asymptotic  arms,  since  the  ordinate  PQ  on  approaching  any 
one  of  the  points  C,E,G,  &c.,  will  decrease  beyond  all  limits,  (3  thus  the  ordinate  PR  will  increase  beyond  all  limits. 
In  this  curve  then  there  will  be  CK,  EL,  GS,  &c.,  all  asymptotes  parallel  to  one  another  &  perpendicular  to  the  base 
AB  ;  this  is  not  necessarily  the  case  in  other  curves,  since  they  may  be  also  inclined  to  one  another  in  any  manner. 
Further  they  will  be  as  many  in  number  as  there  are  points  such  as  C,  E,  G,  &c.,  that  is  to  say,  infinite.  Again,  in 
a  similar  way,  the  several  intersections  of  any  curves  you  please  with  the  axis  give  rise  to  a  pair  of  asymptotic  arms 
in  curves  derived  from  them  according  to  the  same  law  ;  and  these  arms  lie,  either  on  the  same  side  of  the  axis,  as 
in  this  case,  where  the  original  curve  leaves  the  axis  once  more  after  approaching  it,  or  indeed  on  opposite  sides  of  the 
axis,  where  the  original  curve  cuts  W  crosses  it.  Also,  since  it  is  possible  for  the  same  curve  of  higher  orders  to  be 
cut  in  a  large  number  of  points,  or  to  be  touched,  there  will  possibly  be  also  asymptotic  arms  in  this  same  continuous 
curve  equal  to  any  given  number  you  please. 

(k)  For,  from  the  principle  of  geometrical  continuity  itself,  which  I  discussed  in  my  dissertation  De  Lege  Continuitatis 
and  in  the  dissertation  De  Transformatione  Locorum  Geometricorum  appended  to  my  Sectionum  Conicarum 
Elementa,  /  showed  the  necessity  for  the  second  asymptotic  arm  returning  from  infinity.  For  as  often  as  an  algebraical 
curve  has  at  least  one  asymptotic  arm,  it  must  also  have  another  that  corresponds  to  it  y  has  the  same  straight  line 

135 


136  PHILOSOPHIC  NATURALIS  THEORIA 

alios  casus  innumeros,  qui  haberi  possent,  unum  censeo  speciminis  gratia  hie  non  omitten- 
dum  ;  incredibile  enim  est,  quam  ferax  casuum,  quorum  singuli  sunt  notatu  dignissimi, 
unica  etiam  hujusmodi  curva  esse  possit. 

shnufum  curTserte  I7I>  Si  in  %•  H  *n  axe  C'C  sint  segmenta  AA',  A'A"  numero  quocunque,  quorum 

Mundoru'm  mag-  posteriora  sint  in  immensum  majora  respectu  praecedentium,  &  per  singula  transeant, 
donaikfm  propor"  asympto-[79]-ti  AB,  A'B',  A"B"  perpendiculares  axi ;  possent  inter  binas  quasque  asymptotes 
esse  curvae  ejus  formae,  quam  in  fig.  I  habuimus,  &  quae  exhibetur  hie  in  DEFI  &c.,  D'E'F'F, 
&c.,  in  quibus  primum  crus  ED  esset  asymptoticum  repulsivum,  postremum  SV  attractivum, 
in  singulis  vero  intervallum  EN,  quo  arcus  curvae  contorquetur,  sit  perquam  exiguum 
respectu  intervalli  circa  S,  ubi  arcus  diutissime  perstet  proximus  hyperbolae  habenti 
ordinatas  in  ratione  reciproca  duplicata  distantiarum,  turn  vero  vel  immediate  abiret 
in  arcum  asymptoticum  attractivum,  vel  iterum  contorqueretur  utcunque  usque  ad 
ejusmodi  asymptoticum  attractivum  arcum,  habente  utroque  asymptotico  arcu  aream 
infinitam ;  in  eo  casu  collocate  quocunque  punctorum  numero  inter  binas  quascunque 
asymptotes,  vel  inter  binaria  quotlibet,  &  rite  ordinato,  posset  exurgere  quivis,  ut  ita 
dicam,  Mundorum  numerus,  quorum  singuli  essent  inter  se  simillimi,  vel  dissimillimi, 
prout  arcus  EF&cN,  E'F'&cN'  essent  inter  se  similes,  vel  dissimiles,  atque  id  ita,  ut  quivis 
ex  iis  nullum  haberet  commercium  cum  quovis  alio  ;  cum  nimirum  nullum  punctum 
posset  egredi  ex  spatio  incluso  iis  binis  arcubus,  hinc  repulsive,  &  inde  attractive ;  &  ut 
omnes  Mundi  minorum  dimensionum  simul  sumpti  vices  agerent  unius  puncti  respectu 
proxime  majoris,  qui  constaret  ex  ejusmodi  massulis  respectu  sui  tanquam  punctualibus, 
dimensione  nimirum  omni  singulorum,  respectu  ipsius,  &  respectu  distantiarum,  ad  quas 
in  illo  devenire  possint,  fere  nulla  ;  unde  &  illud  consequi  posset,  ut  quivis  ex  ejusmodi 
tanquam  Mundis  nihil  ad  sensum  perturbaretur  a  motibus,  &  viribus  Mundi  illius  majoris, 
sed  dato  quovis  utcunque  magno  tempore  totus  Mundus  inferior  vires  sentiret  a  quovis 
puncto  materiae  extra  ipsum  posito  accedentes,  quantum  libuerit,  ad  aequales,  &  parallelas 
quae  idcirco  nihil  turbarent  respectivum  ipsius  statum  internum. 


Omissis  subiimiori-  172.  Sed  ea  jam  pertinent  ad  applicationem  ad  Physicam,  quae  quidem  hie  innui 

areas pr0greSSUS  ad  tantumm°do,  ut  pateret,  quam  multa  notatu  dignissima  considerari  ibi  possent,  &  quanta 
sit  hujusce  campi  fcecunditas,  in  quo  combinationes  possibiles,  &  possibiles  formae  sunt 
sane  infinities  infinitae,  quarum,  quae  ab  humana  mente  perspici  utcunque  possunt,  ita 
sunt  paucae  respectu  totius,  ut  haberi  possint  pro  mero  nihilo,  quas  tamen  omnes  unico 
intuitu  prsesentes  vidit,  qui  Mundum  condidit,  DEUS.  Nos  in  iis,  quae  consequentur, 
simpliciora  tantummodo  qusedam  plerumque  consectabimur,  quae  nos  ducant  ad  phaeno- 
mena  iis  conformia,  quae  in  Natura  nobis  pervia  intuemur,  &  interea  progrediemur  ad 
areas  arcubus  respondentes. 

Cuicunque  axis  173.  Aream  curvae  propositae  cuicunque,  utcunque  exiguo,  axis  segmento  respondentem 

aream  e  "respondere  Posse  esse  utcunque  magnam,  &  aream  respondentem  cuicunque,  utcunque  magno,  [80] 

utcunque  magnam  posse  esse  utcunque  parvam,  facile  patet.     Sit  in  fig.  15,  MQ  segmentum  axis  utcunque 

secundjT"1  de^non-  parvum,  vel  magnum  ;    ac  detur  area  utcunque  magna,  vel  parva.     Ea  applicata  ad  MQ 

stratio.  exhibebit  quandam  altitudinem  MN  ita,  ut,  ducta  NR  parallela  MQ,  sit  MNRQ  aequalis 

areae  datae,  adeoque  assumpta  QS  dupla  QR,  area  trianguli  MSQ  erit  itidem  aequalis  areae 

datae.     Jam  vero  pro  secundo  casu  satis  patet,  posse  curvam  transire  infra  rectam  NR, 

uti  transit  XZ,  cujus  area  idcirco  esset  minor,  quam  area  MNRQ ;   nam  esset  ejus  pars. 

potest  vel  ex  eadem  parte,  vel  ex  opposita  ;  W  crus  ipsum  jacere  potest  vel  ad  easdem  plagas  partis  utriuslibet  cum  priore 
crure,  vel  ad  oppositas,  adeoque  cruris  redeuntis  ex  infinite  poshiones  quatuor  esse  possunt.  Si  in  fig.  13  crus  ED  abeat 
in  infinitum,  existente  asymptoto  ACA',  potest  regredi  ex  parte  A  vel  ut  HI,  quod  crus  facet  ad  eandem  plagam,  velut 
KL,  quod,  facet  ad  oppositam  ;  y  ex  parte  A',  vel  ut  MN,  ex  eadem  plaga,  vel  ut  OP,  ex  opposita.  In  posteriore  ex 
iis  duabus  dissertationibus  profero  exempla  omnium  ejusmodi  regressuum  ;  ac  secundi,  ($  quarti  casus  exempla  exhibet 
etiam  superior  genesis,  si  curva  generans  contingat  axem,  vel  secet,  ulterius  progressa  respectu  ipsius.  Inde  autem  fit,  ut 
crura  asymptotica  rectilineam  babentia  asymptotum  esse  non  possint,  nisi  numero  part,  ut  &  radices  imaginarite  in 
eequationibus  algebraicis. 

Verum  hie  in  curva  virium,  in  qua  arcus  semper  debet  progredi,  ut  singulis  distantiis,  sive  abscissis,  singula  vires, 
sive  ordinatts  respondeant,  casus  primus,  &  tertius  haberi  non  possunt.  Nam  ordinata  RQ  cruris  DE  occurreret  alicubi 
in  S,  S'  cruribus  etiam  HI,  MN  ,•  adeoque  relinquentur  soli  quartus,  &  secundus,  quorum  usus  erit  infra. 


A  THEORY  OF  NATURAL  PHILOSOPHY 


137 


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PHILOSOPHIC  NATURALIS  THEORIA 


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A  THEORY   OF   NATURAL    PHILOSOPHY  139 

there  is  one,  out  of  an  innumerable  number  of  other  cases  that  may  possibly  happen,  which 
I  think  for  the  sake  of  an  example  should  not  be  omitted  here  ;  for  it  is  incredible  how 
prolific  in  cases,  each  of  which  is  well  worth  mentioning,  a  single  curve  of  this  kind  can  be. 

171.  If,  in  Fig.  14,  there  are  any  number  of  segments  AA',  A'A",  of  which  each  that  A  series  of  similar 
follows  is  immensely  great  with  regard  to  the  one  that  precedes  it ;  &  if  through  each  c"rve.s- wlth  a  s^ies 
point  there  passes  an  asymptote,  such  as  AB,  A'B',  A"B",  perpendicular  to  the  axis  ;  then  tionai  in  magnitude, 
between  any  two  of  these  asymptotes  there  may  be  curves  of  the  form  given  in  Fig.  i. 
These  are  represented  in  Fig.  14  by  DEFI  &c.,  D'E'F'I'  &c. ;  &  in  these  the  first  arm  E 
would  be  asymptotic  &  repulsive,  &  the  last  SV  attractive.  In  each  the  interval  EN, 
where  the  arc  of  the  curve  is  winding,  is  exceedingly  small  compared  with  the  interval 
near  S,  where  the  arc  for  a  very  long  time  continues  closely  approximating  to  the  form 
of  the  hyperbola  having  its  ordinates  in  the  inverse  ratio  of  the  squares  of  the  distances  ; 
&  then,  either  goes  off  straightway  into  an  asymptotic  &  attractive  arm,  or  once  more 
winds  about  the  axis  until  it  becomes  an  asymptotic  attractive  arc  of  this  kind,  the  area 
corresponding  to  either  asymptotic  arc  being  infinite.  In  such  a  case,  if  a  number  of  points 
are  assembled  between  any  pair  of  asymptotes,  or  between  any  number  of  pairs  you  please, 
£  correctly  arranged,  there  can,  so  to  speak,  arise  from  them  any  number  of  universes, 
each  of  them  being  similar  to  the  other,  or  dissimilar,  according  as  the  arcs  EF  .  .  .  .  N, 
E'F'  ....  N'  are  similar  to  one  another,  or  dissimilar  ;  &  this  too  in  such  a  way  that 
no  one  of  them  has  any  communication  with  any  other,  since  indeed  no  point  can  possibly 
move  out  of  the  space  included  between  these  two  arcs,  one  repulsive  &  the  other 
attractive  ;  &  such  that  all  the  universes  of  smaller  dimensions  taken  together  would 
act  merely  as  a  single  point  compared  with  the  next  greater  universe,  which  would 
consist  of  little  point-masses,  so  to  speak,  of  the  same  kind  compared  with  itself,  that  is 
to  say,  every  dimension  of  each  of  them,  compared  with  that  universe  &  with  respect  to 
the  distances  to  which  each  can  attain  within  it,  would  be  practically  nothing.  From 
this  it  would  also  follow  that  any  one  of  these  universes  would  not  be  appreciably  influenced 
in  any  way  by  the  motions  &  forces  of  that  greater  universe  ;  but  in  any  given  time, 
however  great,  the  whole  inferior  universe  would  experience  forces,  from  any  point  of  matter 
placed  without  itself,  that  approach  as  near  as  possible  to  equal  &  parallel  forces  ;  these 
therefore  would  have  no  influence  on  its  relative  internal  state. 

172.  Now  these  matters  really  belong  to  the  application  of  the  Theory  to  physics ;  &  Leaving  out  more 
indeed  I  only  mentioned  them  here  to  show  how  many  things  there  may  be  well  worth  abstruse    matters, 

•  j.         .        -i  •         •  a     i  .,..,.  '        -    9  .     n   i  i      r  •  ••          •      we  pass  on  to  areas. 

considering  in  that  section,  &  how  great  is  the  fertility  of  this  field  of  investigation,  m 
which  possible  combinations  &  possible  forms  are  truly  infinitely  infinite  ;  of  these,  those 
that  can  be  in  any  way  comprehended  by  the  human  intelligence  are  so  few  compared 
with  the  whole,  that  they  can  be  considered  as  a  mere  nothing.  Yet  all  of  them  were  seen 
in  clear  view  at  one  gaze  by  GOD,  the  Founder  of  the  World.  We,  in  what  follows,  will 
for  the  most  part  investigate  only  certain  of  the  more  simple  matters  which  will  lead  us 
to  phenomena  in  conformity  with  those  things  that  we  contemplate  in  Nature  as  far  as 
our  intelligence  will  carry  us ;  meanwhile  we  will  proceed  to  the  areas  corresponding  to 
the  arcs. 

173.  It  is  easily  shown  that  the  area  corresponding  to  any  segment  of  the  axis,  however  To  any  segment  of 
small,  can  be  anything,  no  matter  how  great ;   &  the  area  corresponding  to  any  segment,  corrt'spo^a'rfy 
however  great,  can  be  anything,  no  matter  how  small.     In  Fig.  1 5 ,  let  MQ  be  a  segment  of  the  area,  however 
axis,  no  matter  how  small,  or  great;    &  let  an  area  be  given,  no  matter  how  great,  or  SSi ;  proof^the 
small.     If  this  area  is  applied  to  MQ  a  certain  altitude  MN  will  be  given,  such  that,  if  NR  second  part  of  this 
is  drawn  parallel  to  MQ,  then  MNRQ  will  be  equal  to  the  given  area  ;   &  thus,  if  QS  is  a 

taken  equal  to  twice  QR,  the  area  of  the  triangle  MSQ  will  also  be  equal  to  the  given  area. 
Now,  for  the  second  case  it  is  sufficiently  evident  that  a  curve  can  be  drawn  below  the 
straight  line  NR,  in  the  way  XZ  is  shown,  the  area  under  which  is  less  than  the  area  MNRQ  ; 

as  its  asymptote  ;  &  this  can  take  place  with  either  the  same  part  of  the  line  or  with  the  other  part ;  also  the  arm 
itself  can  lie  either  on  the  same  side  of  either  of  the  two  parts,  or  on  the  opposite  side.  Thus  there  may  be  four  positions 
of  the  arm  that  returns  from  infinity.  If,  in  Fig.  13,  the  arm  ED  goes  off  to  infinity,  the  asymptote  being  ACA, 
it  may  return  from  the  direction  of  A,  either  like  HI,  wheie  the  arm  lies  on  the  same  side  of  the  asymptote  or  as  KL 
which  lies  'on  the  opposite  side  of  it ;  or  from  the  direction  of  A',  either  as  MN,  on  the  same  side,  or  as,  DP,  on  the 
opposite  side.  In  the  second  of  these  two  dissertations,  I  have  given  examples  of  all  regressions  of  this  sort ;  y  the 
method  of  generation  given  above  will  yield  examples  of  the  second  W  fourth  cases,  if  the  generating  curve  touches 
the  axis,  or  cuts  it  &  passes  over  beyond  it.  Further,  it  thus  comes  about  that  asymptotic  arms  having  a  rectilinear 
asymptote  cannot  exist  except  in  pairs,  just  like  imaginary  roots  in  algebraical  equations. 

But  here  in  the  curve  of  forces,  in  which  the  arc  must  always  proceed  in  such  a  manner  that  to  each  distance  or 
abscissa  there  corresponds  a  single  force  or  ordinate,  the  first  £tf  third  cases  cannot  occur.  For  the  ordinate  RQ  of  the 
arm  DE  would  meet  somewhere,  in  S,  S',  the  branches  HI,  MN  as  well.  Hence  only  the  fourth  &  second  cases  are 
left ;  W  these  we  will  make  use  of  later. 


140 


PHILOSOPHIC  NATURALIS  THEORIA 


Demonst  ratio 
primse. 


Aream  asympto- 
ticam  posse  esse 
infinitam,  vel  fini- 
tam  magnitudinis 
cujuscunque. 


Areas  exprimere 
incrementa,  vel 
decrementa  quad- 
ati  velocitatis. 


Quin  immo  licet  ordinata  QV  sit  utcunque  magna  ;  facile  patet,  posse  arcum  MaV  ita 
accedere  ad  rectas  MQ,  QV ;  ut  area  inclusa  iis  rectis,  &  ipsa  curva,  minuatur  infra 
quoscunque  determinatos  limites.  Potest  enim  jacere  totus  arcus  intra  duo  triangula 
QaM,  QaV,  quorum  altitudines  cum  minui  possint, 
quantum  libuerit,  stantibus  basibus  MQ,  QV,  potest 
utique  area  ultra  quoscunque  limites  imminui.  Pos- 
set autem  ea  area  esse  minor  quacunque  data ; 
etiamsi  QV  esset  asymptotus,  qua  de  re  paullo 
inferius. 

174.  Pro  primo  autem  casu  vel  curva  secet  axem 
extra  MQ,  ut  in  T,  vel  in  altero  extremo,  ut  in  M  ; 
fieri  poterit,   ut  ejus  arcus  TV,  vel  MV  transeat  per 
aliquod  punctum  V  jacens   ultra   S,    vel    etiam  per 
ipsum    S    ita,   ut    curvatura     ilium    ferat,     quemad- 
modum  figura   exhibet,  extra   triangulum  MSQ,  quo 
casu  patet,  aream  curvae  respondentem  intervallo  MQ 
fore  majorem,  quam  sit  area  trianguli  MSQ,  adeoque 
quam    sit  area  data  ;    erit  enim   ejus   trianguli   area 
pars    areae    pertinentis   ad    curvam.     Quod   si   curva 
etiam  secaret  alicubi   axem,  ut  in   H  inter   M,  &  Q, 
turn   vero  fieri  posset,   ut   area    respondens   alteri    e 
segmentis   MH,    QH   esset   major,  quam   area   data . 

simul,  &  area  alia  assumpta,  qua  area  assumpta  esset  minor  area    respondens  segmento, 
alteri  adeoque  excessus  prioris  supra  posteriorem  remaneret  major,  quam  area  data. 

175.  Area  asymptotica  clausa  inter  asymptotum,  &  ordinatam  quamvis,  ut  in  fig.  I 
BA#g,  potest  esse  vel  infinita,  vel  finita  magnitudinis  cujusvis  ingentis,  vel  exiguae.     Id 
quidem  etiam  geometrice  demonstrari  potest,  sed  multo  facilius  demonstratur  calculo 
integrali  admodum  elementari ;  &  in  Geometriae  sublimioris  elementis  habentur  theoremata, 
ex  quibus  id  admodum  facile  deducitur  0.     Generaliter   nimi-[8l]-rum   area    ejusmodi 
est  infinita  ;   si  ordinata  crescit  in  ratione  reciproca  abscissarum  simplici,  aut  majore  :   & 
est  finita  ;    si  crescit  in  ratione  multiplicata  minus,  quam  per  unitatem. 

176.  Hoc,  quod  de  areis  dictum  est,  necessarium  fuit  ad  applicationem  ad  Mechanicam, 
ut  nimirum  habeatur  scala  quaedam  velocitatum,  quae  in  accessu  puncti  cujusvis  ad  aliud 
punctum,  vel  recessu  generantur,  vel  eliduntur  ;   prout  ejus  motus  conspiret  directione  vis, 
vel  sit  ipsi  contrarius.     Nam,  quod  innuimus  &  supra  in  adnot.  (/)  ad  num.  118.,  ubi  vires 
exprimuntur  per  ordinatas,  &  spatia  per  abscissas,  area,  quam  texit  ordinata,  exprimit 
incrementum,  vel  decrementum  quadrati  velocitatis,  quod  itidem  ope  Geometrise  demon- 
stratur facile,  &  demonstravi  tam  in  dissertatione  De  Firibus  Vivis,  quam  in  Stayanis 
Supplements ;   sed  multo  facilius  res  conficitur  ope  calculi  integralis.  («) 


M    H 

FIG.  15. 


(1)  Sit  Aa  in  Fig.  I  =x,  ag=y  ;  ac  sit  #"y  =  I  ;  erit  y  =  *-">/",  y  dx  elementum  areee=x~m/*dx,  cujus  integrate 

—  *fn»  +  A,  addita  constanti  A,  sive  ob  x~*>">=y,  habebitur  —?—xy  +  A.  Quoniam  incipit  area  in  A,  in 
n~m  "  n-m 

origine  abscissarum  ;  si  n—m  fuerit  numerus  positivus,  adeoque  n  major,  quam  m  ;  area  erit  finita,  ac  valor  A  =o; 
area  vero  erit  ad  rectangulum  AaXag,  ut  in  ad  n  —  m,  quod  rectangulum,  cum  ag  possit  esse  magna,  &  parva,  ut  libuerit, 
potest  esse  magnitudinis  cujusvis.  Is  valor  fit  infinitus,  si  facto  m  =n,  divisor  evaaat—Q;  adeoque  multo  magis  fit 
infinitus  valor  area,  si  m  sit  major,  quam  n.  Unde  constat,  aream  fore  infiniiam,  quotiescunque  ordinatte  crescent  in 
ratione  reciproca  simplici,  y  majore  ;  secus  fore  finitam. 

(m)  Sit  u  vis,  c  celeritas,  t  tempus,  s  spatium  :  erit  u  at  =  dc,  cum  celeritatis  incrementum  sit  proportionale  vi,  W 
tempusculo  ;  ac  erit  c  dt  =  ds,  cum  spatiolum  confectwm  respondeat  velocitati,  &  tempusculo.  Hinc  eruitur  dt  =— , 

W  pariter  dt  =—,  adeoque—-  =—    W  c  dc  =  u  ds.     Porro  2c  dc  est  incrementum  quadrati  vekcitatis  cc,  i3  u  ds 

c  u         c 

in  bypotbesi,  quod  ordinata  sit  w,  &  spatium  s  sit  abscissa,  est  areola  respondens  spatiolo  ds  confecto.  Igitur  incrementum 
quadrati  velocitatis  conspirante  vi,  adeoque  decrementum  vi  contraria,  respondet  arete  respondent  spatiolo  percurso  quovis 
infinitesimo  tempusculo ;  &  proinde  tempore  etiam  quovis  finito  incrementum,  vel  decrementum  quadrati  velocitatis 
respondet  arece  pertinenti  ad  partem  axis  referentem  spatium  percursum. 

Hinc  autem  illud  sponte  consequitur  :  si  per  aliquod  spatium  vires  in  singulis  punctis  eeedem  permaneant,  mobile  autem 
adveniat  cum  velocitate  quavis  ad  ejus  initium  ;  diferentiam  quadrati  velocitatis  finalis  a  quadrate  velocitatis  initialis 
fore  semper  eandem,  quts  idcirco  erit  tola  velocitas  finalis  in  casu,  in  quo  mobile  initio  illius  spatii  haberet  velocitatem 
nullam.  Quare,  quod  nobis  erit  inferius  usui,  quadratum  velocitatis  finalis,  conspirante  vi  cum  directione  motus,  tzquabitur 
binis  quadratis  binarum  velocitatum,  ejus,  quam  babuit  initio,  W  ejus,.quam  acquisivisset  in  fine,  si  initio  ingressum  fuisset 
sine  ulla  velocitate. 


A  THEORY  OF  NATURAL  PHILOSOPHY 


141 


for  it  is  part  of  it.  Again,  although  the  ordinate  QV  may  be  of  any  size,  however  great, 
it  is  easily  shown  that  an  arc  MoV  can  approach  so  closely  to  the  straight  lines  MQ, 
QV  that  the  area  included  between  these  lines  &  the  curve  shall  be  diminished  beyond 
any  limits  whatever.  For  it  is  possible  for  the  curve  to  lie  within  the  two  triangles  QaM, 
QaV ;  &  since  the  altitudes  of  these  can  be  diminished  as  much  as  you  please,  whilst  the 
bases  MQ,  QV  remain  the  same,  therefore  the  area  can  indeed  be  diminished  beyond  all 
limits  whatever.  Moreover  it  is  possible  for  this  area  to  be  less  than  any  given  area,  even 
although  QV  should  be  an  asymptote  ;  we  will  consider  this  a  little  further  on. 

174.  Again,  for  the  first  case,  either  the  curve  will  cut  the  axis  beyond  MQ,  as  at  T, 
or  at  either  end,  as  at  M.     Then  it  is  possible  for  it  to  happen  that  an  arc  of  it,  TV  or  MV, 
will  pass  through  some  point  V  lying  beyond  S,  or  even  through  S  itself,  in  such  a  way 
that  its  curvature  will  carry  it,  as  shown  in  the  diagram,  outside  the  triangle  MSQ  ;    in 
this  case  it  is  clear  that  the  area  of  the  curve  corresponding  to  the  interval  MQ  will  be 
greater  than  the  area  of  the  triangle  MSQ,  &    therefore   greater    than  the  given    area, 
for  the  area  of  this  triangle  is  part  of  the  area  belonging  to  the  curve.      But  if  the  curve 
should  even  cut  the  axis  anywhere,  as  at  H,  between  M  &  Q,  then  it  would  be  possible 
for  it  to  come  about  that  the  area  corresponding  to  one  of  the  two  segments  MH,  QH  would 
be  greater  than  the  given  area  together  with  some  other  assumed  area ;  &  that  the  area 
corresponding  to  the  other  segment  should  be  less  than  this  assumed  area  ;   and  thus  the 
excess  of  the  former  over  the  latter  would  remain  greater  than  the  given  area. 

175.  An  asymptotic  area,  bounded  by  an  asymptote  &  any   ordinate,  like  BAag  in 
Fig.  i,  can  be  either  infinite,  or  finite  of  any  magnitude  either  very  great  or  very  small. 
This  can  indeed  be  also  proved  geometrically,  but  it  can  be  demonstrated  much  more 
easily  by  an  application  of  the  integral  calculus  that  is  quite  elementary ;  &  in  the  elements 
of  higher  geometry  theorems  are  obtained  from  which  it  is  derived  quite  easily.  0     In 
general,  it  is  true,  an  area  of  this  kind  is  infinite  ;    namely  when  the  ordinate  increases  in 
the  simple  inverse  ratio  of  the  abscissse,  or  in  a  greater  ratio ;   and  it  is  finite,  if  it  increases 
in  this  ratio  multiplied  by  something  less  than  unity. 

176.  What  has  been  said  with  regard  to  areas  was  a  necessary  preliminary  to  the 
application  of  the  Theory  to  Mechanics ;    that  is  to  say,  in  order  that  we  might  obtain  a 
diagrammatic  representation  of  the  velocities,  which,  on  the  approach  of  any  point  to 
another  point,  or  on  recession  from  it,  are  produced  or  destroyed,  according  as  its  motion 
is  in  the  same  direction  as  the  direction  of  the  force,  or  in  the  opposite  direction.     For, 
as  we  also  remarked  above,  in  note  (/)  to  Art.  118,  when  the  forces  are  represented  by 
ordinates  &  the  distances  by  abscissae,  the  area  that  the  ordinate  sweeps  out  represents 
the  increment  or  decrement  of  the  square  of  the  velocity.     This  can  also  be  easily  proved 
by  the  help  of  geometry ;   &  I  gave  the  proof  both  in  the  dissertation  De  Firibus  Fivis 
&  in  the  Supplements  to  Stay's  Philosophy ;   but  the  matter  is  much  more  easily  made 
out  by  the  aid  of  the  integral  calculus. («) 


Proof 
part. 


of   the  first 


An  a  s  y  m  p  totic 
area  may  be  either 
infinite  or  equal  to 
any  finite  area 
whatever. 


The  areas  represent 
the  increments  or 
decrements  of  the 
square  of  the  velo- 
city. 


(1)  In  Fig.  iletAa  =  x,ag  =  y;  y  let  xmy"  =  I.     Then  will  y  —  x~ 


the  element  of  area  y  dx  =  x~m/* 


dx  :   the  integral  of  this  is  -  x  <»-"»/"+  A,  where  a  constant  A  is  added  ;  or,  since  x~m/*=y,  we  shall  have-^—  Xv  +  A 

n-m  n-m    ' 

Now,  since  the  area  is  initially  A,  at  the  origin  of  the  abscissa,  if  n-m  happened  to  be  a  positive  number,  y 
thus  n  greater  than  m,  then  the  area  will  be  finite,  y  the  value  of  A  will  be  =  o.  Also  the  area  will  be  to 
the  rectangle  Aa.ag  as  n  is  to  n-m  ;  y  this  rectangle,  since  ag  can  be  either  great  or  small,  as  you  please,  may  be 
of  any  magnitude  whatever.  The  value  is  infinite,  if  by  making  m  equal  to  n  the  divisor  becomes  equal  to  zero  ;  & 
thus  the  value  of  the  area  becomes  all  the  more  infinite,  if  m  is  greater  than  n.  Hence  it  follows  that  the  area  will 
be  infinite,  whenever  the  ordinates  increase  in  a  simple  inverse  ratio,  or  in  a  greater  ratio  ;  otherwise  it  will  be  finite. 

(m)  Let  u  be  the  force,  c  the  velocity,  t  the  time,  y  s  the  distance.  Then  will  u  dt  —  dc,  since  the  increment 
of  the  velocity  is  proportional  to  the  force,  y  to  the  small  interval  of  time.  Also  c  dt  =  ds,  since  the  distance  traversed 
corresponds  with  the  velocity  W  the  small  interval  of  time.  Hence  it  follows  that  dt  =  dc/u,  y  similarly  dt  —  ds/c, 
y  therefore  dc/u  =  ds/c,  &  c  dc  —  u  ds.  Further,  ^c  dc  is  the  increment  of  the  square  of  the  velocity  c',  y  u  ds, 
on  the  hypothesis  that  the  ordinate  represents  u,  y  the  abscissa  the  distance  s,  is  the  small  area  corresponding  to  the 
small  distance  traversed.  Hence  the  increment  of  the  square  of  the  velocity,  when  in  the  direction  of  the  force,  y 
the  decrement  when  opposite  in  direction  to  the  force,  is  represented  by  the  area  corresponding  to  ds,  the  small  distance 
traversed  in  any  infinitely  short  time.  Hence  also,  in  any  finite  interval  of  time,  the  increment  or  decrement  of  the 
square  of  the  velocity  will  be  represented  by  the  area  corresponding  to  that  part  of  the  axis  which  represents  the  distance 
traversed. 

Hence  also  it  follows  immediately  that,  if  through  any  distance  the  force  on  each  of  the  points  remains  as  before, 
but  the  moving  body  arrives  at  the  beginning  of  it  with  any  velocity,  then  the  difference  between  the  square  of  the  final 
velocity  y  the  square  of  the  initial  velocity  will  always  be  the  same  ;  y  this  therefore  will  be  the  total  final  velocity, 
in  the  case  where  the  moving  body  had  no  velocity  at  the  beginning  of  the  distance.  Hence,  the  square  of  the  final 
velocity,  when  the  motion  is  in  the  same  direction  as  the  force,  will  be  equal  to  the  sum  of  the  squares  of  the  velocity  which 
it  had  at  the  beginning  y  of  the  velocity  it  would  have  acquired  at  the  end,  if  it  had  at  the  beginning  started  without 
any  velocity  ;  a  theorem  that  we  shall  make  use  of  later. 


I42 


PHILOSOPHIC  NATURALIS  THEORIA 


Atque  id  ips  u m, 
licet  segmenta  axis 
sint  dimidia  spatio- 
rum  percursorum  a 
singulis  punctis. 


Si  arese  sint  partim 
attractivae,  partim 
repulsivae,  assumen- 
dam  esse  differen- 
tiam  earundem. 


177.  Duo    tamen  hie    tantummodo    notanda  sunt  ;    primo  quidem  illud  :    si  duo 
puncta  ad  se  invicem  accedant,  vel  a  se  invicem  recedant  in  ea  recta,  quae  ipsa  conjungit, 
segmenta  illius  [82]  axis,  qui  exprimit  distantias,  non  expriment  spatium  confectum  ;   nam 
moveri  debebit  punctum  utrumque  :    adhuc  tamen  ilia  segmenta  erunt  proportionalia  ipsi 
spatio  confecto,  eorum  nimirum  dimidio  ;  quod  quidem  satis  est  ad  hoc,  ut  illae  areae  adhuc 
sint    proportionales    incrementis,    vel    decrementis    quadrati   velocitatum,    adeoque   ipsa 
exprimant. 

178.  Secundo  loco  notandum  illud,  ubi  areae  respondentes  dato  cuipiam  spatio  sint 
partim  attractive,  partim  repulsivae,  earum  differentiam,  quae  oritur  subtrahendo  summam 
omnium  repulsivarum  a  summa  attractivarum,  vel  vice  versa,  exhibituram  incrementum 
illud,  vel  decrementum  quadrati  velocitatis ;    prout  directio  motus  respectivi  conspiret 
cum  vi,  vel  oppositam  habeat  directionem.     Quamobrem  si  interea,  dum  per  aliquod  majus 
intervallum  a  se  invicem  recesserunt  puncta,  habuerint    vires  directionis  utriusque ;    ut 
innotescat,  an  celeritas  creverit,  an  decreverit  &  quantum  ;    erit  investigandum,  an  areas 
omnes  attractivae  simul,  omnes  repulsivas  simul  superent,  an  deficiant,  &  quantum  ;    inde 
enim,  &  a  velocitate,  quae  habebatur  initio,  erui  poterit  quod  quaeritur. 


^e  arcubus,  &  areis ;  nunc  aliquanto  diligentius  considerabimus 
tangentis:  sectio-  ilia  axis  puncta,  ad  quae  curva  appellit.  Ea  puncta  vel  sunt  ejusmodi,  ut  in  iis  curva  axem 
ducT  enera  UmltUm  secet>  cujusmodi  in  fig.  I  sunt  E,G,I,  &c.,  vel  ejusmodi,  ut  in  iis  ipsa  curva  axem  contingat 
tantummodo.  Primi  generis  puncta  sunt  ea,  in  quibus  fit  transitus  a  repulsionibus  ad 
attractiones,  vel  vice  versa,  &  hsec  ego  appello  limites,  quod  nimirum  sint  inter  eas  opposi- 
tarum  directionum  vires.  Sunt  autem  hi  limites  duplicis  generis  :  in  aliis,  aucta  distantia, 
transitur  a  repulsione  ad  attractionem  :  in  aliis  contra  ab  attractione  ad  repulsionem. 
Prioris  generis  sunt  E,I,N,R ;  posterioris  G,L,P  :  &  quoniam,  posteaquam  ex  parte 
repulsiva  in  una  sectione  curva  transiit  ad  partem  attractivam  ;  in  proxime  sequent!  sectione 
debet  necessario  ex  parte  attractiva  transire  ad  repulsivam,  ac  vice  versa  ;  patet,  limites 
fore  alternatim  prioris  illius,  &  hujus  posterioris  generis. 


t  P°rro  linrites  prioris  generis,  a  limitibus  posterioris  ingens  habent  inter  se  dis- 
differant':  limites  crimen.  Habent  illi  quidem  hoc  commune,  ut  duo  puncta  collocata  in  distantia  unius 
cohaesionls'  &  n°n  h'111^8  cujuscunque  nullam  habeant  mutuam  vim,  adeoque  si  respective  quiescebant,  pergant 
itidem  respective  quiescere.  At  si  ab  ilia  respectiva  quiete  dimoveantur  ;  turn  vero  in 
limite  primi  generis  ulteriori  dimotioni  resistent,  &  conabuntur  priorem  distantiam  recu- 
perare,  ac  sibi  relicta  ad  illam  ibunt ;  in  limite  vero  secundi  generis,  utcunque  parum 
dimota,  sponte  magis  fugient,  ac  a  priore  distantia  statim  recedent  adhuc  magis.  Nam 
si  distantia  minuatur  ;  habebunt  in  limite  prioris  generis  vim  repulsivam,  quae  obstabit 
uteriori  accessui,  &  urgebit  puncta  ad  mutuum  recessum,  quern  sibi  relicta  acquirent,  [83] 
adeoque  tendent  ad  illam  priorem  distantiam  :  at  in  limite  secundi  generis  habebunt 
attractionem,  qua  adhuc  magis  ad  se  accedent,  adeoque  ab  ilia  priore  distantia,  quae  erat 
major,  adhuc  magis  sponte  fugient.  Pariter  si  distantia  augeatur,  in  primo  limitum  genere 
a  vi  attractiva,  quse  habetur  statim  in  distantia  majore  ;  habebitur  resistentia  ad  ulteriorem 
recessum,  &  conatus  ad  minuendam  distantiam,  ad  quam  recuperandam  sibi  relicta  tendent 
per  accessum  ;  at  in  limitibus  secundi  generis  orietur  repulsio,  qua  sponte  se  magis  adhuc 
fugient,  adeoque  a  minore  ilia  priore  distantia  sponte  magis  recedent.  Hinc  illos  prioris 
generis  limites,  qui  mutuse  positionis  tenaces  sunt,  ego  quidem  appellavi  limites  coh&sionis, 
&  secundi  generis  limites  appellavi  limites  non  cobasionis. 


Duo    genera 
tactuum. 


181.  Ilia  puncta,  in  quibus  curva  axem  tangit,  sunt  quidem  terminus  quidam  virium, 
quae  ex  utraque  parte,  dum  ad  ea  acceditur,  decrescunt  ultra  quoscunque  limites,  ac  demum 
ibidem  evanescunt  ;  sed  in  iis  non  transitur  ab  una  virium  directione  ad  aliam.  Si  con- 
tactus  fiat  ab  arcu  repulsive  ;  repulsiones  evanescunt,  sed  post  contactum  remanent  itidem 
repulsiones ;  ac  si  ab  arcu  attractive,  attractionibus  evanescentibus  attractiones  iterum 
immediate  succedunt.  Duo  puncta  collocata  in  ejusmodi  distantia  respective  quiescunt ; 


A  THEORY  OF  NATURAL  PHILOSOPHY 


'43 


177.  However,  there  are  here  two  things  that  want  noting  only.     The  first  of  them  The    same   result 
is  this,  that  if  two  points  approach  one  another  or  recede  from  one  another  in  the  straight  holds    good    even 

,....,  ,  r     i  •,.   ,  ,.  i  °         when  the  segments 

line  joining  them,  the  segments  of  the  axis,  which  expresses  distances,  do  not  represent  of  the  axis  are  the 
the  distances  traversed  ;    for  both  points  will  have  to  move.     Nevertheless  the  segments  'ialves  of  the  dis- 

•11      -11  i  •         i          i        T  i        i      i     if      f  •  i  .     .     i       •,         tances  traversed  by 

will  still  be  proportional  to  the  distance  traversed,  namely,  the  half  of  it ;  &  this  indeed  is  single  points, 
sufficient  for  the  areas  to  be  still  proportional  to  the  increments  or  decrements  of  the 
squares  of  the  velocities,  &  thus  to  represent  them. 

178.  In  the  second  place  it  is  to  be  noted  that,  where  the  areas  corresponding  to  any  if   the  areas  are 
given    interval    are    partly    attractive  &  partly  repulsive,  their  difference,  obtained  by    p^ti*tt2SS2  & 
subtracting  the  sum  of  all  those  that  are  repulsive  from  the  sum  of  those  that  are  attractive,  their  difference 
or  vice  versa,  will  represent  the  increment,  or  the  decrement,  of  the  square  of  the  velocity,  must  be  taken- 
according  as  the  direction  of  relative  motion  is  in  the  same  direction  as  the  force,  or  in 

the  opposite  direction.  Hence,  if,  during  the  time  that  the  points  have  receded  from 
one  another  by  some  considerable  interval,  they  had  forces  in  each  direction  ;  then 
in  order  to  ascertain  whether  the  velocity  had  been  increased  or  decreased,  &  by  how 
much,  it  will  have  to  be  considered  whether  all  the  attractive  areas  taken  together  are 
greater  or  less  than  all  the  repulsive  areas  taken  together,  &  by  how  much.  For  from  this, 
&  from  the  velocity  which  initially  existed,  it  will  be  possible  to  deduce  what  is  required. 

179.  So  much  for  the  arcs  &  the  areas;  now  we  must  consider  in  a  rather  more  careful  Approach  of   the 
manner  those  points  of  the  axis  to  which  the  curve  approaches.     These  points  are  either  ^en   it  cSa^or 
such  that  the  curve  cuts  the  axis  in  them,  for  instance,  the  points  E,  G,  I,  &c.  in  Fig.   I  :  touches  it;    two 
or  such  that  the  curve  only  touches  the  axis  at  the  points.     Points  of  the  first  kind  are  u^ns^/'ihnit- 
those  in  which  there  is  a  transition  from  repulsions  to  attractions,  or  vice-versa  ;  &  these  points. 

I  call  limit-points  or  boundaries,  since  indeed  they  are  boundaries  between  the  forces  acting 
in  opposite  directions.  Moreover  these  limit-points  are  twofold  in  kind  ;  in  some,  when 
the  distance  is  increased,  there  is  a  transition  from  repulsion  to  attraction  ;  in  others,  on 
the  contrary,  there  is  a  transition  from  attraction  to  repulsion.  The  points  E,  I,  N,  R 
are  of  the  first  kind,  and  G,  L,  P  are  of  the  second  kind.  Now,  since  at  one  intersection, 
the  curve  passes  from  the  repulsive  part  to  the  attractive  part,  at  the  next  following 
intersection  it  is  bound  to  pass  from  the  attractive  to  the  repulsive  part,  &  vice  versa. 
It  is  clear  then  that  the  limit-points  will  be  alternately  of  the  first  &  second  kinds. 

1 80.  Further,  there  is  a  distinction  between  limit-points  of  the  first  &  those  of  the  in  what  they  agree 
second  kind.     The  former  kind  have  this  property  in  common  ;   namely  that,  if  two  points  *iffj£ .  w^  u*?ty 
are  situated  at  a  distance  from  one  another  equal  to  the  distance  of  any  one  of  these  limit-  points  of  cohesion 
points  from  the  origin,  they  will  have  no  mutual  force  ;    &  thus,  if  they  are  relatively  &  of  non-cohesic«i. 
at  rest  with  regard  to  one  another,  they  will  continue  to  be  relatively  at  rest.     Also,  if 

they  are  moved  apart  from  this  position  of  relative  rest,  then,  for  a  limit-point  of  the  first 
kind,  they  will  resist  further  separation  &  will  strive  to  recover  the  original  distance,  & 
will  attain  to  it  if  left  to  themselves ;  but,  in  a  limit-point  of  the  second  kind,  however 
small  the  separation,  they  will  of  themselves  seek  to  get  away  from  one  another  &  will 
immediately  depart  from  the  original  distance  still  more.  For,  if  the  distance  is  diminished, 
they  will  have,  in  a  limit-point  of  the  first  kind,  a  repulsive  force,  which  will  impede  further 
approach  &  impel  the  points  to  mutual  recession,  &  this  they  will  acquire  if  left  to 
themselves ;  thus  they  will  endeavour  to  maintain  the  original  distance  apart.  But  in  a 
limit-point  of  the  second  kind  they  will  have  an  attraction,  on  account  of  which  they  will 
approach  one  another  still  more  ;  &  thus  they  will  seek  to  depart  still  further  from  the 
original  distance,  which  was  a  greater  one.  Similarly,  if  the  distance  is  increased,  in 
limit-points  of  the  first  kind,  due  to  the  attractive  force  which  is  immediately  obtained 
at  this  greater  distance,  there  will  be  a  resistance  to  further  recession,  &  an  endeavour 
to  diminish  the  distance ;  &  they  will  seek  to  recover  the  original  distance  if  left  to 
themselves  by  approaching  one  another.  But,  in  limit-points  of  the  second  class,  a  repulsion 
is  produced,  owing  to  which  they  try  to  get  away  from  one  another  still  further  ;  &  thus 
of  themselves  they  will  depart  still  more  from  the  original  distance,  which  was  less.  On 
this  account  indeed  I  have  called  those  limit-points  of  the  first  kind,  which  are  tenacious 
of  mutual  position,  limit-points  of  cohesion,  &  I  have  termed  limit-points  of  the  second 
kind  limit-points  of  non-cohesion. 

181.  Those  points  in  which  the  curve  touches  the  axis  are  indeed  end-terms  of  series   Two  kinds  of  con- 
of  forces,  which  decrease  on  both  sides,  as  approach  to  these  points    takes  place,  beyond   tactt 

all  limits,  &  at  length  vanish  there  ;  but  with  such  points  there  is  no  transition  from 
one  direction  of  the  forces  to  the  other.  If  contact  takes  place  with  a  repulsive  arc,  the 
repulsion  vanishes,  but  after  contact  remains  still  a  repulsion.  If  it  takes  place  with  an 
attractive  arc,  attraction  follows  on  immediately  after  a  vanishing  attraction.  Two  points 
situated  such  a  distance  remain  in  a  state  of  relative  rest ;  but  in  the  first  case  they  will 


144 


PHILOSOPHIC  NATURALIS  THEORIA 


pro   forma   curvae 
prope  sectionem. 


sed  in  prime  casu  resistunt  soli  compressioni,  non  etiam  distractioni,  £  in  secundo  resistunt 
huic  soli,  non  illi. 

l^2'  Limites  cohsesionis  possunt  esse  validissimi,  &  languidissimi.  Si  curva  ibi  quasi 
ad  pcrpendiculum  secat  axem,  &  ab  eo  longissime  recedit  ;  sunt  validissimi  :  si  autem 
ipSum  secet  in  angulo  perquam  exiguo,  &  parum  ab  ipso  recedat  ;  erunt  languidissimi. 
Primum  genus  limitum  cohsesionis  exhibet  in  fig.  I  arcus  tNy,  secundum  cNx.  In  illo 
assumptis  in  axe  Nz,  NM  utcunque  exiguis,  possunt  vires  zt,  uy,  &  areae  Nzt,  Nwy  esse 
utcumque  magnas,  adeoque,  mutatis  utcunque  parum  distantiis,  possunt  haberi  vires  ab 
ordinatis  expressae  utcunque  magnae,  quae  vi  comprimenti,  vel  distrahenti,  quantum  libuerit, 
valide  resistant,  vel  areae  utcunque  magnae,  quae  velocitates  quantumlibet  magnas 
respectivas  elidant,  adeoque  sensibilis  mutatio  positionis  mutuae  impediri  potest  contra 
utcunque  magnam  vel  vim  prementem,  vel  celeritatem  ab  aliorum  punctorum  actionibus 
impressam.  In  hoc  secundo  genere  limitum  cohaesionis,  assumptis  etiam  majoribus 
segmentis  Nz,  Nw,  possunt  &  vires  zc,ux,  &  areae  Nzf  ,  N«tf,  esse  quantum  libuerit  exiguae, 
&  idcirco  exigua  itidem,  quantum  libuerit,  resistentia,  quae  mutationem  vetet. 


P°ssunt  autem  hi  Hmites  esse  quocunque,  utcunque  magno  numero ;  cum 
ro,  utcunque  proxi-  demonstratum  sit,  posse  curvam  in  quotcunque,  &  quibuscunque  punctis  axem  secare. 
mos,  vel  remotes  possunt  idcirco  etiam  esse  utcunque  inter  se  proximi,  vel  remoti,  ut  [84]  alicubi  intervallum 
originis'  abscissa-  inter  duos  proximos  limites  sit  etiam  in  quacunque  ratione  majus,  quam  sit  distantia 
ordme  praecedentis  ab  origine  abscissarum  A  ;  alibi  in  intervallo  vel  exiguo,  vel  ingenti  sint  quam- 
plurimi  inter  se  ita  proximi,  ut  a  se  invicem  distent  minus,  quam  pro  quovis  assumpto, 
aut  dato  intervallo.  Id  evidenter  fluit  ex  eo  ipso,  quod  possint  sectiones  curvae  cum  axe 
haberi  quotcunque,  &  ubicunque.  Sed  ex  eo,  quod  arcus  curvae  ubicunque  possint  habere 
positiones  quascunque,  cum  ad  datas  curvas  accedere  possint,  quantum  libuerit,  sequitur, 
quod  limites  ipsi  cohaesionis  possint  alii  aliis  esse  utcunque  validiores,  vel  languidiores, 
atque  id  quocunque  ordine,  vel  sine  ordine  ullo ;  ut  nimirum  etiam  sint  in  minoribus 
distantiis  alicubi  limites  validissimi,  turn  in  majoribus  languidiores,  deinde  itidem  in 
majoribus  multo  validiores,  &  ita  porro  ;  cum  nimirum  nullus  sit  nexus  necessarius  inter 
distantiam  limitis  ab  origine  abscissarum,  &  ejus  validitatem  pendentem  ab  inclinatione, 
&  recessu  arcus  secantis  respectu  axis,  quod  probe  notandum  est,  futurum  nimirum  usui 
ad  ostendendum,  tenacitatem,  sive  cohaesionem,  a  densitate  non  pendere. 


similes. 


Quse  positio  rectae  jg^..  In  utroque  limitum  genere  fieri  potest,  ut  curva  in   ipso  occursu  cum  axe  pro 

infinite3  rarissima!  tangente  habeat  axem  ipsum,  ut  habeat  ordinatam,  ut  aliam  rectam  aliquam  inclinatam. 
quae  frequentissima.  Jn  primo  casu  maxime  ad  axem  accedit,  &  initio  saltern  languidissimus  est  limes  ;  in  secundo 
maxime  recedit,  &  initio  saltern  est  validissimus  ;  sed  hi  casus  debent  esse  rarissimi,  si 
uspiam  sunt  :  nam  cum  ibi  debeat  &  axem  secare  curva,  &  progredi,  adeoque  secari  in 
puncto  eodem  ab  ordinata  producta,  debebit  habere  flexum  contrarium,  sive  mutare 
directionem  flexus,  quod  utique  fit,  ubi  curva  &  rectam  tangit  simul,  &  secat.  Rarissimos 
tamen  debere  esse  ibi  hos  flexus,  vel  potius  nullos,  constat  ex  eo,  quod  flexus  contrarii  puncta 
in  quovis  finito  arcu  datae  curvae  cujusvis  numero  finite  esse  debent,  ut  in  Theoria  curvarum 
demonstrari  potest,  &  alia  puncta  sunt  infinita  numero,  adeoque  ilia  cadere  in  intersectiones 
est  infinities  improbabilius.  Possunt  tamen  saepe  cadere  prope  limites  :  nam  in  singulis 
contorsionibus  curvae  saltern  singuli  flexus  contrarii  esse  debent.  Porro  quamcunque 
directionem  habuerit  tangens,  si  accipiatur  exiguus  arcus  hinc,  &  inde  a  limite,  vel 
maxime  accedet  ad  rectam,  vel  habebit  curvaturam  ad  sensum  aequalem,  &  ad  sensum 
aequali  lege  progredientem  utrinque,  adeoque  vires  in  aequali  distantia  exigua  a  limite 
erunt  ad  sensum  hinc,  &  inde  aequales  ;  sed  distantiis  auctis  poterunt  &  diu  aequalitatem 
retinere,  &  cito  etiam  ab  ea  recedere. 


Transitus  per  infi-  185.  Hi  quidem  sunt  limites  per  intersectionem  curvae  cum  axe,  viribus  evanescentibus 

astlm"toticisribUS  m  *PSO  limite-     At  possunt  [85]  esse  alii  limites,  ac  transitus  ab  una  directione  virium  ad 

aliam  non  per  evanescentiam,  sed  per  vires  auctas  in  infinitum,  nimirum  per  asymptoticos 


A  THEORY  OF  NATURAL  PHILOSOPHY  145 

resist  compression  only,  &  not  separation  ;  and  in  the  second  case  the  latter  only,  but  not 
the  former. 

182.  Limit-points  may  be  either  very  strong  or  very  weak.     If  the  curve  cuts  the  axis  The  limit-points  of 
at  the  point  almost  at  right  angles,  &  goes  off  to  a  considerable  distance  from  it,  they  o°h^eak  ?ccord£f 
are  very  strong.     But  if  it  cuts  the  axis  at  a  very  small  angle  &  recedes  from  it  but  little,  to  the  form  of  the 
then  they  will  be  very  weak.     The  arc  *Ny  in  Fig.   i   represents  the  first  kind  of  limit-  ^Hint  *  iVater- 
points  of  cohesion,  and  the  arc  cNx  the  second  kind.     At  the  point  N,  if  Nz,  N«  are  section. 

taken  along  the  axis,  no  matter  how  small,  the  forces  zt,  uy,  &  the  areas  Nzt,  N«y  may 
be  of  any  size  whatever  ;  &  thus,  if  the  distances  are  changed  ever  so  little,  it  is  possible 
that  there  will  be  forces  represented  by  ordinates  ever  so  great ;  &  these  will  strongly 
resist  the  compressing  or  separating  force,  be  it  as  great  as  you  please ;  also  that  we  shall 
have  areas,  ever  so  large,  that  will  destroy  the  relative  velocities,  no  matter  how  great  they 
may  be.  Thus,  a  sensible  change  of  relative  position  will  be  hindered  in  opposition  to 
any  impressed  force,  however  great,  or  against  a  velocity  generated  by  the  actions  upon 
them  of  other  points.  In  the  second  kind  of  limit-points  of  cohesion,  if  also  segments  Nz, 
Nw  are  taken  of  considerable  size  even,  then  it  is  possible  for  both  the  forces  zc,  ux,  & 
the  areas  Nzc,  Nux  to  be  as  small  as  you  please ;  &  therefore  also  the  resistance  that 
opposes  the  change  will  be  as  small  as  you  please. 

183.  Moreover,  there  can  be  any  number  of  these  limit-points,  no  matter  how  great ;  The     limit-points 
for  it  has  been  proved  that  the  curve  can  cut  the  axis  in  any  number  of  points,  &  anywhere.  are  m<Jefimte   as 

rrM  F  •       •  •  i    i        r  i  i  •    i  i  f  i  i  6  g  cl  i  Cl  S    HUTU  DCr, 

I  herefore  it  is  possible  for  them  to  be  either  close  to  or  remote  from  one  another,  without  their  proximity  to 

any  restriction  whatever,  so  that  the  interval  between  any  two  consecutive  limit-points  one^another^&^th^ 

at  any  place  shall  even  bear  to  the  distance  of  the  first  of  the  two  from  A,  the  origin  of  order  of  their  occur- 

abscissae,  a  ratio  that  is  greater  than  unity.     In  other  words,  in  any  interval,  either  very  ^toe  <SgVof  Pab- 

small  or  very  large,  there  may  be  an  exceedingly  large  number  of  them  so  close  to  one  scissae. 

another,  that  they  are  less  distant  from  one  another  than  they  are  from  any  chosen  or  given 

interval.     This  evidently  follows  from  the  fact  that  the  intersections  of  the  curve  with 

the  axis  can  happen  any  number  of  times  &  anywhere.     Again,  from  the  fact  that  arcs 

of  the  curve  can  anywhere,  owing  to  their  being  capable  of  approximating  as  closely  as 

you  please  to  given  curves,  have  any  positions  whatever,  it  follows  that  these  limit-points 

of   cohesion  can  be  some  of   them  stronger  than  others,  or  weaker,  in  any  manner ;    & 

that  too,  in  any  order,  or  without  order.      So  that,  for  instance,  we  may  have  at  small 

distances  anywhere  very  strong  limit-points,  then    at   greater   distances  weaker   ones,  & 

then  again  at  still  greater  distances  much  stronger  ones,  &  so  on.     That  is  to  say,  since 

there  is  no  necessary  connection  between  the  distance  of  a  limit-point  from  the  origin  of 

abscissae  and  its  strength,  which  depends  on  the  inclination  of  the  intersecting  arc  &  the 

distance  it  recedes  from  the  axis.     It  is  well  that  this  should  be  made  a  note  of ;  for  indeed 

it  will  be  used  later  to  prove  that  tenacity  or  cohesion  does  not  depend  on  density. 

184.  In  each  of  these  kinds  of  limit-points  it  may  happen  that  the  curve,  where  it  What    position  of 
meets  the  axis,  may  have  the  axis  itself  as  its  tangent,  or  the  ordinate,  or  any  other  straight  touchkig^he* curve 
line  inclined  to  the  axis.     In  the  first  case  it  approximates    very  closely  to  the  axis,  &  at  a  limit-point  is 
close  to  the  point  at  any  rate  it  is  a  very  weak  limit-point ;   in  the  second  case,  it  departs  ^at  magt^fr* 
from  the  axis  very  sharply,  &  close  to  the  point  at  any  rate  it  is  a  very  strong  limit-point,  quent ;  small  arcs 
But  these  two  cases  must  be  of  very  rare  occurrence,  if  indeed  they  ever  occur.     For,  since  fTm^tVo'in t*  are 
at  the  point  the  curve  is  bound  to  cut  the  axis  &  go  on,  &  thus  be  cut  in  the  same  point  equal  &  similar, 
by  the  ordinate  produced,  it  is  bound  to  have  contrary-flexure  ;    that  is  to  say,  a  change 

in  the  direction  of  its  curvature,  such  as  always  takes  place  at  a  point  where  the  curve  both 
touches  a  straight  line  &  cuts  it  at  the  same  time.  Yet,  that  these  flexures  must  occur 
very  rarely  at  such  points,  or  rather  never  occur  at  all,  is  evident  from  the  fact  that  in  any 
finite  arc  of  any  given  curve  the  number  of  points  of  contrary-flexure  must  be  finite,  as  can 
be  proved  in  the  theory  of  curves ;  &  other  points  are  infinite  in  number ;  hence  that  the 
former  should  happen  at  the  points  of  intersection  with  the  axis  is  infinitely  improbable. 
On  the  other  hand  they  may  often  fall  close  to  the  limit-points ;  for  in  each  winding  of 
the  curve  about  the  axis  there  must  be  at  least  one  point  of  contrary-flexure.  Further, 
whatever  the  direction  of  the  tangent,  if  a  very  small  arc  of  the  curve  is  taken  on  each  side 
of  the  limit-point,  this  arc  will  either  approximate  very  closely  to  the  straight  line,  or  will 
have  its  curvature  the  same  very  nearly,  &  will  proceed  very  nearly  according  to  the  same 
law  on  each  side  ;  &  thus  the  forces,  at  equal  small  distances  on  each  side  of  the  limit- 
point  will  be  very  nearly  equal  to  one  another  ;  but  when  the  distances  are  increased, 
they  can  either  maintain  this  equality,  for  some  considerable  time,  or  indeed  very  soon 
depart  from  it. 

185.  The  limit-points  so  far  discussed  are  those  obtained  through  the   intersection  Passage     through 
of    the  curve  with  the  axis,  where  the  forces  vanish  at  the   limit-point.     But    there 

may  be  other  limit-points ;  the  transition  from  one  direction  of  the  forces  to  another 

L 


146 


PHILOSOPHIC  NATURALIS  THEORIA 


curvse  arcus.  Diximus  supra  num.  168.  adnot.  (i),  quando  crus  asymptoticum  abit  in 
infinitum,  debere  ex  infinite  regredi  crus  aliud  habens  pro  asymptote  eandem  rectam,  & 
posse  regredi  cum  quatuor  diversis  positionibus  pendentibus  a  binis  partibus  ipsius  rectae, 
&  binis  plagis  pro  singulis  rectae  partibus ;  sed  cum  nostra  curva  debeat  semper  progredi, 
diximus,  relinqui  pro  ea  binas  ex  ejusmodi  quatuor  positionibus  pro  quovis  crure  abeunte 
in  infinitum,  in  quibus  nimirum  regressus  fiat  ex  plaga  opposita.  Quoniam  vero,  progre- 
diente  curva,  abire  potest  in  infinitum  tarn  crus  repulsivum,  quam  crus  attractivum  ,  jam 
iterum  fiunt  casus  quatuor  possibiles,  quos  exprimunt  figurae  16,  17,  18,  &  19,  in  quibus 
omnibus  est  axis  ACS,  asymptotus  DCD',  crus  recedens  in  infinitum  EKF,  regrediens 
ex  infinite  GMH. 


D 


A 


I    C 


B 


FIG.  1 6. 


I    C 


D 


B 


D 

FIG.  17. 


Quatuor     eorum  186.  In  fig.  16.  cruri  repulsivo  EKF  succedit  itidem    repulsivum  GMH  ;  in  fig.  17 

f^nXntesTontac-  repulsivo  attractivum  ;  in  1 8  attractive  attractivum;  in  19  attractive  repulsivum.    Primus 

tibus,  bini    Hmiti-  &  tertius  casus  respondent  contactibus.     Ut  enim  in  illis  evanescebat  vis ;   sed  directionem 

ionU,  alaHerCOhnon  non  mutabat ;  ita  &  hie  abit  quidem  in  infinitum,  sed  directionem  non  mutat.     Repulsioni 

cohaesionis.  IK  in  fig.  1 6  succedit  repulsio  LM  ;   &  attractioni  in  fig.  18  attractio.     Quare  ii  casus  non 

habent  limites  quosdam.      Secundus,  &  quartus  habent  utique  limites ;    nam  in  fig.  17 

repulsion!  IK  succedit  attractio  LM  ;    &  in  fig.   19  attractioni  repulsio  ;    atque  idcirco 

secundus  continet  limitem  cobasionis,  quartus  limitem  non  cohcesionis. 


Nuiium  in  Natura 


vero    eum 
utcunque. 


187.  Ex  istis  casibus  a  nostra  curva  censeo  removendos  esse  omnes  praeter  solum 
quartum  ;  &  in  hoc  ipso  removenda  omnia  crura,  in  quibus  ordinata  crescit  in  ratione 
ipsum  minus,  quam  simplici  reciproca  distantiarum  a  limite.  Ratio  excludendi  est,  ne  haberi 
aliquando  vis  infinita  possit,  quam  &  per  se  se  absurdam  censeo,  &  idcirco  praeterea,  quod 
infinita  vis  natura  sua  velocitatem  infinitam  requirit  a  se  generandam  finito  tempore.  Nam 
in  primo,  &  secundo  casu  punctum  collocatum  in  ea  distantia  ab  alio  puncto,  quam  habet 
I,  ab  origine  abscissarum,  abiret  ad  C  per  omnes  gradus  virium  auctarum  in  infinitum, 
&  in  C  deberet  habere  vim  infinitam  ;  in  tertio  vero  idem  accideret  puncto  collocate  in 
distantia,  quam  habet  L.  At  in  quarto  casu  accessum  ad  C  prohibet  ex  parte  I  attractio 
IK,  &  ex  parte  L  repulsio  LM.  Sed  quoniam,  si  eae  crescant  in  ratione  reciproca  minus, 
quam  simplici  distantiarum  CI,CL  ;  area  FKICD,  vel  GMLCD  erit  finita,  adeoque 
punctum  impulsum  versus  C  velocitate  majore,  quam  quae  respondeat  illi  areae,  debet 
transire  per  omnes  virium  magnitudines  usque  ad  vim  absolute  infinitam  in  C,  quae  ibi 
[86]  praeterea  &  attractiva  esse  deberet,  &  repulsiva,  limes  videlicet  omnium  &  attracti- 
varum,  &  repulsivarum  ;  idcirco  ne  hie  quidem  casus  admitti  debet,  nisi  cum  hac 
conditione,  ut  ordinata  crescat  in  ratione  reciproca  simplici  distantiarum  a  C,  vel  etiam 
majore,  ut  nimirum  area  infinita  evadat,  &  accessum  a  puncto  C  prohibeat. 


A  THEORY  OF  NATURAL  PHILOSOPHY 

may  occur,  not  with  evanescence  of  the  forces,  but  through  the  forces  increasing  indefinitely, 
that  is  to  say  through  asymptotic  arcs  of  the  curve.  We  said  above,  in  Note  (»)  to  Art. 
1 68,  when  an  asymptotic  arm  goes  off  to  infinity,  there  must  be  another  asymptotic  arm 
returning  from  infinity  having  the  same  straight  line  for  an  asymptote  ;  &  it  may  return 
in  four  different  positions,  which  depend  on  the  two  parts  of  the  straight  line  &  the  two 
sides  of  each  part  of  the  straight  line.  But,  since  our  curve  must  always  go  forward,  we 
said  that  for  it  there  remained  only  two  out  of  these  four  positions,  for  any  arm  going  off 
to  infinity ;  that  is  to  say,  those  in  which  the  return  is  made  on  the  opposite  side  of  the 
straight  line.  However,  since,  whilst  the  curve  goes  forward,  either  a  repulsive  or  an 
attractive  arm  can  go  off  to  infinity,  here  again  we  must  have  four  possible  cases,  represented 
in  Figs.  16,  17,  18,  19,  in  all  of  which  ACB  is  the  axis,  DCD'  the  asymptote,  EKF  the 
arm  going  off  to  infinity,  &  GMH  the  arm  returning  from  infinity. 


I   C 


B 


n  cr 

FIG  19. 


1 86.  In  Fig.  1 6, to  a  repulsive  arm  EKF  there  succeeds  an  arm  that  is  also  repulsive; 
in  Fig.  17,  to  a  repulsive  succeeds  an  attractive  ;   in  Fig.  18,  to  an  attractive  succeeds  an 
attractive  ;  and  in  Fig.  19,  to  an  attractive  succeeds  a  repulsive.     The  first  &  third  cases 
correspond  to  contacts.     For,  just  as  in  contact,  the  force  vanished,  but  did  not  change 
its  direction,  so  here  also  the  force  indeed  becomes  infinite  but  does  not  change  its  direction. 
In  Fig.  1 6,  to  the  repulsion  IK  there  succeeds  the  repulsion  LM,  &  in  Fig.   18  to  an 
attraction    an    attraction  ;    &  thus    these  two  cases  cannot  have  any  limit-points.     But 
the  second  &  fourth  cases  certainly  have  limit-points ;    for,  in  Fig.  17,  to  the  repulsion 
IK    there  succeeds  the   attraction   LM,  &  in  Fig.   19  to  an  attraction  a  repulsion  ;    & 
thus  the   second  case  contains  a  limit-point    of   cohesion,  &  the  fourth  a  limit-point  of 
non-cohesion. 

187.  Out  of  these  cases  I  think  that  all  except  the  last  must  be  barred  from  our  curve  ; 
&  even  with  that  all  arms  must  be  rejected  for  which  the  ordinates  increase  in  a  ratio 
less  than  the  simple  reciprocal  of  the  distances  from  the  limit-point.      My  reasons  for 
excluding  these  are  to  avoid  the  possibility  of  there  being  at  any  time  an  infinite  force 
(which  of  itself  I  consider  to  be  impossible),  &  because,  in  addition  to  that,  an  infinite 
force,  by  its  very  nature  necessitates  the  creation  by  it  of  an  infinite  velocity  in  a  finite  time. 
For,  in  the  first  &  second  cases,  a  point,  situated  at  the  distance  from  another  point  equal 
to  that  which  I  has  from  the  origin  of  abscissae,  would  go  off  to  C  through  all  stages  of 
forces  increased  indefinitely,  &  at  C  would  be  bound  to  have  an  infinite  force.     In  the 
third  case,  too,  the  same  thing  would  happen  to  a  point  situated  at  a  distance  equal  to  that 
of  L.     Now,  in  the  fourth  case,  the  approach  to  C  is  restrained,  from  the  side  of  I  by  the 
attraction  IK,  &  from  the  side  of  L  by  the  repulsion  LM.     However,  since,  if  these 
forces  increase  in  a  ratio  that  is  less  than  the  simple  reciprocal  ratio  of  the  distances  CI, 
CL,  then  the  area  FKICD  or  the  area  GMLCD  will  be  finite ;  £  thus  the  point,  being 
impelled  towards  C  with  a  velocity  that  is  greater  than  that  corresponding  to  the  area, 
must  pass  through  all  magnitudes  of  the  forces  up  to  a  force  that  is  absolutely  infinite  at 
C  ;  and  this  force  must  besides  be  both  attractive  &  repulsive,  the  limit  so  to  speak  of  all 
attractive   &    repulsive   forces.     Hence  not  even  this  case  is  admissible,  unless  with  the 
condition  that  the  ordinate  increases  in  the  simple  reciprocal  ratio  of  the  distances  from  C, 
or  in  a  greater  ;    that  is  to  say,  the  area  must  turn  out  to  be  infinite  and  so  restrain  the 
approach  towards  the  point  C. 


Four  kinds  of 
them  ;  two  corre- 
sponding to  contact, 
&  two  to  limit- 
points,  of  which  the 
one  is  a  limit-point 
of  cohesion  &  the 
other  of  non-cohe- 
sion. 


None  of  these  ex- 
cept the  last  admis- 
sible in  Nature  ;  & 
not  even  that  in 
general. 


148  PHILOSOPHIC  NATURAL!  S  THEORIA 

Transitus  per  eum  188.  Quando  habeatur  hie  quartus  casus  in  nostra  curva  cum  ea  conditione  ;    turn 

bills:  teqaibai  quidem  nullum  punctum  collocatum  ex  alters  parte  puncti  C  poterit  ad  alteram  transilire, 
distantiis    constet,  quacunque  velocitate  ad  accessum  impellatur  versus  alterum  punctum,  vel  ad  recessum 

eum  non  haben.  u    •  •  j  •  •  i  •         •    £    •  i    •    £    •  •  TJ 

ab  ipso,  impediente  transitum  area  repulsiva  mnnita,  vel  innnita  attractiva.  Inde  vero 
facile  colligitur,  eum  casum  non  haberi  saltern  in  ea  distantia,  quae  a  diametris  minimarum 
particularum  conspicuarum  per  microscopia  ad  maxima  protenditur  fixarum  intervalla 
nobis  conspicuarum  per  telescopia  :  lux  enim  liberrime  permeat  intervallum  id  omne. 
Quamobrem  si  ejusmodi  limites  asymptotici  sunt  uspiam,  debent  esse  extra  nostrae  sensibi- 
litatis  sphaeram,  vel  ultra  omnes  telescopicas  fixas,  vel  citra  microscopicas  moleculas. 

Transitus  ad  puncta  189.  Expositas  hisce,  quae  ad  curva  virium  pertinebant,  aggrediar  simpliciora  quaadam, 

iae,  &  massas.  maxime  notatu  digna  sunt,  ac  pertinent  ad  combinationem  punctorum  primo  quidem 


duorum,  turn  trium,  ac  deinde  plurium  in  massa  etiam  coalescentium,  ubi  &  vires  mutuas, 
&  motus  quosdam,  &  vires,  quas  in  alia  exercent  puncta,  considerabimus. 

in  limitibus  ;  icp.  Duo  puncta  posita  in  distantia  aequali  distantiae  limitis  cujuscunque  ab  origine 

?1  ^  '  abscissarum,  ut  in  fig.  i.  AE,  AG,  AI,  &c,  (immo  etiam  si  curva  alicubi  axem  tangat,  aequali 
distantiae  contactus  ab  eodem),  ac  ibi  posita  sine  ulla  velocitate,  quiescent,  ut  patet,  quia 
nullam  habebunt  ibi  vim  mutuam  :  posita  vero  extra  ejusmodi  limites,  incipient  statim 
ad  se  invicem  accedere,  vel  a  se  invicem  recedere  per  intervalla  aequalia,  prout  fuerint  sub 
arcu  attractivo,  vel  repulsive.  Quoniam  autem  vis  manebit  semper  usque  ad  proximum 
limitem  directionis  ejusdem  ;  pergent  progredi  in  ea  recta,  quae  ipsa  urgebat  prius,  usque 
ad  distantiam  limitis  proximi,  motu  semper  accelerate,  juxta  legem  expositam  num.  176, 
ut  nimirum  quadrata  velocitatum  integrarum,  quae  acquisitae  jam  sunt  usque  ad  quodvis 
momentum  (nam  velocitas  initio  ponitur  nulla)  respondeant  areis  clausis  inter  ordinatam 
respondentem  puncto  axis  terminanti  abscissam,  quae  exprimebat  distantiam  initio  motus, 
&  ordinatam  respondentem  puncto  axis  terminanti  abscissam,  quae  exprimit  distantiam 
pro  eo  sequent!  momento.  Atque  id  quidem,  licet  interea  occurrat  contactus  aliquis  ; 
quamvis  enim  in  eo  vis  sit  nulla,  tamen  superata  distantia  per  velocitatem  jam  acquisitam, 
statim  habentur  iterum  [87]  vires  ejusdem  directionis,  quae  habebatur  prius,  adeoque 
perget  acceleratio  prioris  motus. 


Motus  post  proxi-  lqI    prOximus  limes  erit  ems  generis,  cuius  generis  diximus  limites  cohaesioms,  in  quo 

mum  limitem  super-      .     .    7         .     ,.  .  ,J.  i  j   ^  •  •  •         i 

atum,  &  osciiiatio.  nimirum  si  distantia  per  repulsionem  augebatur,  succedet  attractio ;  si  vero  minuebatur 
per  attractionem,  succedet  e  contrario  repulsio,  adeoque  in  utroque  casu  limes  erit  ejusmodi, 
ut  in  distantiis  minoribus  repulsionem,  in  majoribus  attractionem  secum  ferat.  In  eo 
limite  in  utroque  casu  recessus  mutui,  vel  accessus  ex  praecedentibus  viribus,  incipiet, 
velocitas  motus  minui  vi  contraria  priori,  sed  motus  in  eadem  directione  perget  ;  donee 
sub  sequent!  arcu  obtineatur  area  curvae  aequalis  illi,  quam  habebat  prior  arcus  ab  initio 
motus  usque  ad  limitem  ipsum.  Si  ejusmodi  aequalitas  obtineatur  alicubi  sub  arcu 
sequente  ;  ibi,  extincta  omni  praecedenti  velocitate,  utrumque  punctum  retro  reflectet 
cursum ;  &  si  prius  accedebant,  incipient  a  se  invicem  recedere  ;  si  recedebant,  incipient 
accedere,  atque  id  recuperando  per  eosdem  gradus  velocitates,  quas  amiserant,  usque  ad 
limitem,  quern  fuerant  prsetergressa  ;  turn  amittendo,  quas  acquisiverant  usque  ad  dis- 
tantiam, quam  habuerant  initio  ;  viribus  nimirum  iisdem  occurrentibus  in  ingressu,  & 
areolis  curvae  iisdem  per  singula  tempuscula  exhibentibus  quadratorum  velocitatis  incre- 
menta,  vel  decrementa  eadem,  quae  fuerant  antea  decrementa,  vel  incrementa.  Ibi  autem 
iterum  retro  cursum  reflectent,  &  oscillabunt  circa  ilium  cohaesionis  limitem,  quern  fuerant 
praetergressa,  quod  facient  hinc,  &  inde  perpetuo,  nisi  aliorum  externorum  punctorum 
viribus  perturbentur,  habentia  velocitatem  maximam  in  plagam  utramlibet  in  distantia 
ipsius  illius  limitis  cohaesionis. 


Casus 


osdiiationis          jo.2.  Quod  si  ubi  primum  transgressa  sunt  proximum  limitem  cohaesionis,  offendant 

'  S  arcum  ita  minus  validum  praecedente,  qui  arcus  nimirum  ita  minorem  concludat  aream, 

quam  praecedens,  ut  tota  ejus  area  sit  aequalis,  vel  etiam  minor,  quam  ilia  praecedentis 

arcus  area,  quae  habetur  ab  ordinata  respondente  distantiae  habitse  initio  motus,  usque  ad 


A  THEORY  OF  NATURAL  PHILOSOPHY  149 

1 88.  When,  if  ever,  this  fourth  case  occurs  in  our  curve,  then  indeed  no  point  situated  Passage  through  a 
on  either  side  of  the  point  C  will  be  able  to  pass  through  it  to  the  other  side,  no  matter  L^dTslmpo'ssibiel 
what  the  velocity  with  which  it  is  impelled  to  approach  towards,  or  recede  from,  the  other  distances  at  which 
point ;    for  the  infinite  repulsive  area,  or  the  infinite  attractive  area,  will  prevent  such  $,ere  are^iio  such 
passage.     Now,  it  can  easily  be  derived  from  this,  that  this  case  cannot  happen  at  any  rate  limit-points. 

in  the  distance  lying  between  the  diameters  of  the  smallest  particles  visible  under  the 
microscope  &  the  greatest  distances  of  the  stars  visible  to  us  through  the  telescope ;  for 
light  passes  with  the  greatest  freedom  through  the  whole  of  this  interval.  Therefore,  if 
there  are  ever  any  such  asymptotic  limit-points,  they  must  be  beyond  the  scope  of  our 
senses,  either  superior  to  all  telescopic  stars,  or  inferior  to  microscopic  molecules. 

1 89.  Having  thus  set  forth  these  matters  relating  to  the  curve  of  forces,  I  will  now  w?  n°w  pass  on  to 
discuss  some  of  the  simpler  things  that  are  more  especially  worth  mentioning  with  regard  ^^s.of  matter>  & 
to  combination  of  points ;  &  first  of  all  I  will  consider  a  combination  of  two  points,  then 

of  three,  &  then  of  many,  coalescing  into  masses ;  &  with  them  we  will  discuss  their 
mutual  forces,  &  certain  motions,  and  forces,  which  they  exercise  on  other  points. 

190.  Two  points  situated  at  a  distance  apart  equal  to  the  distance  of  any  limit-point  Rest  at  Hmit- 
from  the  origin  of  abscissae,  like  AE,  AG,  AI,  &c.  in  Fig.  I   (or  indeed  also  where  the  r 

curve  touches  the  axis  anywhere,  equal  to  the  distance  of  the  point  of  contact  from  the  without  them. 

origin),  &  placed  in  that  position  without  any  velocity,  will  be  relatively  at  rest ;    this  is 

evident  from  the  fact  that  they  have  then  no  mutual  force  ;   but  if  they  are  placed  at  any 

other  distance,  they  will  immediately  commence  to  move  towards  one  another  or  away 

from  one  another  through  equal  intervals,  according  as  they  lie  below  an  attractive  or  a 

repulsive  arc.     Moreover,  as  the  force  always  remains  the  same  in  direction  as  far  as  the 

next  following  limit-point,  they  continue  to  move  in  the  same  straight  line  which  contained 

them  initially  as  far  as  the  distance  apart  equal  to  the  distance  of  the  next  limit-point 

from  the  origin,  with  a  motion  that  is  continually  accelerated  according  to  the  law  given 

in  Art.  176  ;  that  is  to  say,  in  such  a  manner  that  the  squares  of  the  whole  velocities  which 

have  been  already  acquired  up  to  any  instant  (for  the  velocity  at  the  commencement  is 

supposed  to  be  nothing)  will  correspond  to  the  areas  included  between  the  ordinate 

corresponding  to  the  point  of  the  axis  terminating  the  abscissa  which  the  distance  traversed 

since  motion  began  and  the  ordinate  corresponding  to  the  point  on  the  axis  terminating 

the  abscissa  which  expresses  the  distance  for  the  next  instant  after  it.     This  is  still  the  case, 

even  if  a  contact  should  occur  in  the  meantime.     For,  although  at  a  point  where  contact 

occurs  the  force  is  nothing,  yet,  this  distance  being  passed  by  the  velocity  already  acquired, 

immediately  afterwards  there  will  be  forces  having  the  same  direction  as  before  ;   and  thus 

the  acceleration  of  the  former  motion  will  proceed. 

191.  The  next  limit-point  will  be  one  of  the  kind  we  have  called  limit-points  of  cohesion,  Motion    after  the 
namely,  one  in  which,  if  the  distance  is  increased  by  repulsion,  then  attraction  follows ;  passed^osc^Sion5 
but  if  the  distance  is  diminished  by  attraction,  then  on  the  contrary  repulsion  will  follow ; 

&  thus,  in  either  case,  the  limit-point  will  be  of  such  a  kind,  that  it  gives  a  repulsion  at 
smaller  distances  &  an  attraction  at  larger.  In  this  limit-point,  in  either  case,  the  separation 
or  approach,  due  to  the  forces  that  have  preceded,  will  be  changed,  &  the  velocity  of  motion 
will  begin  to  be  diminished  by  a  force  opposite  to  the  original  force,  but  the  motion  will 
continue  in  the  same  direction  ;  until  an  area  of  the  curve  under  the  arc  that  follows  the 
limit-point  becomes  equal  to  the  area  under  the  former  arc  from  the  commencement  of 
the  motion  as  far  as  the  limit-point.  If  equality  of  this  kind  is  obtained  somewhere  under 
the  subsequent  arc,  then,  the  whole  of  the  preceding  velocity  being  destroyed,  both  the 
points  will  return  along  their  paths ;  &  if  at  the  start  they  approached  one  another,  they 
will  now  begin  to  recede  from  one  another,  or  if  they  originally  receded  from  one  another, 
they  will  now  commence  to  approach  ;  and  as  they  do  this,  they  will  regain  by  the  same 
stages  the  velocities  which  they  lost,  as  far  as  the  limit-point  which  they  passed  through  ; 
then  they  will  lose  those  which  they  had  acquired,  until  they  reach  the  distance 
apart  which  they  had  at  the  commencement.  That  is  to  say,  the  same  forces  occur  on 
the  return  path,  &  the  same  little  areas  of  the  curve  for  the  several  short  intervals  of  time 
represent  increments  or  decrements  of  the  squares  of  the  velocities  which  are  the  same 
as  were  formerly  decrements  or  increments.  Then  again  they  will  once  more  retrace  their 
paths,  &  they  will  oscillate  about  the  limit-point  of  cohesion  which  they  had  passed  through  ; 
&  this  they  will  do,  first  on  this  side  &  then  on  that,  over  &  over  again,  unless  they  are  disturbed 
by  forces  due  to  other  points  outside  them  ;  &  their  greatest  velocity  in  either  direction 
will  occur  at  a  distance  apart  equal  to  that  of  the  limit-point  of  cohesion  from  the  origin. 

192.  But  if,  when  they  first  passed  through  the  nearest  limit-point  of  cohesion,  they  The  case  of  a  larger 
happened  to  come  to  an  arc  representing  forces  so  much  weaker  than  those  of  the  preceding 

•ti-ii  f  •  ,  °  11  i  ft  T 

arc  that  the  whole  area  of  it  was  equal  to,  or  even  less  than,  the  area  of  the  preceding  arc, 
reckoning  from  the  ordinate  corresponding  to  the  distance  apart  at  the  commencement 


150  PHILOSOPHIC  NATURALIS  THEORIA 

limitem  ipsum ;  turn  vero  devenient  ad  distantiam  alterius  limitis  proximi  priori,  qui 
idcirco  erit  limes  non  cohaesionis.  Atque  ibi  quidem  in  casu  sequalitatis  illarum  arearum 
consistent,  velocitatibus  prioribus  elisis,  &  nulla  vi  gignente  novas.  At  in  casu,  quo  tota 
ilia  area  sequentis  arcus  fuerit  minor,  quam  ilia  pars  areae  praecedentis,  appellent  ad  dis- 
tantiam ejus  limitis  motu  quidem  retardate,  sed  cum  aliqua  velocitate  residua,  quam 
distantiam  idcirco  praetergressa,  &  nacta  vires  directionis  mutatse  jam  conspirantes  cum 
directione  sui  motus,  non,  ut  ante,  oppositas,  accelerabunt  motum  usque  ad  distantiam 
limitis  proxime  sequentis,  quam  praetergressa  precedent,  sed  motu  retardato,  ut  in  priore  ; 
&  si  area  sequentis  arcus  non  sit  par  extinguendae  ante  suum  finem  toti  [88]  velocitati, 
quae  fuerat  residua  in  appulsu  ad  distantiam  limitis  praecedentis  non  cohaesionis,  &  quae 
acquisita  est  in  arcu  sequent!  usque  ad  limitem  cohsesionis  proximum ;  turn  puncta 
appellent  ad  distantiam  limitis  non  cohaesionis  sequentis,  ac  vel  ibi  sistent,  vel  progredientur 
itidem,  eritque  semper  reciprocatio  quaedam  motus  perpetuo  accelerati,  turn  retardati ; 
donee  deveniatur  ad  arcum  ita  validum,  nimirum  qui  concludat  ejusmodi  aream,  ut  tota 
velocitas  acquisita  extinguatur  :  quod  si  accidat  alicubi,  &  non  accidat  in  distantia  alicujus 
limitis ;  cursum  reflectent  retro  ipsa  puncta,  &  oscillabunt  perpetuo. 


Velocitatis    muta-  193.  Porro  in  hujusmodi  motu  patet  illud,  dum  itur  a  distantia  limitis  cohaesionis 

^"^abeat^maxU  a^  distantiam  limitis  non   cohaesionis,  velocitatem    semper    debere    augeri  ;    turn    post 

mum,  &  minimum  transitum  per  ipsam  debere  minui,  usque  ad  appulsum  ad  distantiam  limitis  non  cohaesionis, 

extmgui  possit  adeOque  habebitur  semper  in  ipsa  velocitate  aliquod  maximum  in  appulsu  ad  distantiam 

limitis  cohaesionis,  &  minimum  in  appulsu  ad  distantiam  limitis  non  cohaesionis.     Quamo- 

brem  poterit  quidem  sisti  motus  in  distantia  limitis  hujus  secundi  generis  ;   si  sola  existant 

ilia  duo  puncta,  nee  ullum  externum  punctum  turbet  illorum  motum  :    sed  non  poterit 

sisti  in  distantia  limitis  illius  primi  generis  ;  cum  ad  ejusmodi  distantias  deveniatur  semper 

motu  accelerate.     Praeterea  patet  &  illud,  si  ex  quocunque  loco  impellantur  velocitatibus 

aequalibus  vel  alterum  versus  alterum,  vel  ad  partes  oppositas,  debere  haberi  reciprocationes 

easdem  auctis  semper  aeque  velocitatibus  utriusque,  dum  itur  versus  distantiam  limitis 

primi  generis,  &  imminutis,  dum  itur  versus  distantiam  limitis  secundi  generis. 


oscMatlo°S  ma/**?  X94'  Patet  &  illud,  si  a  distantia  limitis  primi  generis  dimoveantur  vi  aliqua,  vel  non 

esse  debeat,  &  unde  ita  uigenti  velocitate  impressa,  oscillationem  fore  perquam  exiguam,  saltern  si  quidam 
" 


CJUS  mag  vah"dus  fuerit  limes  ;  nam  velocitas  incipiet  statim  minui,  &  ei  vi  statim  vis  contraria 
invenietur,  ac  puncta  parum  dimota  a  loco  suo,  turn  sibi  relicta  statim  retro  cursum  reflect- 
ent. At  si  dimoveantur  a  distantia  limitis  secundi  generis  vi  utcunque  exigua  ;  oscillatio 
erit  multo  major,  quia  necessario  debebunt  progredi  ultra  distantiam  sequentis  limitis 
primi  generis,  post  quern  motus  primo  retardari  incipiet.  Quin  immo  si  arcus  proximus 
hinc,  &  inde  ab  ejusmodi  limite  secundi  generis  concluserit  aream  ingentem,  ac  majorem 
pluribus  sequentibus  contrariae  directionis,  vel  majorem  excessu  eorundem  supra  areas 
interjacentes  directionis  suae  ;  turn  vero  oscillatio  poterit  esse  ingens  :  nam  fieri  poterit, 
ut  transcurrantur  hinc,  &  inde  limites  plurimi,  antequam  deveniatur  ad  arcum  ita  validum, 
ut  velocitatem  omnem  elidat,  &  motum  retro  reflectat.  Ingens  itidem  oscillatio  esse 
poterit,  si  cum  ingenti  vi  dimoveantur  puncta  a  distantia  limitum  generis  utriuslibet  ;  ac 
res  tota  pendet  a  velocitate  initiali,  &  ab  areis,  quae  post  oc-[8Q]-currunt,  &  quadratum 
velocitatis  vel  augent,  vel  minuunt  quantitate  sibi  proportionali. 


Accessum     debere  195.  Utcunque  magna  sit  velocitas,  qua  dimoveantur  a  distantia  limitum  ilia  duo 

swt^  saltern  a^pmno  puncta>  utcunque  validos  inveniant  arcus  conspirantes  cum  velocitatis  directione,  si  ad 
recess um  posse  se  invicem  accedunt,  debebunt  utique  alicubi  motum  retro  reflectere,  vel  saltern  sistere, 
cas^'^o^bms  1™  saltern  advenient  ad  distantias  illas  minimas,  quae  respondent  arcui  asymptotico, 
exiguae  differentiae  cujus  area  est  capax  cxtinguendse  cujuscunque  velocitatis  utcunque  magnae.  At  si 
velocitatis  ingentis.  rece(jant  a  se  mvicem,  fieri  potest,  ut  deveniant  ad  arcum  aliquem  repulsivum  validissimum, 
cujus  area  sit  major,  quam  omnis  excessus  sequentium  arearum  attractivarum  supra  repul- 


A  THEORY  OF  NATURAL  PHILOSOPHY  151 

of  the  motion  up  to  the  limit-point ;  then  indeed  they  will  arrive  at  a  distance  apart  equal 
to  that  of  the  limit-point  next  following  the  first  one,  which  will  therefore  be  a  limit-point  of 
non-cohesion.  Here  they  will  stop,  in  the  case  of  equality  between  the  areas  in  question ; 
for  the  preceding  velocities  have  been  destroyed  &  no  fresh  ones  will  be  generated.  But 
in  the  case  when  the  whole  of  the  area  under  the  second  arc  is  less  than  the  said  part  of 
the  first  area,  they  will  reach  a  distance  apart  equal  to  that  of  the  limit-point  with  a  motion 
that  is  certainly  diminished  ;  but  some  velocity  will  be  left,  &  this  distance  will  therefore 
be  passed,  &  the  points,  coming  under  the  influence  of  forces  changed  in  direction  so  that  they 
now  act  in  the  same  sense  as  their  own  motion,  will  accelerate  their  motion  as  far  as  the 
next  following  limit-point ;  &  having  passed  through  this  they  will  go  on,  but  with 
retarded  motion  as  in  the  first  case.  Then,  if  the  area  of  the  subsequent  arc  is  not  capable 
before  it  ends  of  destroying  the  whole  of  the  velocity  which  remained  on  attaining  the 
distance  of  the  preceding  limit-point  of  non-cohesion,  &  that  which  was  acquired  in  the 
arc  that  followed  it  up  to  the  next  limit-point  of  cohesion,  then  the  points  will  move  to  a 
distance  apart  equal  to  that  of  the  next  following  limit-point  from  the  origin,  &  will  either 
stop  there  or  proceed  ;  &  there  will  always  be  a  repetition  of  the  motion,  continually 
accelerated  &  retarded.  Until  at  length  it  comes  to  an  arc  so  strong,  that  is  to  say,  one 
under  which  the  area  is  such,  that  the  whole  velocity  acquired  is  destroyed  ;  &  when  this 
happens  anywhere,  &  does  not  happen  at  a  distance  equal  to  that  of  any  limit-point,  then 
the  points  will  retrace  their  paths  &  oscillate  continuously. 

193.  Further  in  this  kind  of  motion  it  is  clear  that  along  the  path  from  the  distance  Alternate   changes 
of  a  limit-point  of  cohesion  to  a  limit-point  of  non-cohesion  the  velocity  is  bound  to  be  of  velocity ;  where 
always  increasing  ;  then  after  passing  through  the  latter  it  must  decrease  up  to  its  arrival  at  the  value!  &  ""a^mln? 
distance  of  a  limit-point  of  non-cohesion.     Thus,  there  will  always  be  in  the  velocity  a  Pum  value ;  where 
maximum  on  arrival  at  a  distance  equal  to  that  of  a  limit-point  of  cohesion,  &  a  minimum  '  maybe  estr°yed. 
on  arrival  at  a  distance  of  a  limit-point  of  non-cohesion.     Hence  indeed  the  motion  may 

possibly  cease  at  a  limit-point  of  this  second  kind,  if  the  two  points  exist  by  themselves, 
&  no  other  point  influences  their  motion  from  without.  But  it  cannot  cease  at  a  distance 
of  a  limit-point  of  the  first  kind ;  for  it  will  always  arrive  at  distances  of  this  kind  with 
an  accelerated  motion.  Moreover  it  is  also  clear  that,  if  they  are  urged  from  any  given 
position  with  equal  velocities,  either  towards  one  another  or  in  opposite  directions,  the 
same  alternations  must  be  had  as  before,  the  velocities  being  increased  equally  for  each 
point  whilst  they  are  moving  up  to  a  distance  of  a  limit-point  of  the  first  kind,  &  diminished 
whilst  they  are  moving  up  to  a  distance  of  a  limit-point  of  the  second  kind. 

194.  It  is  evident  also  that,  if  the  points  are  moved  from  a  distance  apart  equal  to  that  of  The  limit-points 
a  limit-point  of  the  first  kind  by  some  force  (especially  when  the  velocity  thus  impressed  oscfflation  mus^be 
is  not  extremely  great),  then  the  oscillation  will  be  exceedingly  small,  at  least  so  long  as  the  krger;  &  the  thing 
limit-point  is  a  fairly  strong  one.     For  the  velocity  will  commence  to  be  diminished  tude" 
immediately,  &  to  the  force  another  force  will  be  obtained  at  once,  acting  in  opposition 

to  it ;  &  the  points,  being  moved  but  little  from  their  original  position,  will  immediately 
afterwards  retrace  their  paths  if  left  to  themselves.  But  if  they  are  moved  from  a  distance 
apart  equal  to  that  of  a  limit-point  of  the  second  kind  by  any  force,  no  matter  how  small, 
then  the  oscillation  will  be  much  greater  ;  for,  of  necessity,  they  are  bound  to  go  on  beyond 
the  distance  equal  to  that  of  the  next  following  limit-point  of  the  first  kind  ;  &  not  until 
this  has  been  done,  will  the  motion  begin  to  be  retarded.  Nay,  if  the  next  arc  on  each 
side  of  such  a  limit-point  of  the  second  kind  should  include  a  very  large  area,  and  one  that 
is  greater  than  several  of  those  subsequent  to  them,  which  are  opposite  in  direction,  or 
greater  than  the  excess  of  these  over  the  intervening  areas  that  are  in  the  same  direction, 
then  indeed  the  oscillation  may  be  exceedingly  large.  For  it  may  be  that  very  many 
limit-points  on  either  side  are  traversed  before  an  arc  is  arrived  at,  which  is  sufficiently 
strong  to  destroy  the  whole  of  the  velocity  &  reverse  the  direction  of  motion.  A  very 
large  oscillation  will  also  be  possible,  if  the  points  are  moved  from  a  distance  apart  equal  to 
that  of  a  limit-point  of  either  kind  by  an  exceedingly  large  force.  The  whole  thing  depends 
on  the  initial  velocity  &  the  areas  which  occur  subsequently,  &  either  increase  or  decrease 
the  square  of  the  velocity  by  a  quantity  that  is  proportional  to  the  areas  themselves. 

195.  However  great  the  velocity  may  be,  with  which  the  two  points  are  moved  from  Approach  is  bound 
a  distance  equal  to  that  of  any  limit-point,  no  matter  how  strong  are  the  arcs  they  come  *^gS \* ttahney %££ 
upon,  which  are  in  the  same  direction  as  that  of  the  velocity ;   yet,  if  they  approach  one  repulsive  arc,  but 
another,  they  are  bound  somewhere  to  have  their  motion  reversed,  or  at  least  to  come  onpaSde°nnitety ;  Sa 
to  rest ;  for,  at  all  events,  they  must  finally  attain  to  those  very  small  distances  that  correspond  noteworthy    case 
to  an  asymptotic  arm,  the  area  of  which  is  capable  of  destroying  any  velocity  whatever,  arSerencTfar  aSvery 
no  matter  how  great.     But,  if  they  recede  from  one  another,  it  may  happen  that  they  come  great  velocity. 

to  some  very  strong  repulsive  arc,  the  area  of  which  is  greater  than  the  whole  of  the  excess 
of  the  subsequent  attractive  arcs  above  those  that  are  repulsive,  as  far  as  the  very  weak 


152 


PHILOSOPHIC  NATURALIS  THEORIA 


sivas,  usque  ad  languidissimum  ilium  arcum  postremi  cruris  gravitatem  exhibentis.  Turn 
vero  motus  acquisitus  ab  illo  arcu  nunquam  poterit  a  sequentibus  sisti,  &  puncta  ilia  recedent 
a  se  invicem  in  immensum  :  quin  immo  si  ille  arcus  repulsivus  cum  sequentibus  repulsivis 
ingentem  habeat  areae  excessum  supra  arcus  sequentes  attractivos ;  cum  ingenti  velocitate 
pergent  puncta  in  immensum  recedere  a  se  invicem  ;  &  licet  ad  initium  ejus  tarn  validi 
arcus  repulsivi  deveniant  puncta  cum  velocitatibus  non  parum  diversis ;  tamen  velocitates 
recessuum  post  novum  ingens  illud  augmentum  erunt  parum  admodum  discrepantes  a 
se  invicem  :  nam  si  ingentis  radicis  quadrate  addatur  quadratum  radicis  multo  minoris, 
quamvis  non  exiguae  ;  radix  extracta  ex  summa  parum  admodum  differet  a  radice  priore. 


Demonstratio     ad- 
modum simplex. 


A 


BD 


FIG.  20. 


Quid  accidat  binis 
punctis,  cum  sunt 
sola,  quid  possit 
accidere  actionibus 
aliorum  externis. 


Si  limites  sint  a  se 
invicem  r  e  m  o  t  i, 
m  u  t  a  t  a  multum 
distantia  r  e  d  i  ri 
retro  :  secus,  si 
sint  proximi. 


196.  Id  quidem  ex  Euclidea  etiam  Geometria  manifestum  fit.     Sit  in  fig.  20  AB 
linea  longior,  cui  addatur  ad  perpendiculum  BC,  multo  minor,  quam  fit  ipsa  ;  turn  centre  A, 
intervallo  AC,   fiat    semicirculus    occurrens   AB   hinc,  &   inde  in   E,  D.     Quadrate   AB 
addendo  quadratum  BC  habetur  quadratum  AC,  sive  AD ;    &  tamen  haec  excedit  prsece- 
dentem  radicem  AB  per  solam  BD,  quae  semper  est  minor,  quam  BC,  &  est  ad  ipsam,  ut 
est  ipsa  ad  totam  BE.     Exprimat  AB  velocitatem,  quam  in  punctis  quiescentibus  gigneret 
arcus  ille  repulsivus  per  suam  aream,  una  cum  differentia  omnium  sequentium  arcuum 
repulsivorum  supra  omnes  sequentes  attractivos  :  exprimat  autem  BC  velocitatem,  cum 
qua     advenitur     ad    distantiam    respondentem 

initio  ejus  arcus  :  exprimet  AC  velocitatem, 
qu33  habebitur,  ubi  jam  distantia  evasit  major, 
&  vis  insensibilis,  ac  ejus  excessus  supra 
priorem  AB  erit  BD,  exiguus  sane  etiam  re- 
spectu  BC,  si  BC  fuerit  exigua  respectu  AB, 
adeoque  multo  magis  respectu  AB  ;  &  ob 
eandem  rationem  perquam  exigua  area  sequentis 
cruris  attractivi  ingentem  illam  jam  acqui- 
sitam  velocitatem  nihil  ad  sensum  mutabit,  quae 
permanebit  ad  sensum  eadem  post  recessum  in 
immensum. 

197.  Haec  accident  binis  punctis  sibi  relictis,  vel  impulsis  [90]  in  recta,  qua  junguntur, 
cum  oppositis  velocitatibus   aequalibus,   quo  casu  etiam  facile  demonstratur,   punctum, 
quod  illorum  distantiam  bifariam  secat,   debere  quiescere  ;    nunquam  in  hisce  casibus 
poterit  motus  extingui  in  adventu  ad  distantiam  limitis  cohaesionis,  &  multo  minus  poterunt 
ea  bina  puncta  consistere  extra  distantiam  limitis  cujuspiam,  ubi  adhuc  habeatur  vis  aliqua 
vel  attractiva,  vel  repulsiva.     Verum  si  alia  externa  puncta  agant  in  ilia,  poterit  res  multo 
aliter  se  habere.     Ubi  ex.  gr.  a  se  recedunt,  &  velocitates  recessus  augeri  deberent  in  accessu 
ad  distantiam  limitis  cohaesionis ;    potest  externa  compressio  illam  velocitatem  minuere, 
&  extinguere  in  ipso  appulsu  ad  ejusmodi  distantiam.     Potest  externa  compressio  cogere 
ilia  puncta  manere  immota  etiam  in  ea  distantia,  in  qua  se  validissime  repellunt,  uti  duae 
cuspides  elastri    manu    compressae  detinentur  in  ea  distantia,  a  qua  sibi    relictas    statim 
recederent  :   &  simile  quid  accidere  potest  vi  attractivae  per  vires  externas  distrahentes. 

198.  Turn  vero  diligenter  notandum  discrimen  inter  casus  varies,  quos  inducit  varia 
arcuum  curvae  natura.     Si  puncta  sint  in  distantia  alicujus  limitis  cohassionis,  circa  quern 
sint  arcus  amplissimi,  ita,  ut  proximi  limites  plurimum  inde  distent,  &  multo  magis  etiam, 
quam  sit  tota  distantia  proximi  citerioris  limitis  ab  origine  abscissarum  ;    turn  poterunt 
externa  vi  comprimente,  vel  distrahente  redigi  ad  distantiam  multis  vicibus  minorem, 
vel  majorem  priore  ita,  ut  semper  adhuc  conentur  se  restituere  ad  priorem  positionem 
recedendo,  vel  accedendo,  quod  nimirum  semper  adhuc  sub  arcu  repulsive  permaneat,  vel 
attractive.     At  si  ibi  frequentissimi  limites,  curva  saepissime  secante  axem  ;    turn  quidem 
post  compressionem,  vel  distractionem  ab  externa  vi  factam,  poterunt  sisti  in  multo  minore, 
vel  majore  distantia,  &  adhuc  esse  in  distantia  alterius  limitis  cohaesionis  sine  ullo  conatu 
ad  recuperandum  priorem  locum. 


Superiorum  usus  in  199.  Haec  omnia  aliquanto  fusius  considerare  libuit,  quia  in  applicatione  ad  Physicam 

magno  usui  erunt  infra  haec  ipsa,  &  multo  magis  hisce  similia,  quae  massis  respondent 
habentibus  utique  multo  uberiores  casus,  quam  bina  tantummodo  habeant  puncta.  Ilia 
ingens  agitatio  cum  oscillationibus  variis,  &  motibus  jam  acceleratis,  jam  retardatis,  jam 
retro  reflexis,  fermentationes,  &  conflagrationes  exhibebit  :  ille  egressus  ex  ingenti  arcu 


A  THEORY  OF  NATURAL  PHILOSOPHY  153 

arc  of  the  last  branch  which  represents  gravity.  Then  indeed  the  velocity  acquired  through 
that  arc  can  never  be  stopped  by  the  subsequent  arcs,  &  the  points  will  recede  from  one 
another  to  an  immense  distance.  Nay  further,  if  that  repulsive  arc  taken  together  with 
the  subsequent  repulsive  arcs  has  a  very  great  excess  of  area  over  the  subsequent  attractive 
arcs,  then  the  points  will  continue  to  recede  to  an  immense  distance  from  one  another 
with  a  very  great  velocity  ;  &,  although  points  arrive  at  this  repulsive  arc,  which  is  so  strong, 
with  considerably  different  velocities,  yet  the  velocities  after  this  fresh  &  exceedingly  great 
increase  will  be  very  little  different  from  one  another.  For,  if  to  the  square  of  a  very 
great  number  there  is  added  the  square  of  a  number  that  is  much  less,  although 
not  in  itself  very  small,  the  square  root  of  the  sum  differs  very  little  from  the  first 
number. 

196.  This  indeed  is  very  evident  from  Euclidean   geometry  even.     In    Fig.  20,  let  The  demonstration 
AB  be  a  fairly  long  line,  to  which  is  added,   perpendicular  to  it,  BC,  which  is  much  less  ls  Perfectly  sunPIe- 
than  AB.     Then,  with  centre  A,  &  radius  AC,  describe  a  semicircle  meeting  AB  on  either 

side  in  E  &  D.  On  adding  the  square  on  BC  to  the  square  on  AB,  we  get  the  square  on 
AC  or  AD  ;  &  yet  this  exceeds  the  former  root  AB  by  BD  only,  which  is  always  less  than 
BC,  bearing  the  same  ratio  to  it  as  BC  bears  to  the  whole  length  BE.  Suppose  that 
AB  represents  the  velocity  which  the  repulsive  arc,  owing  to  the  area  under  it,  would 
generate  in  points  initially  at  rest,  together  with  the  difference  for  all  the  subsequent 
repulsive  arcs  over  all  the  subsequent  attractive  arcs  ;  also  let  BC  represent  the  velocity 
with  which  the  distance  corresponding  to  the  beginning  of  this  arc  is  reached  ;  then  AC 
will  represent  the  velocity  which  is  obtained  when  the  distance  has  already  become  of 
considerable  amount,  &  the  force  insensible.  Now  the  excess  of  this  above  the  former 
velocity  AB  will  be  represented  by  BD  ;  &  this  is  really  very  small  compared  with  BC, 
if  BC  were  very  small  compared  with  AB  ;  &  therefore  much  more  so  with  regard  to  AB. 
For  the  same  reason,  the  very  small  area  under  the  subsequent  attractive  branch  will  not 
sensibly  change  the  very  great  velocity  acquired  so  far ;  this  will  remain  sensibly  the  same 
after  recession  to  a  huge  distance. 

197.  These  things  will  take  place  in  the  case  of  two  points  left  to  themselves,  or  impelled  What  may  happen 
along  the  straight  line  joining  them  with  velocities  that  are  equal  &  opposite  ;    in  such  the^areT^then? 
a  case  it  can  be  easily  proved  that  the  middle  point  of  the  distance  between  them  is  bound  selves ;  what  may 
to  remain  at  rest.     The  motion  in  the  cases  we  have  discussed  can  never  be  destroyed  ^teif11  under  ththe 
altogether  on  arrival  at  a  distance  equal  to  that  of  a  limit-point  of  cohesion,  &  much  less  actions    of    other 
will  the  two  points  be  able  to  stop  at  a  distance  apart  that  is  not  equal  to  that  of  some  them3  extemal  to 
limit-point,  as  far  as  which  there  is  some  force  acting,  either  attractive  or  repulsive.     But 

if  other  external  points  act  upon  them,  we  may  have  altogether  different  results.  For 
instance,  in  a  case  where  they  recede  from  one  another,  &  the  velocities  would  therefore 
be  bound  to  be  increased  as  they  approached  a  distance  equal  to  that  of  a  limit-point  of 
cohesion,  an  external  compression  may  diminish  that  velocity,  &  completely  destroy  it 
as  it  approaches  the  distance  of  that  limit-point.  An  external  compression  may  even 
force  the  points  to  remain  motionless  at  a  distance  for  which  they  repel  one  another  very 
strongly  ;  just  as  the  two  ends  of  a  spring  compressed  by  the  hands  are  kept  at  a  distance 
from  which  if  left  to  themselves  they  will  immediately  depart.  A  similar  thing  may  come 
about  in  the  case  of  an  attractive  force  when  there  are  external  tensile  forces. 

198.  Now,  a  careful  note  must  be  made  of  the  distinctions  between  the  various  cases,  if  the  limit-points 
which  arise  from  the  various  natures  of  the  arcs  of  the  curve.     If  our  points  are  at  a  distance  fje  far  apart,  there 

.,_.  i       •  i  111  is     a    tenaency     to 

of  any  limit-point  of  cohesion,  on  each  side  of  which  the  arcs  are  very  wide,  so  that  the  return  if  the   dis- 
nearest  limit-points  are  very  far  distant  from  it,  &  also  much  more  so  than  the  nearest  j^sfderabSH-han^e; 
limit-point  to  the  left  is  distant  from  the  origin  of  abscissae  ;   they  may,  under  the  action  but  this  is  not  the' 
of  an  external  force  causing  either  compression  or  tension,  be  reduced  after  many  alternations  l^tsare  very'dose 
to  a  distance,  either  less,  or  greater,  than  the  original  distance,  in  such  a  way  that  they  together, 
will  always  strive  however  to  revert  to  their  old  position  by  receding  from  or  approaching 
towards  one  another  ;  for  indeed  they  will  still  always  remain  under  a  repulsive,  or  an 
attractive  arc.     But  if,  near  the  limit-point  in  question,  the  limit-points  on  either  side 
occur  at  very  frequent  intervals  ;    then  indeed,  after  compression,  or  separation,   caused 
by  an  external  force,  they  may  stop  at  a  much  less,  or  a  much  greater,  distance  apart,  & 
still  be  at  a  distance  equal  to  that  of  another  limit-point  of  cohesion,  without  there  being 
any  endeavour  to  revert  to  their  original  position. 

199.  All  these  considerations  I  have  thought  it  a  good  thing  to  investigate  somewhat  The  use  of  the 
at  length  ;  for  they  will  be  of  great  service  later  in  the  application  of  the  Theory  to  physics,  ^°ve  facts  "* phy" 
both  these  considerations,  &  others  like  them  to  an  even  greater  degree  ;    namely  those 

that  correspond  to  masses,  for  which  indeed  there  are  far  more  cases  than  for  a  system 
of  only  two  points.  The  great  agitation,  with  its  various  oscillations  &  motions  that  are 
sometimes  accelerated, sometimes  retarded,  &  sometimes  reversed,  will  represent  fermentations 


154  PHILOSOPHIC  NATURALIS  THEORIA 

repulsive  cum  velocitatibus  ingentibus,  quas  ubi  jam  ad  ingentes  deventum  est  distantias, 
parum  admodum  a  se  invicem  differant,  nee  ad  sensum  mutentur  quidquam  per  immensa 
intervalla,  luminis  emissionem,  &  propagationem  uniformem.,  ac  ferme  eandem  celeritatem 
in  quovis  ejusdem  speciei  radio  fixarum,  Solis,  flammse,  cum  exiguo  discrimine  inter 
diversos  coloratos  radios ;  ilia  vis  permanens  post  compressionem  ingentem,  vel  diffractionem 
elasticitati  explicandae  in-[9i]-serviet ;  quies  ob  frequentiam  limitum,  sine  conatu  ad 
priorem  recuperandam  figuram,  mollium  corporum  ideam  suggeret ;  quae  quidem  hie 
innuo  in  antecessum,  ut  magis  haereant  animo,  prospicienti  jam  nine  insignes  eorum  usus. 


Motus    binorum  2OO.  Quod  si  ilia  duo  puncta  proiiciantur  oblique  motibus  contrarns,  &  aequalibus 

punctorum  oblique  j-          •  •          -11      j  i          rr    • 

projectorum.  Per  directiones,  quae  cum  recta  jungente  ipsa  ilia  duo  puncta  angulos  aequales  efficiant ; 

turn  vero  punctum,  in  quo  recta  ilia  conjungens  secatur  bifariam,  manebit  immotum  ; 
ipsa  autem  duo  puncta  circa  id  punctum  gyrabunt  in  curvis  lineis  aaqualibus,  &  contrariis, 
quae  data  lege  virium  per  distantias  ab  ipso  puncto  illo  immoto  (uti  daretur,  data  nostra 
curva  virium  figurae  i,  cujus  nimirum  abscissae  exprimunt  distantias  punctorum  a  se  invicem, 
adeoque  eorum  dimidiae  distantias  a  puncto  illo  medio  immoto)  invenitur  solutione  pro- 
blematis  a  Newtono  jam  olim  soluti,  quod  vocant  inversum  problema  virium  centralium, 
cujus  problematis  generalem  solutionem  &  ego  exhibui  syntheticam  eodem  cum  Newtoniana 
recidentem,  sed  non  nihil  expolitam,  in  Stayanis  Supplementis  ad  lib.  §  19. 

Casus,  in  quo  duo  201.  Hie  illud  notabo  tantummodo,   inter  infinita   curvarum  genera,  quae  describi 

sc1Hberedebe&pira<ies  Possunt5  cum  nulla  sit  curva,  quas  assumpto  quovis  puncto  pro  centre  virium  describi 
circa  medium  im-  non  possit  cum  quadam  virium  lege,  quae  definitur  per  Problema  directum  virium 
centralium,  esse  innumeras,  quas  in  se  redeant,  vel  in  spiras  contorqueantur.  Hinc  fieri 
potest,  ut  duo  puncta  delata  sibi  obviam  e  remotissimis  regionibus,  sed  non  accurate  in 
ipsa  recta,  quae  ilia  jungit  (qui  quidem  casus  accurati  occursus  in  ea  recta  est  infinities 
improbabilior  casu  deflexionis  cujuspiam,  cum  sit  unicus  possibilis  contra  infinites),  non 
recedant  retro,  sed  circa  punctum  spatii  medium  immotum  gyrent  perpetuo  sibideinceps 
semper  proxima,  intervallo  etiam  sub  sensus  non  cadente  ;  qui  quidem  casus  itidem 
diligenter  notandi  sunt,  cum  sint  futuri  usui,  ubi  de  cohaesione,  &  mollibus  corporibus 
agendum  erit. 

Theorema  de  statu  202.  Si  utcunque  alio  modo  projiciantur  bina  puncta  velocitatibus  quibuscunque ; 

enSfterm1nU'maSs  P°test  facile  ostendi  illud  :    punctum,  quod  est  medium  in  recta  jungente  ipsa,  debere 

centri  gravitatis  quiescere,  vel  progredi  uniformiter  in  directum,  &  circa  ipsum  vel  quietum,  vel  uniformiter 

progrediens,   debere    haberi   vel  illas   oscillationes,   vel  illarum  curvarum   descriptiones. 

Verum  id  generalius  pertinet  ad  massas  quotcunque,  &   quascunque,  quarum  commune 

gravitatis  centrum  vel  quiescit,  vel  progreditur  uniformiter  in  directum  a  viribus  mutuis 

nihil  turbatum.     Id  theorema  Newtonus  proposuit,  sed  non  satis  demonstravit.     Demon- 

strationem  accuratissimam,  ac  generalem  simul,&  non  per  casuum  inductionem  tantummodo, 

inveni,  ac  in  dissertation  e  De  Centra  Gravitatis  proposui,  quam  ipsam  demonstrationem 

hie  etiam  inferius  exhibebo. 

Accessum    aiterius  [92]  203.     Interea     hie    illud    postremo    loco    adnotabo,    quod     pertinet    ad    duorum 

quodvis ad  aiterius  punctorum  motum  ibi  usui  futurum  :  si  duo  puncta  moveantur  viribus  mutuis  tantummodo, 

aequari  recessui  ex  &  ultra  ipsa  assumatur  planum  quodcunque  ;    accessus  aiterius  ad  illud  planum  secundum 

directionem   quamcunque,    aequabitur   recessui   aiterius.     Id   sponte   consequitur   ex   eo, 

quod  eorum  absoluti  motus  sint  aequales,  &  contrarii ;    cum  inde  fiat,  ut  ad  directionem 

aliam  quamcunque  redacti  aequales  itidem  maneant,  &  contrarii,  ut  erant  ante.     Sed  de 

aequilibrio,  &  motibus  duorum  punctorum  jam  satis. 

Transitus  ad  syste-  204.  Deveniendo  ad  systema  trium  punctorum,  uti  etiam  pro  punctis  quotcunque, 

trium  "binagene™  res>  si  generaliter  pertractari  deberet,  reduceretur  ad  haac  duo  problemata,  quorum  alterum 
alia probiemata.  pertinet  ad  vires,  &  alterum  ad  motus  :  I.  Data  positions,  £5?  distantia  mutua  eorum  punc- 
torum, invenire  magnitudinem,  &  directionem  vis,  qua  urgetur  quodvis  ex  ipsis,  composites  a 
viribus,  quibus  urgetur  a  reliquis,  quarum  singularum  virium  lex  communis  datur  per  curvam 
figure  primce.  2.  Data  ilia  lege  virium  figures  -primes  invenire  motus  eorum  punctorum, 
quorum  singula  cum  datis  velocitatibus  projiciantur  ex  datis  locis  cum  datis  directionibus. 
Primum  facile  solvi  potest,  &  potest  etiam  ope  curva;  figurae  i  determinari  lex  virium 


A  THEORY  OF  NATURAL  PHILOSOPHY  155 

&  conflagrations.  The  starting  forth  from  a  very  large  repulsive  arc  with  very  great 
velocities,  which,  as  soon  as  very  great  distances  have  been  reached,  are  very  little  different 
from  one  another  ;  nor  are  they  sensibly  changed  in  the  slightest  degree  for  very  great 
intervals ;  this  will  represent  the  emission  &  uniform  propagation  of  light,  &  the  approximately 
equal  velocities  in  any  ray  of  the  same  kind  from  the  stars,  the  sun,  and  a  flame,  with  a 
very  slight  difference  between  rays  of  different  colours.  The  force  persisting  after 
compression,  or  separation,  will  serve  to  explain  elasticity.  The  lack  of  motion  due  to 
the  frequent  occurrence  of  limit-points,  without  any  endeavour  towards  recovering  the 
original  configuration,  will  suggest  the  idea  of  soft  bodies.  I  mention  these  matters  here 
in  anticipation,  in  order  that  they  may  the  more  readily  be  assimilated  by  a  mind  that 
already  sees  from  what  has  been  said  that  there  is  an  important  use  for  them. 

200.  But  if  the  two  points  are  projected  obliquely  with  velocities  that  are  equal  and  The  motion  of  two 
opposite  to  one  another,  in  directions  making  equal  angles  with  the  straight  line  joining  obikmei    pro^ected 
the    two    points  ;    then,  the  point  in  which  the    straight  line   joining    them  is  bisected 

will  remain  motionless ;  the  two  points  will  gyrate  about  this  middle  point  in  equal  curved 
paths  in  opposite  directions.  Moreover,  if  the  law  of  forces  is  given  in  terms  of  the  distances 
from  that  motionless  point  (as  it  will  be  given  when  our  curve  of  forces  in  Fig.  i  is  given, 
where  the  abscissae  represent  the  distances  of  the  points  from  one  another,  &  therefore 
the  halves  of  these  abscissae  represent -the  distances  from  the  motionless  middle  point), 
then  we  arrive  at  a  solution  of  the  problem  already  solved  by  Newton  some  time  ago,  which 
is  called  the  inverse  -problem  of  central  forces.  Of  this  problem  I  also  gave  a  general  synthetic 
solution  that  was  practically  the  same  thing  as  that  of  Newton,  not  altogether  devoid  of 
neatness,  in  the  Supplements  to  Stay's  Philosophy,  Book  3,  Art.  19. 

201.  At  present  I  will  only  remark  that,  amongst  the  infinite  number  of  different  The  case  in  which 
curves  that  can  be  described,  there  are  an  innumerable  number  which  will  either  re-enter  boumT^o^teswibe 
their  paths,  or  wind  in  spirals ;   for  there  is  no  curve  that,  having  taken  any  point  whatever  spirals  about   the 
for  the  centre  of  forces,  cannot  be  described  with  some  law  of  forces,  which  is  determined  j^°^°nless    mlddle 
by  the  direct  problem  of  central  forces.     Hence  it  may  happen  that  two  points  approaching 

one  another  from  a  long  way  off,  but  not  exactly  in  the  straight  line  joining  them — and 
the  case  of  accurate  approach  along  the  straight  line  joining  them  is  infinitely  more  improbable 
than  the  case  in  which  there  is  some  deviation,  since  the  former  is  only  one  possible  case 
against  an  infinite  number  of  others — then  the  points  will  not  reverse  their  motion  and 
recede,  but  will  gyrate  about  a  motionless  middle  point  of  space  for  evermore,  always 
remaining  very  near  to  one  another,  the  distance  between  them  not  being  appreciable  by 
the  senses.  These  cases  must  be  specially  noted  ;  for  they  will  be  of  use  when  we  come 
to  consider  cohesion  &  soft  bodies. 

202.  If  two  points  are  projected  in  any  manner  whatever  with  any  velocities  whatever,  Theorem    on    the 
it  can  readily  be  proved  that  the  middle  point  of  the  line  joining  them  must  remain  at  steady  state  of  the 

1       . , r      ,  .   ,      , .  r      ,     ,  ,  ,J .          P  ,       ,         .     .  central     point     &, 

rest  or  move  uniformly  in  a  straight  line  ;   and  that  about  this  point,  whether  it  is  at  rest  more     generally, 
or  is  moving  uniformly,  the  oscillations  or  descriptions  of  the  curved  paths,  referred  to  of  the  .cen,tre    of 

i        '      T>          i  •  11       •  i     •      r  c  gravity  in  the  case 

above,  must  take  place.     But  this,  more  generally,  is  a  property  relating  to  masses,  of  any  of  masses. 

number  or  kind,  for  which  the  common  centre  of  gravity  is  either  at  rest  or  moves  uniformly 

in  a  straight  line,  in  no  wise  disturbed  by  the  mutual  forces.     This  theorem  was  enunciated 

by  Newton,  but  he  did  not  give  a  satisfactory  proof  of  it.     I  have  discovered  a  most  rigorous 

demonstration,  &  one  that  is  at  the  same  time  general,  &  I   gave  it  in   the   dissertation 

De  Centra   Gravitatis ;    this   demonstration  I  will  also  give  here  in   the   articles    that 

follow. 

203.  Lastly,  I  will  here  mention  in  passing  something  that  refers  to  the  motion  of  The    approach   of 
two  points,  which  will  be  of    use  later,  in  connection  with  that  subject.     If  two  points  poLts  towards^y 
move  subject  to  their  mutual  forces  only,  &  any  plane  is  taken  beyond  them  both,  then  plane  is  equal  to 
the  approach  of  one  of  them  to  that  plane,  measured  in  any  direction,  will  be  equal  to  the  other^rom1  it  'on 
recession  of  the  other.     This  follows  immediately  from  the  fact  that  their  absolute  motions  account    of  'the 
are  equal  &  opposite  ;   for,  on  that  account,  it  comes  about  that  the  resolved  parts  in  any  mutual  force- 
other  direction  also  remain  equal  &  opposite,  as  they  were  to  start  with.     However,  I 

have  said  enough  for  the  present  about  the  equilibrium  &  motions  of  two  points. 

204.  When  we  come  to  consider  systems  of  three  points,  as  also  systems  of  any  number  Extension    to    a 
of  points,  the  whole  matter  in  general  will  reduce  to  these  two  problems,  of  which  the  system  of  three 

tt  11  .  T,    .  .   .  ,  points ;  two  general 

one  refers  to  forces  and  the  other  to  motions,     i.  Being  given  the  position  and  the  mutual  problems. 

distance  of  the  points,  it  is  required  to  find  the  magnitude  and  direction  of  the  force,  to  which 

any  one  of  them  is  subject ;    this  force  being  the  resultant  of  the  forces  due  to  the  remaining 

points,  and  each  of  these  latter  being  found  by  a  general  law  which  is  given  by  the  curve  of  Fig.  i. 

2.  Being  given  the  law  of  forces  represented  by  Fig.  I,  it  is  required  to  find  the  motions  of 

the  points,  when  each  of  them  is  projected  with  known  velocities  from  given  initial  positions 

in  given  directions.     The  first  of  these  problems  is  easily  solved  ;    and  also,  by  the  aid  of 


156  PHILOSOPHIC  NATURALIS  THEORIA 

generaliter  pro  omnibus  distantiis  assumptis  in  quavis  recta  positionis  datae,'  a  que  id  tarn 
geometrice  determinando  per  puncta  curvas,  quae  ejusmodi  legem  exhibeant,  ac  determinent 
sive  magnitudinem  vis  absolutae,  sive  magnitudines  binarum  virium,  in  quas  ea  concipiatur 
resoluta,  &  quarum  altera  sit  perpendicularis  data?  illi  rectse,  altera  secundum  illam  agat  ; 
quam  exhibendo  tres  formulas  analyticas,  quas  id  praestent.  Secundum  omnino  generaliter 
acceptum,  &  ita,  ut  ipsas  curvas  describendas  liceat  definire  in  quovis  casu  vel  constructione, 
vel  caculo,  superat  (licet  puncta  sint  tantummodo  tria)  vires  methodorum  adhuc  cognit- 
arum  :  &  si  pro  tribus  punctis  substituantur  tres  massae  punctorum,  est  illud  ipsum 
celeberrimum  problema  quod  appellant  trium  corporum,  usque  adeo  qusesitum  per  haec 
nostra  tempora,  &  non  nisi  pro  peculiaribus  quibusdam  casibus,  &  cum  ingentibus  limita- 
tionibus,  nee  adhuc  satis  promoto  ad  accurationem  calculo,  solutum  a  paucissimis  nostri 
asvi  Geometris  primi  ordinis,  uti  diximus  num.  122. 
Theorema  de  motu  205.  Pro  hoc  secundo  casu  illud  est  notissimum,  si  tria  puncta  sint  in  fig.  21  A,  C,  B, 

puncti  habentis  ac-    „      •,.      J    .        .  T.     i  •>•    •  IT-  •      T->  i 

tionem   cum    aiiis  &  distantia   AB  duorum  divisa  semper   bifanam   in  D,  ac  ducta    CD,   &   assumpto  ejus 
binis-  triente  DE,  utcunque   moveantur  eadem  puncta 

motibus    compositis     a     projectionibus   quibus- 

cunque,  &  mutatis  viribus ;    punctum  E  debere 

vel  quiescere  semper,  vel  progredi  in  directum 

motu    uniformi.     Pendet  id   a  general!  theore- 

mate    de     centre    gravitatis,    cujus   &  superius 

injecta    est    mentio,  &    de     quo    age-[93]-mus 

infra  pro  massis  quibuscunque.     Hinc  si  sibi  re- 

linquantur,     accedet      C    ad    E,    &    rectae    AB 

punctum    medium   D    ibit   ipsi    obviam    versus 

ipsum  cum  velocitate  dimidia  ejus,  quam  ipsum 

habebit,  vel  contra  recedent,  vel  hinc,  aut  inde 

movebuntur  in  latus,  per  lineas  tamen  similes, 

atque  ita,  ut  C,  &  D  semper  respectu  puncti  E 

immoti  ex  adverse  sint,  in  quo  motu  tam  directio      «  —.  _ 

rectae  AB,  quam  directio  rectae  CD,  &  ejus  incli-     "  _  *^ 

natio  ad  AB,  plerumque  mutabitur. 
Determmatio    vis  2o6.  Quod  pertinet  ad  inveniendam  vim  pro  quacunque  positione  puncti  C  respectu 

ejusdem  composite  A      „     T>  r      -i        •      •  •  T       r  •  • 

e  binis  viribus.  punctorum  A,  &  B,  ea  facile  sic  mvemetur.  In  fig.  i  assumendae  essent  abscissae  in  axe 
asquales  rectis  AC,  BC  figurae  21,  &  erigendae  ordinatas  ipsis  respondentes,  quae  vel  ambae 
essent  ex  parte  attractiva,  vel  ambae  ex  parte  repulsiva  ;  vel  prima  attractiva,  &  secunda 
repulsiva  ;  vel  prima  repulsiva  &  secunda  attractiva.  In  primo  casu  sumendae  essent  CL, 
CK  ipsis  aequales  (figura  21  exhibet  minores,  nenimis  excrescat)  versus  A,  &B  ;  in  secundo 
CN,  CM  ad  partes  oppositas  A,B  :  in  tertio  CL  versus  A,  &  CM  ad  partes  oppositas  B ;  in 
quarto  CN  ad  partes  oppositas  A,  &  CK  versus  B.  Tam  complete  parallelogrammo  LCKF, 
vel  MCNH,  vel  LCMI,  vel  KCNG,  diameter  CF,  vel  CH,  vel  CI,  vel  CG  exprimeret 
directionem,  &  magnitudinem  vis  compositae,  qua  urgetur  C  a  reliquis  binis  punctis. 


Methodus  constru-  207.  Hinc  si  assumantur  ad  arbitrium  duo  loca  qusecunque  punctorum  A,  &  B,  ad 

expi?  <lu3e  referendum  sit  tertium  C  ;  ducta  quavis  recta  DEC  indefinita,  ex  quovis  ejus  puncto 


mat  vim  ejusmodi.  posset  erigi  recta  ipsi  perpendicularis,  &  asqualis  illi  diametro,  ut  CF  in  primo  casu,  ac 
haberetur  curva  exprimens  vim  absolutam  puncti  in  eo  siti,  &  solicitati  a  viribus,  quas 
habet  cum  ipsis  A,  &  B.  Sed  satis  esset  binas  curvas  construere,  alteram,  quae  exprimeret 
vim  redactam  ad  directionem  DC  per  perpendiculum  FO,  ut  CO  ;  alteram,  quae  exprimeret 
vim  perpendicularem  OF  :  nam  eo  pacto  haberentur  etiam  directiones  vis  absolutae  ab 
iis  compositae  per  ejusmodi  binas  ordinatas.  Oporteret  autem  ipsam  ordinatam  curvas 
utriuslibet  assumere  ex  altera  plaga  ipsius  CD,  vel  ex  altera  opposita  ;  prout  CO  jaceret 
versus  D,  vel  ad  plagam  oppositam  pro  prima  curva;  &  prout  OF  jaceret  ad  alteram  partem 
rectae  DC,  vel  ad  oppositam,  pro  secunda. 


Expressio    magis  208.  Hoc  pacto  datis  locis  A,  B  pro  singulis  rectis  egressis  e  puncto  medio  D  duas 

*115  Per  super"  haberentur  diversae  curvae,  quae  diversas  admodum  exhiberent  virium  leges ;  ac  si  quasre- 
retur  locus  geometricus  continuus,  qui  exprimeret  simul  omnes  ejusmodi  leges  pertinentes 
ad  omnes  ejusmodi  curvas,  sive  indefinite  exhiberet  omnes  vires  pertinentes  ad  omnia 


A  THEORY  OF  NATURAL  PHILOSOPHY  157 

the  curve  given  in  Fig.  i,  the  law  of  forces  can  be  determined  in  general  for  any  assumed 
distances  along  any  straight  line  given  in  position.  Moreover,  this  can  be  effected  either 
by  constructing  geometrically  curves  through  sets  of  points,  which  represent  a  law  of  this 
sort  &  give  either  the  magnitude  of  the  absolute  force,  or  the  magnitudes  of  the  pair  of 
forces  into  which  it  may  be  considered  to  be  resolved,  the  one  acting  perpendicularly  to 
the  given  straight  line  &  the  other  in  its  direction  ;  or  else  by  writing  down  three  analytical 
formulae,  which  will  represent  its  value.  The  second,  if  treated  perfectly  generally,  & 
in  such  a  manner  that  the  curves  to  be  described  can  be  assigned  in  any  case  whatever, 
either  by  construction  or  by  calculation,  is  (even  when  there  are  only  three  points  in  question) 
beyond  the  power  of  all  methods  known  hitherto.  Further,  if  instead  of  three  points 
we  have  three  masses  of  points,  then  we  have  the  well-known  problem  that  is  called  "  the 
problem  of  three  bodies."  The  solution  of  this  problem  is  still  sought  after  in  our  own 
times ;  &  has  only  been  solved  in  certain  special  cases,  with  great  limitations  by  a  very 
few  of  the  geometricians  of  our  age  belonging  to  the  highest  rank,  &  even  then  with  insufficient 
accuracy  of  calculation  ;  as  was  pointed  out  in  Art.  122. 

205.  As  for  this  second  case,  it  is  very  well  known  that,  if  in  Fig.  21,  A,C,B,  are  three  Theorem  with  re- 
points,  &  the  distance  between  two  of  them,  A  &  B,  is  always  bisected  at  D,  &  CD  is  joined,  g'Vyjj  m°^r 
&  DE  is  taken  equal  to  one  third  of  DC,  then,  however  these  points  move  under  the  influence  the  action  of  two 
of  the  forces  compounded  from  the  forces  of  any  projection  whatever  &  the  mutual  forces,  other  P°mts- 

the  point  E  must  always  remain  at  rest  or  proceed  in  a  straight  line  with  uniform  motion. 
This  depends  on  a  general  theorem  with  regard  to  the  centre  of  gravity,  about  which 
passing  mention  has  already  been  made,  &  with  which  we  shall  deal  in  what  follows  for  the 
case  of  any  masses  whatever.  From  this  it  follows  that,  if  they  are  left  to  themselves, 
the  point  C  will  approach  the  point  E,  &  D,  the  middle  point  of  the  straight  line  AB,  will 
move  in  the  opposite  direction  towards  E  with  half  the  velocity  of  C  ;  or,  on  the  contrary, 
both  C  &  D  will  recede  from  E  ;  or  they  will  move,  one  in  one  direction  &  the  other  in 
the  opposite  direction  :  nevertheless  they  will  follow  similar  paths,  in  such  a  manner  that 
C  &  D  will  always  be  on  opposite  sides  of  the  stable  point  E  ;  &  in  this  motion,  the  direc- 
tion of  the  straight  line  AB,  that  of  the  straight  line  DE,  &  the  inclination  of  the  latter 
to  AB  will  usually  be  altered. 

206.  As  regards  the  determination  of  the  force  for  any  position  of  the  point  C  with  Determination   o  f 
regard  to  the  points  A  &  B,  that  is  easily  effected  in  the  following  manner.     Take,  in  Fig.  I,  compound^ 
abscissa  measured  along  the  axis  equal  to  the  straight  lines  AC  &  BC  of  Fig.  21  ;    draw  two  forces. 

the  ordinates  corresponding  to  them,  which  may  be  either  both  on  the  attractive  side  of 
the  axis,  or  both  on  the  repulsive  side  ;  or  the  first  on  the  attractive  &  the  second  on  the 
repulsive  ;  or  the  first  on  the  repulsive  &  the  second  on  the  attractive  side.  In  the  first 
case,  take  CL,  CK,  equal  to  these  ordinates  (in  Fig.  21  they  are  reduced  so  as  to  prevent 
the  figure  from  being  too  large)  ;  let  them  be  taken  in  the  direction  of  A  &  B  ;  similarly, 
in  the  second  case,  take  CN  &  CM  in  the  opposite  directions  to  those  of  A  &  B  ;  and,  in 
the  third  case,  take  CL  in  the  direction  of  A,  &  CM  in  the  direction  opposite  to  that  of  B  ; 
whilst,  in  the  fourth  case,  take  CN  in  the  direction  opposite  to  that  of  A,  &  CK  in  the 
direction  of  B.  Then,  completing  the  parallelogram  LCKF,  or  MCNH,  or  LCMI,  or 
KCNG,  the  diagonal  CF,  or  CH,  or  CI,  or  CG,  will  represent  the  direction  &  the  magnitude 
of  the  resultant  force,  which  is  exerted  upon  the  point  C  by  the  remaining  two  points. 

207.  Hence,  if  any  two  positions  are  taken  at  random  as  those  of  the  points  A  &  B,  The   method  of 
&  to  these  the  third  point  C  is  referred  ;  &  if  any  straight  line  DEC  is  drawn  of  indefinite  curve' wWch^m  in 
length  ;    then  from  any  point  of  it  a  straight  line  can  be  erected  perpendicular  to  it,  &  general  express   a 
equal  to  the  diagonal  of  the  parallelogram,  for  instance  CF  in  the  first  case.     From  these  force  of  this  sort' 
perpendiculars  a  curve  will  be  obtained,  which  will  represent  the  absolute  force  on  a  point 

situated  in  the  straight  line  DEC,  &  under  the  action  of  the  forces  exerted  upon  it  by  the 
points  A  &  B.  However,  it  would  be  more  satisfactory  if  two  curves  were  constructed  ; 
one  of  which  would  represent  the  force  resolved  along  the  direction  DC  by  means  of  a 
perpendicular  FO,  such  as  CO  ;  &  the  other  to  represent  the  perpendicular  force  OF. 
For,  in  this  way,  we  should  also  obtain  the  directions  of  the  absolute  forces  compounded 
from  these  resolved  parts,  by  means  of  the  two  ordinates  of  this  kind.  Moreover, 
we  ought  to  take  these  ordinates  of  either  of  the  curves  on  the  one  side  or  the  other  of 
the  straight  line  CD,  according  as  CO  would  be  towards  D,  or  away  from  it,  in  the  first 
curve,  &  according  as  OF  would  be  away  from  the  straight  line  CD,  on  the  one  side  or  on  the 
other,  in  the  second  curve. 

208.  In  this  way,  given  the  positions  of  A  &  B,  for  each  straight  line  drawn  through  A     m°re    general 
the  point  D,  we  should  obtain  distinct  curves ;  &  these  would  represent  altogether  different  of ^ surface.71 
laws    of    forces.     If    then    a    continuous    geometrical    locus    is    required,  which   would 
simultaneously  represent  all  the  laws  of   this  kind  relating  to  every  curve  of   this  sort, 

or  express  in  general  all  the  forces  pertaining  to  all  points  such  as  C,  wherever  they  might 


I58 


PHILOSOPHIC  NATURALIS  THEORIA 


puncta  C,  ubicunque  collocata  ;  oporteret  erigere  in  omnibus  punctis  C  rectas  normales 
piano  ACB,  alteram  aequalem  CO,  [94]  alteram  OF,  &  vertices  ejusmodi  normalium 
determinarent  binas  superficies  quasdam  continuas,  quarum  altera  exhiberet  vires  in 
directione  CD  attractivas  ad  D,  vel  repulsivas  respectu  ipsius,  prout,  cadente  O  citra,  vel 
ultra  C,  normalis  ilia  fuisset  erecta  supra,  vel  infra  planum ;  &  altera  pariter  vires  perpen- 
diculares.  Ejusmodi  locus  geometricus,  si  algebraice  tractari  deberet,  esset  ex  iis,  quos 
Geometrse  tractant  tribus  indeterminatis  per  unicam  aequationem  inter  se  connexis ;  ac 
data  aequatione  ad  illam  primam  curvam  figurse  I,  posset  utique  inveniri  tam  sequatio  ad 
utramlibet  curvam  respondentem  singulis  rectis  DC,  constans  binis  tantum  indeterminatis, 
quam  sequatio  determinans  utramlibet  superficiem  simul  indefinite  per  tres  indetermin- 
atas.  (») 

Methpdus  determi-  [gel  20Q.  Si  pro  duobus   punctis  tantummodo  agentibus  in  tertium   daretur  numerus 

nandi  vim  composi-    L7TJ  .  r  .       ,      .    .      .  .     ., 

tam  ex  viribus  re-  quicunque  punctorum  positorum  in  datis  locis,  ac  agentium  in  idem  punctum,  posset  utique 
spicientibus  puncta  constructione  simili  inveniri  vis,  qua  sineula  agunt  in  ipsum  collocatum  in  quovis  assumpto 

quotcunque.  ,      .  .  '   ~*.       .   ..  °  .       r  ,    .,     .  .? 

loci  puncto,  ac  vis  ex  ejusmodi  viribus  composita  denniretur  tam  directione,  quam 
magnitudine,  per  notam  virium  compositionem.  Posset  etiam  analysis  adhiberi  ad  expri- 
mendas  curvas  per  asquationes  duarum  indeterminatarum  pro  rectis  quibuscunque,  &  (") 
si  omnia  puncta  jaceant  in  eodem  piano,  superficies  per  asquationem  trium.  [96]  Mirum 
autem,  quanta  inde  diversarum  legum  combinatio  oriretur.  Sed  &  ubi  duo  tantummodo 
puncta  agant  in  tertium,  incredibile  dictu  est,  quanta  diversitas  legum,  &  curvarum  inde 
erumpat.  Manente  etiam  distantia  AB,  leges  pertinentes  ad  diversas  inclinationes  rectae 
DC  ad  AB,  admodum  diversse  obveniunt  inter  se  :  mutata  vero  punctorum  A,  B  distantia 


(n)  Stantibus  in  fig.  22  punctis  ADBCKFLO,  ut  in  fig.  21,  ducantur  perpendicula  BP,  AQ  in  CD,  qute  dabuntur 
data  inclinations  DC,  y  punctis  B,  A,  ac  pariter  dabuntur  y  DP,  DQ.  Dicatur  prtsterea  DC  =  x,  y  dabuntur  analytice 
CQ,  CP.  Quare  ob  angulos  rectos  P,  Q,  dabuntur  etiam  analytice  CB,  CA.  Denominentur  CK=«,  CL  =z,  CF  =y. 
Quoniam  datur  AB,  y  dantur  analytice  AC,  CB  ;  dabitur  analytice  ex  applicatione  Algebrte  ad  Trigonometriam 
sinus  anguli  ACB  per  x,  y  datas  quantitates,  qui  est  idem,  ac  sinus  anguli  CKF  complements  ad  duos  rectos.  Datur 
autem  idem  ex  datis  analytice  valoribus  CK  =  «,  KF  =  CL  =z,  CF  =y  ;  quare  habetur  ibi  una  tequatio  per  x,  y, 
z,  u,  y  constantes.  Si  pr<eterea  valor  CB  ponatur  pro  valore  abscissae  in  tequatione  curvte  figurte  I  ;  acquiritur  altera 
tequatio  per  valores  CK,  CB,  sive  per  x,u,  y  constantes.  Eodem  facto  invenietur  ope  tequationis  curvte  figure  I  tertia 
tequatio  per  AC,  &  CL,  adeoque  per  x,  z,  y  constantes.  Quare  jam  babebuntur  tequationes  tres  per  x,u,z,y,  y  con- 
stantes, qute,  eliminates  u,  y  z,  reducentur  ad  unicam  per  x,y,  &  constantes,  ac  ea  primam  illam  curvam  definiet. 

Quod  si   queeratur'  tequatio   ad  secundam   curvam,   cujus  ordinata  est  CO,   vel  tertiam,   cujus  ordinata    OF, 

T>p 

inveniri  itidem    poterit.     Nam  datur  analytice  sinus  anguli  DCB   =  ™,  W   *»    trianguk  FCK    datur  analytice 

\sD 
•pTT 

sinus  FCK  ===-  X  sin  CKF.     Quare  datur  analytice  etiam  sinus 
Cr 

differentite  OCF,  adeoque  &  ejus  cosinus,  &  inde,  ac  ex  CF  datur 
analytice  OF,  vel  CO.  Sz  igitur  altera  ex  illis  dicatur  p,  acquiri- 
tur nova  tequatio,  cujus  ope  una  cum  superioribus  eliminari 
poterit  pristerea  una  alia  indeterminata  ;  adeoque  eliminata 
CF  =y,  habebitur  unica  tequatio  per  x,p,  y  constantes,  qua 
exhibebit  utramlibet  e  reliquis  curvis  determinantibus  legem 
virium  CO,  vel  OF. 

Pro  tequatione  cum  binis  indeterminatis,  quts  exhibebit  locum- 
ad  superficiem,  ducatur  CR  perpendicularis  ad  AB,  y  dicatur 
DR  —x,  RC  =  q,  denominatis,  ut  prius,  CK  =«,  CL  =  z, 
CF  =  v  ;  y  quoniam  dantur  AD,  DB  ;  dabuntur  analytice  per  x, 
y  constantes  AR,  RB,  adeoque  per  x,  q,  &  constantes  AC,  CB,  W 
factis  omnibus  reliquis,  ut  prius,  kabebuntur  quatuor  tequationes 
per  x,q,u,z,y,p,  y  constantes,  qute  eliminatis  valoribus  u,z,y, 
reducentur  ad  unicam  datam  per  constantes,  y  tres  indeterminatas 
x,p,q,  sive  DR,  RC,  y  CO,  vel  OF,  qute  exhibebit  qutesitum 
locum  ad  superficiem. 

Calculus  quidem  esset  immensus,  sed  patet  methodus,  qua  deveniri  possit  ad  tequationem  qutesitam.  Mirum  autem, 
quanta  curvarum,  y  superficierum,  adeoque  y  legum  virium  varietas  obvenerit,  mutata  tantummodo  distantia  AB  binorum 
punctorum  agentium  in  tertium,  qua  mutata,  mutatur  tola  lex,  y  tequatio. 

(o)  Htec  conditio  punctorum  jacentium  in  eodem  piano  necessaria  fuit  pro  loco  ad  superficiem,  y  pro  tequatione,  qute 
legem  virium  exhibeat  per  tequationem  indeterminatarum  tantummodo  trium  :  at  si  puncta  sint  plura,  y  in  eodem  piano 
non  jaceant,  quod  punctis  tantummodo  tribus  accidere  omnino  non  potest ;  turn  vero  locus  ad  superficiem,  y  tequatio  trium 
indeterminatarum  non  sufficit,  sed  ad  earn  generaliter  exprimendam  legem  Geometria  omnis  est  incapax,  y  analysis  indiget 
tequatione  indeterminatarum  quatuor.  Primum  patet  ex  eo,  quod  si  manentibus  punctis  A,  B,  exeat  punctum  C  ex  data 
quodam  piano,  pro  quo  constructus  sit  locus  ad  superficiem  ;  liceret  converters  circa  rectam  AB  planum  illud  cum  superficie 
curva  legem  virium  determinate,  donee  ad  punctum  C  deveniret  planum  ipsum :  turn  enim  erecto  perpendiculo  usque  ad 
superficiem  illam  curvam,  definiretur  per  ipsum  vis  agens  secundum  rectam  CD,  vel  ipsi  perpendicularis,  prout  locus  ille 
ad  curvam  superficiem  constructus  fuerit  pro  altera  ex  iis. 


A  THEORY  OF  NATURAL  PHILOSOPHY  159 

be  situated  ;  we  should  have  to  erect  at  every  point  C  normals  to  the  plane  ACB,  one  of 
them  equal  to  CO  &  the  other  to  OF.  The  ends  of  these  normals  would  determine  two 
continuous  surfaces  ;  &  of  these,  the  one  would  represent  the  forces  in  the  direction  CD, 
attractive  or  repulsive  with  respect  to  the  point  D,  according  as  the  normal  was  erected 
above  or  below  this  plane,  whether  C  fell  on  the  near  side  or  on  the  far  side  of  D  ;  & 
similarly  the  other  would  represent  the  perpendicular  forces.  A  geometrical  locus  of  this 
kind,  if  it  has  to  be  treated  algebraically,  is  such  as  geometricians  deal  with  by  means  of 
three  unknowns  connected  together  by  a  single  equation  ;  &,  if  the  equation  to  the  primary 
curve  of  Fig.  i  is  given,  it  would  in  all  cases  be  possible  to  find,  not  only  the  equations  to 
the  two  curves  corresponding  to  each  &  every  straight  line  DC,  involving  only  two  unknowns, 
but  also  the  equations  for  both  the  surfaces  corresponding  to  the  general  determination, 
by  means  of  three  unknowns. («) 

209.  If  instead  of  only  two  points  acting  upon  a  third  we  are  given  any  number  of  The     method     of 
points  situated  in  given  positions,  &  acting  on  the  same  point,  it  would  be  possible,  by  a  force  ""compounded 
similar  construction  in  each  case,  to  find  the  force,  with  which  each  acts  on  the  point  from  the  forces  due 
situated  in  any  chosen  position  ;   &  the  force  compounded  from  forces  of  this  kind  would  points7  The1  great 
be  determined,  both  in  position  &  magnitude,  by  the  well-known  method  for  composition  £"]^sr  &  variety 
of  forces.     Also  analysis  could  be  employed  to  represent  the  curves  by  equations  involving 
two  unknowns  for  any  straight  lines ;   &  (»)  provided  that  all  the  points  were  in  the  same 
plane,  the  surface  could  be  represented  by  an  equation  involving  three  unknowns.     But 
it  is  marvellous  what  a  huge  number  of  different  laws  arise.     But,  indeed,  it  is  incredible, 
even  when  there  are  only  two  points  acting  on  a  third,  how  great  a  number  of  different 
laws  &  curves  are  produced  in  this  way.     Even  if  the  distance  AB  remains  the  same,  the 
laws  with  respect  to  different  inclinations  of  the  straight  line  CD  to  the  straight  line  AB, 
come  out  quite  different  to  one  another.     But  when  the  distance  of  the  points  A  &  B  from 

(n)  In  Fig.  22,  let  the  -points  A,D,B,C,K,F,L,O  be  in  the  same  positions  as  in  Fig.  21,  y  let  BP,  AQ  be  drawn 
•perpendicular  to  CD  ;  then  these  will  be  known,  if  the  inclination  of  CD  y  the  positions  of  A  y  B  are  known:  y 
so  also  will  DP  W  DQ  be  known.  Further,  suppose  DC  =  x,  then  CQ  y  CP  will  be  given  analytically.  Hence  on 
account  of  the  right  angles  at  P  y  Q,  CB  y  CA  will  also  be  given  analytically.  Suppose  CK  =  w,  CL  =  z,  CF  =  y. 
Since  AB  is  known,  y  AC,  CB  are  given  analytically,  by  an  application  of  algebra  to  trigonometry,  the  sine  of  the 
angle  ACB  is  also  known  analytically  in  terms  of  x  y  known  quantities  ;  y  this  is  the  same  thing  as  the  sine  of 
the  supplementary  angle  CKF.  Moreover  the  same  thing  will  be  given  in  terms  of  the  known  analytical  values  of 
3K  =  u,  KF  =  CL  =  z,  CF  =  y.  Hence  there  is  obtained  in  this  case  an  equation  involving  x,y,z,u,  y  constants. 
If,  in  addition,  the  value  CB  is  substituted  for  the  value  of  the  abscissa  in  the  equation  of  the  curve  in  Fig.  I,  another 
equation  will  be  obtained  in  terms  of  the  values  of  CK,  CB,  i.e.  in  terms  of  x,  u,  y  constants.  In  a  similar  way  by  the  help 
of  the  equation  of  the  curve  of  Fig.  I,  there  can  be  found  a  third  equation  in  terms  of  AC  y  CL,  i.e.,  in  terms  of 
#,z,  y  constants.  Now,  snce  there  will  be' thus  obtained  three  equations  in  terms  of  x,y,z,u,  y  constants,  these,  on 
eliminating  u,z,  will  reduce  to  a  single  equation  involving  x,y,  y  constants  ;  y  this  will  be  the  equation  defining 
the  first  curve. 

Again,  «/  the  equation  to  the  second  curve  is  required,  of  which  the  ordinate  is  CO,  or  of  a  third  curve  for  which 
the  ordinate  is  CF,  it  will  be  possible  to  find  either  of  these  as  well.  For  the  sine  of  the  angle  DCB  is  analytically 
given,  being  equal  to  BP/CB  ;  y  from  the  triangle  FCK,  the  sine  of  the  angle  FCK  is  given,  being  equal  to 
«'»CKF.(FK/CF).  There  fore  the  sine  of  the  difference  OOP  is  also  given  analytically,  y therefore  also  its  cosine;  y 
from  this  y  the  value  of  CF,  the  value  of  OF  or  CO  will  be  given  analytically.  If  then  one  or  the  other  of  them  is 
denoted  by  p,  a  new  equation  will  be  obtained:  by  the  help  of  this  y  one  of  the  equations  given  above,  another  of  the 
unknowns  can  be  eliminated.  If  then,  we  eliminate  CF  =  y,  a  single  equation  will  be  obtained  in  terms  of  x,p,  y 
constants,  which  will  be  that  of  one  or  other  of  the  remaining  curves  determining  the  law  of  forces  for  CO  or  OF. 

For  an  equation  in  three  unknowns,  which  will  represent  the  surface,  draw  CR  perpendicular  to  AB,  y  let  DR=# 
RC  =  q  ;  y,  as  before,  let  CK  =  it,  CL  =  z,  CF  =  y.  Then,  since  AD,  DB  are  given,  AR  y  RB  are  also  given 
analytically  in  terms  of  x  y  constants :  y  therefore  AC  y  CB  are  given  in  terms  of  x,q,  y  constants  :  y  if  all 
the  rest  of  the  work  is  done  as  before,  four  equations  will  be  obtained  in  terms  of  x,q,u,z,y,p,  y  constants.  These,  on 
eliminating  the  values  u,z,y,  will  reduce  to  a  single  equation  in  terms  of  constants  y  the  three  unknowns  x,p,q,  or  DR, 
RC,  y  CO  or  OF  ;  this  equation  will  represent  the  surface  required. 

The  calculation  would  indeed  be  enormous  ;  but  the  method,  by  which  the  required  equation  might  be  obtained  is 
perfectly  clear.  But  it  is  wonderful  what  a  great  number  of  curves  y  surfaces,  y  therefore  of  laws  of  force,  would  be 
met  with,  if  merely  the  distance  between  A  y  B,  the  two  points  which  act  upon  the  third,  is  changed ;  for  if  this 
alone  is  changed,  the  whole  law  is  altered  y  so  too  is  the  equation. 

(o)  This  condition,  that  the  points  should  all  lie  in  the  same  plane,  is  necessary  for  the  determination  of  the  surface, 
y  for  the  equation,  which  will  express  the  law  of  the  forces  by  an  equation  involving  only  three  unknowns.  If  the  points 
are  numerous,  y  they  do  not  all  lie  in  the  same  plane  (which  is  quite  impossible  in  the  case  of  only  three  points),  then 
indeed  a  surface  locus,  y  an  equation  in  three  unknowns,  will  not  be  sufficient;  indeed,  to  express  the  law  generally, 
the  whole  of  geometry  is  powerless,  y  analysis  requires  an  equation  in  four  unknowns.  The  first  point  is  clear  from 
the  fact  that  if,  whilst  the  points  A  y  B  remain  where  they  were,  the  point  C  moves  out  of  the  given  plane,  with 
regard  to  which  the  construction  for  the  surface  locus  was  made,  it  would  be  right  to  rotate  about  the  straight  line  AB 
that  plane  together  with  its  curved  surface,  which  determines  the  law  of  forces,  until  the  plane  passes  through  the  point 
C.  For  then,  if  a  perpendicular  is  drawn  to  meet  the  curved  surface,  this  would  define  the  force  acting  along  the 
straight  line  CD,  or  perpendicular  to  it,  according  as  the  locus  to  the  curved  surface  had  been  constructed  for  the  one 
or  for  the  other  of  them. 


160 


PHILOSOPHIC  NATURALIS  THEORIA 


a  se  invicem,  leges  etiam  pertinentes  ad  eandem  inclinationem  DC  differunt  inter  se 
plurimum  ;  &  infinitum  esset  singula  persequi  ;  quanquam  earum  variationum  cognitio, 
si  obtineri  utcunque  posset,  mirum  in  modum  vires  imaginationis  extenderet,  &  objiceret 
discrimina  quamplurima  scitu  dignissima,  &  maximo  futura  usui,  atque  incredibilem 
Theoriae  foecunditatem  ostenderet. 


distantiis1  2IO>  ^8°  ^c  simpliciora  quaedam,  ac  faciliora,  &  usum  habitura  in  sequentibus,  ac  in 

ac  ejus  usus  pro  applicatione  ad  Physicam  inprimis  attingam  tantummodo  ;  sed  interea  quod  ad  generalem 
nu'iiaVinUs  s^mma  Pertinet  determinationem  expositam,  duo  adnotanda  proponam.  Primo  quidem  in  ipsa 
virium  simpiicium.  trium  punctorum  combinatione  occurrit  jam  hie  nobis  praeter  vim  determinantem  ad 
accessum,  &  recessum,  vis  urgens  in  latus,  ut  in  fig.  21,  praeter  vim  CF,  vel  CH,  vis  CI, 
vel  CG.  Id  erit  infra  magno  usui  ad  explicanda  solidorum  phaenomena,  in  quibus, 
inclinato  fundo  virgse  solidae,  tola  virga,  &  ejus  vertex  moventur  in  latus,  ut  certam  ad 
basim  positionem  acquirant.  Deinde  vero  illud  :  haec  omnia  curvarum,  &  legum  discrimina 
tam  quae  [97]  pertinent  ad  diversas  directiones  rectarum  DC,  data  distantia  punctorum 
A,  B,  quam  quae  pertinent  ad  diversas  distantias  ipsorum  punctorum  A,  B,  data  etiam 
directione  DC,  ac  hasce  vires  in  latus  haberi  debere  in  exiguis  illis  distantiis,  in  quibus 
curva  figurae  I  circa  axem  contorquetur,  ubi  nimirum  mutata  parum  admodum  distantia, 
vires  singulorem  punctorum  mutantur  plurimum,  &  e  repulsivis  etiam  abeunt  in  attractivas, 
ac  vice  versa,  &  ubi  respectu  alterius  puncti  haberi  possit  attractio,  respectu  alterius  repulsio, 
quod  utique  requiritur,  ut  vis  dirigatur  extra  angulum  ACB,  &  extra  ipsi  ad  verticem 
oppositum.  At  in  majoribus  distantiis,  in  quibus  jam  habetur  illud  postremum  crus 
figurae  I  exprimens  arcum  attractivum  ad  sensum  in  ratione  reciproca  duplicata  distantiarum, 
vis  in  punctum  C  a  punctis  A,  B  inter  se  proximis,  utcunque  ejusmodi  distantia  mutetur, 
&  quaecunque  fuerit  inclinatio  CD  ad  AB,  erit  semper  ad  sensum  eadem,  directa  ad  sensum 
ad  punctum  D,  ad  sensum  proportionalis  reciproce  quadrato  distantiae  DC  ab  ipso  puncto 
D,  &  ad  sensum  dupla  ejus,  quam  in  curva  figurae  i  requireret  distantia  DC. 


At  secundum  sit  manifestum  ex  eo,  quod  si  puncta  agenda  sint  etiam  omnia  in  eodem  piano,  y  punctum,  cufus  vis 
composita  quteritur,  in  quavis  recta  posita  extra  ipsum  planum,  relationes  omnes  distantiarum  a  reliquis  punctis,  ac 
directionum,  a  quibus  pendent  vires  singulorum,  y  compositio  ipsarum  virium,  longe  alia  essent,  ac  in  quavis  recta  in  eodem 
piano  posita,  uti  facile  videre  est.  Hinc  pro  quovis  puncto  loci  ubicunque  assumpto  sua  responderet  vis  composita,  y  quarta 
aliqua  plaga,  seu  dimensio,  prater  longum,  latum,  &  profundum,  requireretur  ad  ducendas  ex  omnibus  punctis  spatii  rectas 
•  Us  viribus  proportionales,  quarum  rectarum  vertices  locum  continuum  aliqucm  exhiberent  determinantem  virium  legem. 

Sed  quod  Geometria  non  assequitur,  assequeretur  quarta  alia  dimensio  mente  concepta,  ut  si  conciperetur  spatium  totum 
plenum  materia  continua,  quod  in  mea  sententia  cogitatione  tantummodo  effingi  potest,  W  ea  esset  in  omnibus  spatii  punctis 
densitatis  diverse,  vel  diversi  pretii  ;  turn  ilia  diversa  densitas,  vel  illud  pretium,  vel  quidpiam  ejusmodi,  exhibere  posset 
legem  virium  ipsi  respondentium,  ques  nimirum  ipsi  essent  proportionales.  Sed  ibi  iterum  ad  determmandam  directtonem 
vis  composite  non  esset  satis  resolutio  in  duas  vires,  alteram  secundum  rectam  transcuntem  per  datum  punctum  ;  altcram 
ipsi  perpendicularem  ;  ed  requirerentur  tres,  nimirum  vel  omnes  secundum  tres  datas  directiones,  vel  tendentes  per  rectas, 
qua  per  data  tria  puncta  transeant,  vel  quavis  alia  certa  lege  definitas :  adeoque  tria  loca  ejusmodi  ad  spatium,  quarta 
aliqua  dimensione,  vel  qualitate  affectum  requirerentur,  qu<e  tribus  ejusmodi  plusquam  Geometricis  legibus  vis  composite 
legem  definirent,  turn  quod  pertinet  ad  ejus  magnitudinem,  turn  quod  ad  directionem. 


quod  non  assequitur  Geometria,  assequeretur  Analysis  ope  aquationis  quatuor  indeterminatarum  :    si  enim 
planum,  quod  libuerit,  ut  ACB,  y  in  eo  quavis  recta  AB,  ac  in  ipsa  recta  quodvis  punitum  D  ;    turn  quovis 


Ferum 

conciperetur  .  M 

hujus  segmento  DR  appellate  x,  quavis  recta  RC  ipsi  perpendiculari  y,  quavis  tertia  perpendicular!  ad  totum  planum  z, 
per  hasce  tres  indeterminatas  involveretur  positio  puncti  spatii  cujuscumque,  in  quo  collocatum  esset  punctum  materiel, 
cufus  vis  quteritur. 

Punctorum  agentium  utcunque  collocatorum  ubicunque  vel  intra  id  planum,  vel  extra,  possent  definiri  positiones  per 
ejusmodi  tres  rectas,  datas  utique  pro  singulis,  si  eorum  positiones  dentur.  Per  eas,  y  per  illas  x,y,z,  posset  utique  haberi 
distantia  cujuscumque  ex  Us  punctis  agentibus,  y  positione  datis,  a  puncto  indefinite  accepto  ;  adeoque  ope  aquationis 
figurtz  I  posset  haberi  analytice  per  aquationes  quasdam,  ut  supra,  vis  ad  singula  agentia  puncta  pertinens,  y  per  easdem 
rectas  ejus  etiam  directio  resoluta  in  tres  parallelas  illis  x,y,z.  Hinc  haberetur  analytice  omnium  summa  pro  singulis 
ejusmodi  directionibus  per  aliam  aquationem  derivatam  ab  ejus  summa  denominatione,  ea  nimirum  facia  =  u,  ac  expunctis 
omnibus  subsidiariis  valoribus,  methodo  non  absimili  ei,  quam  adhibuimus  superius  pro  loco  ad  superficiem,  deveniretur  ad 
unam  aquationem  constitutam  illis  quatuor  indeterminatis  x,y,z,u,  y  constantibus  ;  ac  tres  ejusmodi  aquationes  pro  tribus 
directionibus  vim  omnem  compositam  definirent.  Sed  hac  innuisse  sit  satis,  qua  nimirum  y  altiora  sunt,  y  ob  ingentem 
complicationcm  casuum,  ac  nostra  humantf  mentis  imbecillitatem  nulli  nobis  inferius  futura  sunt  usui. 


A  THEORY  OF  NATURAL  PHILOSOPHY  161 

one  another  is  also  changed,  the  laws  corresponding  to  the  same  inclination  of  DC  are 
altogether  different  to  one  another  ;  &  it  would  be  an  interminable  task  to  consider  them 
all,  case  by  case.  However,  a  comprehensive  insight  into  their  variations,  if  it  could  be 
obtained,  would  enlarge  the  powers  of  imagination  to  a  marvellous  extent ;  it  would  bring 
to  the  notice  a  very  large  number  of  characteristics  that  would  be  well  worth  knowing  & 
most  useful  for  further  work ;  &  it  would  give  a  demonstration  of  the  marvellous  fertility 
of  my  Theory. 

210.  First  of  all,  therefore,  I  will  here  only  deal  slightly  with  certain  of  the  more  simple  The  lateral  force  at 
cases,  such  as  will  be  of  use  in  what  follows,  &  later  when  considering  the  application  to  tances,  Tits  use  \n 
Physics.  But  meanwhile,  I  will  enunciate  two  theorems,  applying  to  the  general  deter-  t]}e  consideration 
mination  set  forth  above,  which  should  be  noted.  Firstly,  in  the  case  of  the  combination  absence^  this 

of  three  points,  we  have  here  already  met  with,  in  addition  to  a  force  inducing  approach  f?rce  at  great 
.          .    '     .      T-,.  .         IT.  ,.  X-,T-<  ,"ITT  ^-.T          2Z^~.rr  distances,  the  sum 

&  recession,  i.e.,  in  rig.  21,  in  addition  to  a  force  CF  or  CH,  a  force  CI  or  CG,  urging  of  the  simple  forces 

the  point  C  to  one  side.     This  will  be  of  great  service  to  us  in  explaining  certain  phenomena  in  the  latter  case- 

of  solids ;    for  instance,  the  fact  that,  if  the  bottom  of  a  solid  rod  is  inclined,  the  whole 

rod,  including  its  top,  is  moved  to  one  side  &  takes  up  a  definite  position  with  respect  to 

the  base.     Secondly,  there  is  the  fact  that  we  are  bound  to  have  all  these  differences  of 

curves  &  laws,  not  only  those  corresponding  to  different  directions  of  the  straight  lines  DC 

when  the  distance  between  the  points  A  &  B  is  given,  but  also  those  corresponding  to 

different  distances  of  the  points  A  &  B  when  the  direction  of  DC  is  given  ;   &  that  we  are 

bound  to  have  these  lateral  forces  for  very  small  distances,  for  which  the  curve  in  Fig.  I 

twists  about  the  axis ;   for  then  indeed,  if  the  change  in  distance  is  very  slight,  the  change 

in  the  forces  corresponding  to  the  several  points  is  very  great,  &  even  passes  from  repulsion 

to  attraction  &  vice  versa  ;   &  also  there  may  be  attraction  for  one  point  &  repulsion  for 

another  ;  &  this  must  be  the  case  if  the  direction  of  the  force  has  to  be  without  the  angle 

ACB,  or  the  angle  vertically  opposite  to  it.     But,  at  distances  that  are  fairly  large,  for 

which  we  have  already  seen  that  there  is  a  final  branch  of  the  curve  of  Fig.  i  that  represents 

attraction  approximately  in  the  ratio  of  the  inverse  square  of  the  distance,  the  force  on  the 

point  C,  due  to  two  points  A  &  B  very  near  to  one  another,  will  be  approximately  the 

same,  no  matter  how  this  distance  may  be  altered,  or  what  the  inclination  of  CD  to  AB 

may  be  ;   its  direction  is  approximately  towards  D  ;  &  its  magnitude  will  be  approximately 

in  inverse  proportion  to  the  square  of  DC,  its  distance  from  the  point  D  ;  that  is  to  say,  it 

will  be  approximately  double  of  that  to  which  in  Fig.  I  the  distance  DC  would  correspond. 

The  second  point  is  evident  from  the  fact  that,  if  all  the  points  acting  are  all  in  the  same  plane,  £5?  the  point  for 
which  the  resultant  farce  is  required,  lies  in  any  straight  line  situated  without  that  plane,  even  then  all  the  relations 
between  the  distances  from  the  remaining  points  as  well  as  between  their  directions,  will  be  altogether  different  from 
those  for  any  straight  line  situated  in  the  same  plane,  as  can  be  easily  seen.  Hence,  for  any  point  of  space  chosen  at 
random  there  would  be  a  corresponding  force  ;  W  a  fourth  region,  or  dimension,  in  addition  to  length,  breadth,  &  depth, 
would  be  required,  in  order  to  draw  through  each  point  of  space  straight  lines  proportional  to  these  forces,  the  ends  of 
which  straight  lines  would  give  a  continuous  locus  determining  the  law  for  the  forces. 

But  'what  can  not  be  attained  by  the  use  of  geometry,  could  be  attained,  by  imagining  another,  a  fourth,  dimension 
(just  as  if  the  whole  of  space  were  imagined  to  be  full  of  eontinuous  matter,  which  in  my  opinion  can  only  be  a  mental 
fiction)  ;  W  this  would  be  of  different  density,  or  different  value,  at  all  points  of  space.  Then  the  different  density,  or 
value,  or  something  of  that  kind,  might  represent  the  law  of  forces  corresponding  to  it,  these  indeed  being  proportional 
to  it.  But  here  again,  in  order  to  find  the  direction  of  the  resultant  force,  resolution  into  two  forces,  the  one  along  the 
straight  line  passing  through  the  given  -point,  y  the  other  perpendicular  to  it,  would,  not  be  sufficient.  Three  resolved 
parts  would  be  required,  either  all  in  three  given  directions,  or  along  straight  lines  passing  through  three  given  points, 
or  defined  by  some  other  fixed  law.  Thus,  three  regions  of  this  kind  in  space  possessed  of  some  fourth  dimension  or  quality 
would  be  required  ;  y  these  would  define,  by  three  ultra-geometrical  laws  of  this  sort,  the  law  of  the  resultant  force 
both  as  regards  magnitude  &  direction. 

But  what  cannot  be  obtained  with  the  help  of  geometry  could  be  obtained  by  the  aid  of  analysis  by  employing  an 
equation  with  four  unknowns.  For,  if  we  take  any  arbitrary  plane,  as  ACB,  y  in  it  any  straight  line  AB,  y  in  this 
straight  line  any  point  D  /  then,  calling  any  segment  of  it  x,  any  straight  line  perpendicular  to  it  y,  y  any  third 
straight  line  perpendicular  to  the  whole  plane  z,  there  would  be  contained  in  these  three  unknowns  the  position  of  any 
point  in  space,  at  which  is  situated  a  point  of  matter,  for  which  the  force  is  required. 

The  positions  of  the  acting  points,  however  W  wherever  they  may  be  situated,  either  within  the  plane  or  without 
it,  could  be  defined  by  three  straight  lines  of  this  sort ;  y  these  would  in  all  cases  be  known  for  each  point,  if  the  positions 
of  the  points  are  given.  By  means  of  these,  y  the  former  straight  lines  denoted  by  x,y,z,  there  could  he  obtained  in 
all  cases  the  distance  of  each  of  the  acting  points,  that  are  given  in  position,  from  any  point  assumed  indefinitely.  Thus 
by  the  help  of  the  equation  to  the  curve  of  Fig.  I,  there  could  be  obtained  analytically,  by  means  of  certain  equations 
similar  to  those  above,  the  force  corresponding  to  each  of  the  acting  points ;  also  from  the  same  straight  lines,  its 
direction  as  well,  by  resolving  along  three  parallels  to  x,  y,  y  z.  Hence  there  could  be  obtained  analytically  the  sum 
of  all  of  them  for  each  of  these  directions,  by  means  of  another  equation  derived  from  the  symbol  used  for  the  sum  (for 
instance,  let  this  be  called  u]  ;  y,  eliminating  all  the  subsidiary  values,  by  a  method  not  unlike  that  which  was  used 
above  for  the  surface  locus,  we  should  arrive  at  a  single  equation  in  terms  of  the  four  unknowns,  x,  y,  z,  u,  y  constants. 
Three  equations  of  this  sort,  one  for  each  of  the  three  directions,  would  determine  the  resultant  force  completely.  But 
let  it  suffice  merely  to  have  mentioned  these  things  ;  for  indeed  they  are  too  abstruse,  y,  on  account  of  the  enormous 
Complexity  of  cares,  y  the  disability  of  the  human  intelligence,  will  not  be  of  any  use  to  us  later. 


1  62 


PHILOSOPHIC  NATURALIS  THEORIA 


Demonstratio  post- 

remi     theorematis. 


2u.  Id  quidem  facile  demonstratur.     Si  enim  AB  respectu  DC  sit  perquam  exigua, 

,         .  ,-,«  ^   .  .  /-.T^      i  i  •  r      •  i  •  A  n 

anguius  AL.D  erit  perquam  exiguus,  &  a  recta  CL)  ad  sensum  bitanam  sectus  :  distantias  AC, 
CB  erunt  ad  se  invicem  ad  sensum  in  ratione  sequalitatis,  adeoque  &  vires  CL,  CK  ambae 
attractive  debebunt  ad  sensum  aequales  esse  inter  se,  &  proinde  LCKF  ad  sensum  rhombus, 
diametro  CF  ad  sensum  secante  angulum  LCK  bifariam,  quae  rhombi  proprietas  est,  & 
ipsa  CF  congruente  cum  CO,  ac  (ob  angulum  FCK  insensibilem,  &  CKF  ad  sensum 
aequalem  duobus  rectis)  aequali  ad  sensum  binis  CK,  KF,  sive  CK,  CL,  simul  sumptis  ; 
quae  singulae  cum  sint  quam  proxime  in  ratione  reciproca  duplicata  distantiarum  CB, 
BA  ;  erunt  &  eadem,  &  earum  summa  ad  sensum  in  ratione  reciproca  duplicata  distantiae 
CD. 


suia 


summa 
tarum 

massae, 


ingens  212.  Porro  id  quidem  commune  est  etiam  massulis  constantibus  quocunque  punctorum 

qu;is     mas-  _  _  -^  _  •...  •  •  »••**«*  • 

exercet    in  numero.     Mutata   alarum  combmatione,   vis   composita   a   vinbus   singulorum  agens   in 
oM-  punctum  distans  a  massula  ipsa  per  intervallum    perquam  exiguum,    nimirum  ejusmodi, 
in    remo-  in  quo  curva  figurae  I  circa  axem  contorquetur,  debet  mutare  plurimum  tarn  intensitatem 
^us^  quse  suanij  quam  directionem,  &  fieri  utique  potest,  quod  infra  etiam  in  aliquo  simpliciore 
&  reciproce,  casu  trium  punctorum  videbimus,  ut  in  alia  combinatione  punctorum  massulae  pro  eadem 
dlS    distantia  a  medio  repulsiones  praevaleant,  in  alia  attractiones,  in  alia  oriatur  vis  in  latus  ad 
perpendiculum,  ac  in  eadem  constitutione  massulae  pro  diversis  directionibus  admodum 
diversae  sint  vires  pro  eadem  etiam  distantia  a  medio.     At  in  magnis  illis  distantiis,  in 
quibus  singulorum  punctorum  vires  jam  attractive  sunt  omnes,  &  directiones,  ob  molem 
massulae  tarn  exiguam  respectu  ingentis  distantias,  ad  sensum  conspirant,  vis  com-[98] 
-posita  ex  omnibus  dirigetur  necessario  ad  punctum  aliquod  intra  massulam  situm,  adeoque 
ad  sensum  ejus  directio  erit  eadem,  ac  directio  rectae  tendentis  ad  mediam  massulam,  & 
aequabitur  vis  ipsa  ad  sensum  summae  virium  omnium  punctorum  constituentium  ipsam 
massulam,  adeoque  erit  attractiva  semper,  &  ad  sensum  proportionalis  in  diversis  etiam 
massulis  numero  punctorum  directe,  &  quadrate  distantias  a  medio  massulae  ipsius  reciproce  ; 
sive  generaliter  erit  in  ratione  composita  ex  directa  simplici  massarum,  &  reciproca  duplicata 
distantiarum.     Multo  autem  majus  erit  discrimen  in  exiguis  illis  distantiis,  si  non  unicum 
punctum  a  massula  ilia  solicitetur,  sed  massula  alia,  cujus  vis  componatur  e  singulis  viribus 
singulorum  suorum  punctorum,  quod  tamen  in  massula  etiam  respectu  massulae  admodum 
remotae  evanescet,  singulis  ejus  punctis  vires  habentibus  ad  sensum  aequales,  &  agentes 
in  eadem  ad  sensum  directione  ;    unde  net,  ut  vis  motrix  ejus  massulae  solicitatae,  orta  ab 
actionibus  illius  alterius  remotae  massulae,  sit  ad  sensum  proportionalis  numero  punctorum, 
quas  habet  ipsa,  numero  eorum,  quae  habet  altera,  &  quadrate  distantiae,  quaecunque  sit 
diversa  dispositio  punctorum  in  utralibet,  quicunque  numerus. 


Unde  necessaria  213.  Mirum  sane,  quantum  in  applicatione  ad  Physicam  haec  animadversio  habitura 

unTf^mitTsTn  s^  usum  '•>  nam  inde  constabit,  cur  omnia  corporum  genera  gravitatem  acceleratricem 
gravitate,  differ-  habeant  proportionalem  massae,  in  quam  tendunt,  &  quadrato  distantiae,  adeoque  in 
a-  superficie  Terrae  aurum,  &  pluma  cum  aequali  celeritate  descendant  seclusa  resistentia,  vim 
autem  totam,  quam  etiam  pondus  appellamus,  proportionalem  praeterea  massae  suae,  adeoque 
in  ordine  ad  gravitatem  nullum  sit  discrimen,  quascunque  differentia  habeatur  inter  corpora, 
quae  gravitant,  &  in  quae  gravitant,  sed  ad  solam  demum  massam,  &  distantiam  res  omnis 
deveniat  ;  at  in  iis  proprietatibus,  quae  pendent  a  minimis  distantiis,  in  quibus  nimirum 
fiunt  reflexionis  lucis,  &  refractiones  cum  separatione  colorum  pro  visu,  vellicationes  fibrarum 
palati  pro  gustu,  incursus  odoriferarum  particularum  pro  odoratu,  tremor  communicatus 
particulis  aeris  proximis,  &  propagatus  usque  ad  tympanum  auriculare  pro  auditu,  asperitas, 
ac  aliae  sensibiles  ejusmodi  qualitates  pro  tactu,  tot  cohaesionum  tarn  diversa  genera, 
secretiones,  nutritionesque,  fermentationes,  conflagrationes,  displosiones,  dissolutiones. 
prascipitationes,  ac  alii  effectus  Chemici  omnes,  &  mille  alia  ejusmodi,  quae  diversa  corpora 
a  se  invicem  discernunt,  in  iis,  inquam,  tantum  sit  discrimen,  &  vires  tarn  variae,  ac  tarn 


A  THEORY  OF  NATURAL  PHILOSOPHY  163 

211.  The  latter  theorem  can  be  easily  demonstrated.     For,  if  AB  is  very  small  compared  S[oof  of  thelattcr 
with  DC,  the  angle  ACB  will  be  very  small,  &  will  be  very  nearly  bisected  by  the  straight 

line  CD.  The  distances  AC,  CB  will  be  approximately  equal  to  one  another  ;  &  thus 
the  forces  CL,  CK,  which  are  both  attractive,  must  be  approximately  equal  to  one  another. 
Hence,  LCKF  is  approximately  a  rhombus,  &  the  diagonal  CF  very  nearly  bisects  the 
angle  LCK,  that  being  a  property  of  a  rhombus ;  CF  will  fall  along  CO,  &,  because  the 
angle  FCK  is  exceedingly  small  &  CKF  very  nearly  two  right  angles,  CF  will  be  very 
nearly  equal  to  CK  &  KF,  or  CK  &  CL,  taken  together.  Now  each  of  these  are  as 
nearly  as  possible  in  the  inverse  ratio  of  the  square  of  the  distances  CB,  CA ;  &  these  will 
be  the  same,  &  their  sum  therefore  approximately  inversely  proportional  to  the  square 
of  the  distance  DC. 

212.  Further  this  theorem  is  also  true  in  general  for  little  masses  consisting  of  points,  J.*iere   is  .a    hufe 

,.  T-,,        ,  jjf  it-          difference    in    the 

whatever  their  number  may  be.      Ine  force  compounded  from  the  several  forces  acting  forces  which  a  small 
on  a  point,  whose  distance  from  the  mass  is  very  small,  i.e.,  such  a  distance  as  that  for  which,  mass,  exerts  on,  a 

™r  .-11  7  11  i     T    i  i  •        •         small      mass     that 

in  Fig.  i,  the  curve  is  twisted  about  the  axis,  must  be  altered  very  greatly  if  the  combination  is  very  near  to  it; 

of  the  points  is  altered  ;  &  this  is  so,  both  as  regards  its  intensity,  &  as  regards  its  direction.  possibie^nrformH* 

It  may  even  happen,  as  will  be  seen  later  in  the  more  simple  case  of  three  points,  that  in  in  the  forces  due 

one  combination  of  the  points  forming  the  little  mass,  &  for  one  &  the  same  distance  from  these'var6  dlrectf' 

the  mean  point,  repulsions  will  preponderate,  in  another  case  attractions,  &  in  another  as  the   masses,  & 

case  there  will  arise  a  perpendicular  lateral  force.     Also  for  the  same  constitution  of  the  squaref'of  the  dls6 

mass,  for  the  same  distance  from  the  mean  point,  there  may  be  altogether  different  forces  tances. 

for  different  directions.     But,  for  considerable  distances,  where  the  forces  due  to  the  several 

points  are  now  attractive,  &  their  directions  practically  coincide  owing  to  the  dimensions 

of  the  little  mass  being  so  small  compared  with  the  greatness  of  the  distance,  the  force 

compounded  from  all  of  them  will  necessarily  be  directed  towards  some  point  within  the 

mass  itself ;   &  thus  its  direction  will  be  approximately  the  same  as  the  straight  line  drawn 

through  the  mean  centre  of  the  mass ;   &  the  force  itself  will  be  equal  approximately  to 

the  sum  of  all  the  forces  due  to  the  points  composing  the  little  mass.     Hence,  it  will  always 

be  an  attractive  force  ;   &  in  different  masses,  it  will  be  approximately  proportional  to  the 

number  of  points  directly,  &  to  the  square  of  the  distance  from  the  mean  centre  of  the  mass 

inversely.     That  is,  in  general,  it  will  be  in  the  ratio  compounded  of  the  simple  direct 

ratio  of  the  masses  &  the  inverse  duplicate  ratio  of  the  distances.     Further,  the  differences 

will  be  far  greater,  in  the  case  of  very  small   distances,  if  not  a  single  point  alone,  but 

another  mass,  is  under  the  action  of  the  little  mass  under  consideration  ;   for  in  this  case, 

the  force  is  compounded  from  the  several  forces  on  each  of  the  points  that  constitute  it  ; 

&  yet  these  differences  will  also  disappear  in  the  case  of  a  mass  acted  on  by  a  mass  considerably 

remote  from  it,  since  each  of  the  points  composing  it  is  under  the  influence  of  forces  that 

are  approximately  equal  &  act  in  practically  the  same  direction.     Hence  it  comes  about  that 

the  motive  force  of  the  mass  acted  upon,  which  is  produced  by  the  action  of  the  other 

mass  remote  from  it,  is  approximately  proportional  to  the  number  of  points  in  itself,  to 

the  number  of  points  in  the  other  mass,  &  to  the  square  of  the  distance  between  them, 

whatever  the  difference  in  the  disposition  of  the  points,  or  their  number,  may  be  for  either 

mass. 

213.  It  is  indeed  wonderful  what  great  use  can  be  made  of  this  consideration  in  the  Heice   we  have 
application  of  my  Theory  to  Physics ;   for,  from  it  it  will  be  clear  why  all  classes  of  bodies  bodies?  uniformity 
have  an  accelerating  gravity,  proportional  to  the  mass  on  which  they  act,  &  to  the  square  m    ^e    c*se    ?f 
of  the  distance  [inversely]  ;  &  hence  that,  on  the  surface  of  the  Earth,  a  piece  of  gold  &  a  uniformity  in  the 
feather  will  descend  with  equal  velocity,  when  the  resistance  of  the  air  is  eliminated.     It  cafe  of  numerous 
will  be  clear  also  that  the  whole  force,  which  we  call  the  weight,  is  in  addition  proportional 

to  the  mass  itself  ;  &  thus,  without  exception,  there  is  no  difference  as  regards  gravity, 
no  matter  what  difference  there  may  be  between  the  bodies  which  gravitate,  or  towards 
which  they  gravitate  ;  the  whole  matter  reducing  finally  to  a  consideration  of  mass  & 
distance  alone.  However,  for  those  properties  that  depend  on  very  small  distances,  for 
instance,  where  we  have  reflection  of  light,  &  refraction  with  separation  of  colours,  with 
regard  to  sight,  the  titillation  of  the  nerves  of  the  palate,  with  regard  to  taste,  the  inrush 
of  odoriferous  particles  where  smell  is  concerned,  the  quivering  motion  communicated  to 
the  nearest  particles  of  the  air  &  propagated  onwards  till  it  reaches  the  drum  of  the  ear 
for  sound,  roughness  &  other  such  qualities  as  may  be  felt  in  the  case  of  touch,  the  large 
number  of  kinds  of  cohesion  that  are  so  different  from  one  another,  secretion,  nutrition, 
fermentation,  conflagration,  explosion,  solution,  precipitation,  &  all  the  rest  of  the 
effects  met  with  in  Chemistry,  &  a  thousand  other  things  of  the  same  sort,  which 
distinguish  different  bodies  from  one  another  ;  for  these,  I  say,  the  differences  become 
as  great,  the  forces  and  the  motions  become  as  different,  as  the  differences  in  the  phenomena, 


1  64 


PHILOSOPHISE  NATURALIS  THEORIA 


vis  in  duo  puncta 

puacti    positi  in 

recta  jungente 
ipsa.  vei  in   recta 

secante     hanc     bi- 

fariam,  &  ad  angu- 
los  rectos  directa 

secundum    eandem 

rectam. 


varii  motus,  qui  tarn  varia  phaenomena,  &  omnes  specificas  tot  corporum  differentias 
inducunt,  consensu  Theoriae  hujus  cum  omni  Natura  sane  admirabili.  Sed  hsec,  quas 
hue  usque  dicta  sunt  ad  massas  pertinent,  &  ad  amplicationem  ad  Physicam  :  interea 
peculiaria  quaedam  persequar  ex  innumeris  iis,  quas  per-[99]-tinent  ad  diversas  leges  binorum 
punctorum  agentium  in  tertium. 

214.  Si  libeat  considerare  illas  leges,  quas  oriuntur  in  recta  perpendiculari  ad  AB 

,  T       _         .  .  ,  _.  .  .  .    °     •     T-  .  .         ......        r     r.  ,  ...     ,     ,. 

ducta  per  D,  vel  m  ipsa  AB  hmc,  &  inde  producta,  mprimis  facile  est  videre  mud,  direc- 
tionem  vis  compositas  utrobique  fore  eandem  cum  ipsa  recta  sine  ulla  vi  in  latus,  &  sine  ulla 

.      .  .  "    ,  .          .   .  _  ,    *,     .  .  _ 

declinatione  a  recta,  quas  tendit  ad  ipsum  D,  vel  ab  ipso.  Pro  recta  AB  res  constat  per 
sese  .  nam  vjres  Qjgg  qU33  ad  bina  ea  puncta  pertinent,  vel  habebunt  directionem  eandem, 

,    •  t  »  *«  .  .  r     ,.  •      -1  j 

vel  oppositas,  jacente  ipso  tertio  puncto  in  directum  cum  utroque  e  pnonbus  :  unde 
fit,  ut  vis  composita  asquetur  summae,  vel  differentias  virium  singularum  componentium, 
quae  in  eadem  recta  remaneat.  Pro  recta  perpendiculari  facile  admodum  demonstratur. 
Si  enim  in  fig.  23  recta  DC  fuerit  perpendicularis  ad  AB  sectam  bifariam  in  D,  erunt  AC, 
BC  aequales  inter  se.  Quare  vires,  quibus  C  agitatur  ab  A,  &  B,  sequales  erunt,  &  proinde 
vel  ambae  attractivae,  ut  CL,  CK,  vel  ambae  repulsivae,  ut  CN,  CM.  Quare  vis  composita 
CF,  vel  CH,  erit  diameter  rhombi,  adeoque  secabit  bifariam  angulum  LCK,  vel  NCM  ; 
quos  angulos  cum  bifariam  secet  etiam  recta  DC,  ob  asqualitatem  triangulorum  DCA,  DCB, 
patet,  ipsas  CF,  CH  debere  cum  eadem  congruere.  Quamobrem  in  hisce  casibus  evane- 
scit  vis  ilia  perpendicularis  FO,  quae  in  prsecedentibus  binis  figuris  habebatur,  ac  in  iis 
per  unicam  aequationem  res  omnis  absolvitur  (f),  quarum  ea,  quae  ad  posteriorem  casum 
pertinet,  admodum  facile  invenitur. 


exhfbentis° 
casus  posterioris. 


2I5'  ^egem  pro  recta  perpendiculari  rectae  jungenti  duo  puncta,  &  asque  distanti  ab 
utroque  exhibet  fig.  24,  quse  vitandae  confusionis  causa  exhibetur,  ubi  sub  numero  24 
habetur  littera  B,  sed  quod  ad  ejus  constructionem  pertinet,  habetur  separatim,  ubi  sub 
num.  24  habetur  littera  A  ;  ex  quibus  binis  figuris  fit  unica  ;  si  puncta  XYEAE'  censeantur 
utrobique  eadem.  In  ea  X,  Y  sunt  duo  materiae  puncta,  &  ipsam  XY  recta  CC*  secat 
bifariam  in  A.  Curva,  quae  vires  compositas  ibi  exhibet  per  ordinatas,  constructa  est  ex 
fig.  I,  quod  fieri  potest,  inveniendo  vires  singulas  singulorum  punctorum,  turn  vim  com- 
positam  ex  iis  more  consueto  juxta  [100]  generalem  constructionem  numeri  205  ;  sed 
etiam  sic  facilius  idem  praestatur  ;  centro  Y  intervallo  cujusvis  abscissae  Ad  figurae  I  in- 
veniatur  in  figura  24  sub  littera  A  in  recta  CC'  punctum  d,  sumaturque  de  versus  Y 
aequalis  ordinatae  dh  figuras  i  ,  ductoque  ea  perpendiculo  in  CA,  erigatur  eidem  CA  itidem 
perpendicularis  dh  dupla  da  versus  plagam  electam  ad  arbitrium  pro  attractionibus,  vel 
versus  oppositam,  prout  ilia  ordinata  in  fig.  I  attractionem,  vel  repulsionem  expresserit, 
&  erit  punctum  h  ad  curvam  exprimentem  legem  virium,  qua  punctum  ubicunque 
collocatum  in  recta  C'C  solicitatur  a  binis  X,  Y. 


de 


proprietates.  ° 


Demonstratio  facilis  est  :  si  enim  ducatur  dX,  &  in  ea  sumatur  dc  aequalis  de, 
ac  compleatur  rhombus  debc  ;  patet  fore  ejus  verticem  b  in  recta  dA  secante  angulum 
XdY  bifariam,  cujus  diameter  db  exprimet  vim  compositam  a  binis  de,  dc,  quae  bifariam 
secaretur  a  diametro  altera  ec,  &  ad  angulos  rectos,  adeoque  in  ipso  illo  puncto  a  ;  &  dh, 
dupla  da,  aequabitur  db  exprimenti  vim,  quae  respectu  A  erit  attractiva,  vel  repulsiva,  prout 
ilia  dh  figurae  I  fuerit  itidem  attractiva,  vel  repulsiva. 

2I7-  Porro  ex  ipsa  constructione  patet,  si  centro  Y,  intervallis  AE,  AG,  AI  figuras  i 
inveniantur  in  recta  CAC'  hujus  figurae  positae  sub  littera  B  puncta  E,  G,  I,  &c,  ea  fore 
limites  respectu  novae  curvas  ;  &  eodem  pacto  reperiri  posse  limites  E',  G',  Y,  &c.  ex  parte 
opposita  A  ;  in  iis  enim  punctis  evanescente  de  figuras  ejusdem  positae  sub  A,  evadit  nulla 
da,  &  db.  Notandum  tamen,  ibi  in  figura  posita  sub  B  mutari  plagam  attractivam  in 


(p)  Ducta  enim  LK  in  Fig.  23.  ipsam  FC  secabit  alicubi  in  I  bifariam,  W  ad  angulos  rectos  ex  rhombi  natura. 


Dicatur  CD  =  x,  CF  =  y,  DB  =  a,  W  erit  CB  =  Vaa  +  xx,  fcf  CD  =  *.CB  =  Vaa  +  xx  :  :  CI  =  Jy.CK  =— 


\/aa  +  xx,  quo  valore  posito  in  tequatione  curvie  figura  I  pro  valore  ordinata,  y  vaa  +  xx  ffo  valore  abscissa,  habebitur 
immediate  cfquatio  nova  per  x,  y,  W  constants,  qua  ejusmodi  curvam  deUrminabit, 


A  THEORY  OF  NATURAL  PHILOSOPHY 


165 


0 


O 


0 


i66 


PHILOSOPHIC  NATURALIS  THEORIA 


o 


^> 


A  THEORY  OF  NATURAL  PHILOSOPHY  167 

&  all  the  specific  differences  between  the  large  number  of  bodies  which  they  yield ;  the 
agreement  between  the  Theory  &  the  whole  of  Nature  is  truly  remarkable.  But  what 
has  so  far  been  said  refers  to  masses,  &  to  the  application  of  the  Theory  to  Physics.  Before 
we  come  to  this,  however,  I  will  discuss  certain  particular  cases,  out  of  an  innumerable 
number  of  those  which  refer  to  the  different  laws  concerning  the  action  of  two  points  on 
a  third. 

214.  If  we  wish  to  consider  the  laws  that  arise  in  the  case  of  a  straight  line  drawn  The  force  exerted 
through  D  perpendicular  to  AB,  or  in  the  case  of  AB  itself  produced  on  either  side,  first  by.  two  points  on  a 
of  all  it  is  easily  seen  that  the  direction  of  the  resultant  force  in  either  case  will  coincide  the?*  straight*1  line 
with  the  line  itself  without  any  lateral  force  or  any  declination  from  the  straight  line  which  Joining  them,  or  in 
is  drawn  towards  or  away  from  D.     In  the  case  of  AB  itself  the  matter  is  self-evident ;  whicifblsects  it'at 
for  the  forces  which  pertain  to  the  two  points  either  have  the  same  direction  as  one  another,  "8ht  angles. 

or  are  opposite  in  direction,  since  the  third  point  lies  in  the  same  straight  line  as  each  of 
the  two  former  points.  Whence  it  comes  about  that  the  resultant  force  is  equal  to  the 
sum,  or  the  difference,  of  the  two  component  forces ;  &  it  will  be  in  the  same  straight 
line  as  they.  In  the  case  of  the  line  at  right  angles,  the  matter  can  be  quite  easily 
demonstrated.  For,  if  in  Fig.  23  the  straight  line  DC  were  perpendicular  to  AB,  passing 
through  its  middle  point,  then  will  AC,  BC  be  equal  to  one  another.  Hence,  the  forces, 
by  which  C  is  influenced  by  A  &  B,  will  also  be  equal ;  secondly,  they  will  either  be  both 
attractive,  as  CL,  CK,  or  they  will  be  both  repulsive,  as  CN,  CM.  Hence  the  resultant 
force,  CF,  or  CH,  will  be  the  diagonal  of  a  rhombus,  &  thus  it  will  bisect  the  angle  LCK, 
or  NCM.  Now  since  these  angles  are  also  bisected  by  the  straight  line  DC,  on  account 
of  the  equality  of  the  triangles  DCA,  DCB,  it  is  evident  that  CF,  CH  must  coincide  with 
DC.  Therefore,  in  these  cases  the  perpendicular  force  FO,  which  was  obtained  in  the 
two  previous  figures,  will  vanish.  Also  in  these  cases,  the  whole  matter  can  be  represented 
by  a  single  equation  (?)  ;  &  the  one,  which  refers  to  the  latter  case,  can  be  found  quite 
easily. 

215.  The  law  in  the  case  of  the  straight  line  perpendicular  to  the  straight  line  joining  Construction  for 
the  two  points,  &  equally  distant  from  each,  is  graphically  given  in  Fig.  24  ;    to  avoid  *he  curve .  eiv^s 

.     .  .  ^  1r.'     .  •      -r-  i  -i        i    r  •        r       •     •        •  .the     law     in      the 

confusion  the  curve  itself  is  given  in  rig.  243,  whilst  the  construction  for  it  is  given  separately  second  case. 

in  Fig.  24A.     These  two  figures  are  but  one  &  the  same,  if   the  points  X,Y,E,A,E'  are 

supposed  to  be  the  same  in  both.     Then,  in  the  figure,  X,Y  are  two  points  of  matter,  & 

the  straight  line  CC'  bisects  XY  at  A.     The  curve,  which  here  gives  the  resultant  forces 

by  means  of  the  ordinates  drawn  to  it,  is  constructed  from  that  of  Fig.  i  :  &  this  can  be 

done,  by  finding  the  forces  for  the  points,  each  for  each,  then  the  force  compounded  from 

them  in  the  usual  manner  according  to  the  general  construction  given  in  Art.  205.     But 

the  same  thing  can  be  more  easily  obtained  thus  :, — With  centre  Y,  &  radius  equal  to  any 

abscissa  Ad  in  Fig.  i,  construct  a  point  d  in  the  straight  line  CC',  of  Fig.  24A,  &  mark  off 

de  towards  Y  equal  to  the  ordinate  db  in  Fig.  i  ;    draw  ea  perpendicular  to  CA,  &  erect 

a  perpendicular,  dh,  to  the  same  line  CA  also,  so  that  dh  =  2ae  ;  this  perpendicular  should 

be  drawn  towards  the  side  of  CA  which  is  chosen  at  will  to  represent  attractions,  or  towards 

the  opposite  side,  according  as  the  ordinate  in  Fig.  i  represents  an  attraction  or  a  repulsion  ; 

then  the  point  h  will  be  a  point  on  the  curve  expressing  the  law  of  forces,  with  which  a 

point  situated  anywhere  on  the  line  CC'  will  be  influenced  by  the  two  points  X  &  Y. 

.    216.  The  demonstration  is  easy.     For,  if  dX  is  drawn,  &  in  it  dc  is  taken  equal  to  de,  Proof  of  the  fore- 
&  the  rhombus  debc  is  completed,  then  it  is  clear  that  the  point  b  will  fall  on  the  straight  g°»ng  construction, 
line  dA.  bisecting  the  angle  X/Y ;   &  the  diagonal  of  this  rhombus  represents  the  resultant 
of  the  two  forces  de,  dc.     Now,  this  diagonal  is  bisected  at  right  angles  by  the  other  diagonal 
ec,  &  thus,  at  the  point  a  in  it.     Also  dh,  being  double  of  da,  will  be  equal  to  db,  which 
expresses  the  resultant  force  ;  this  will  be  attractive  with  respect  to  A,  or  repulsive,  according 
as  the  ordinate  dh  in  Fig.  I  is  also  attractive  or  repulsive. 

217.  Further,  from    the    construction,  it  is  evident    that,  if  with  centre  Y  &  radii  Further  properties 
respectively  equal  to  AE,  AG,  AI  in  Fig.  i,  there  are  found  in  the  straight  line  CAC'  of  sort.  °' 
Fig.  248  the  points  E,  G,  I,  &c,  then  these  will  be  limit-points  for  the  new  curve  ;    & 
that  in  the  same  way  limit-points  E',  G',  I',  &c.  may  be  found  on  the  opposite  side  of  A. 
For,  since  at  these  points,  in  Fig.  24A,  de  vanishes,  it  follows  that  da  &  db  become  nothing 
also.     Yet  it  must  be  noted  that,  in  this  case,  in  Fig.  248,  there  is  a  change  from  the  attractive 

(p)  For,  if  in  Fig.  23,  LK  is  drawn,  it  will  cut  FC  somewhere,  in  I  say  ;    &  it  will  be  at  right  angles  to  it 
on  account  of  the  nature  of  a  rhombus.     Sup-pose  CD  =  x,  CF  =  y,  DB  =  a  ;  then  CB  =  •\/(az+  x2),  W  toe  have 

CD   (or  x)  :  CB  (or   ^(a*  +  x*)  =  CI   (or  Jy)  :   CK,    /.   CK  =  y.yV  +  x*)/2x ; 

y  if  this  value  is  substituted  in  the  equation  of  the  curve  in  Fig.  I  instead  of  the  ordinate,  W  ^/  (az  +  xz)  for  the 
abscissa,  we  shall  get  straightaway  a  new  equation  in  x,  y,  \£  constants  ;  &  Ms  will  determine  a  curve  of  the  kind 
under  consideration. 


1 68  PHILOSOPHIC  NATURALIS  THEORIA 

repulsivam,  &  vice  versa  ;  nam  in  toto  tractu  CA  vis  attractiva  ad  A  habet  directionem 
CC',  &  in  tractu  AC'  vis  itidem  attractiva  ad  A  habet  directionem  oppositam  C'C.  Deinde 
facile  patebit,  vim  in  A  fore  nullam,  ubi  nimirum  oppositae  vires  se  destruent,  adeoque 
ibi  debere  curvam  axem  secare  ;  ac  licet  distantiae  AX,  AY  fuerint  perquam  exiguse,  ut 
idcirco  repulsiones  singulorum  punctorum  evadant  maximse  ;  tamen  prope  A  vires  erunt 
perquam  exiguae  ob  inclinationes  duarum  virium  ad  XY  ingentes,  &  contrarias ;  &  si  ipsae 
AY,  AX  fuerint  non  majores,  quam  sit  AE  figurae  I  ;  postremus  arcus  EDA  erit  repulsivus  ; 
secus  si  fuerint  majores,  quam  AE,  &  non  majores,  quam  AG,  atque  ita  porro  ;  cum  vires 
in  exigua  distantia  ab  A  debeant  esse  ejus  directionis,  quam  in  fig.  I  requirunt  abscissas 
paullo  majores,  quam  sit  haec  YA.  Postrema  crura  T/>V,T"/>'V,  patet,  fore  attractiva  ; 
&  si  in  figura  I  fuerint  asymptotica,  fore  asymptotica  etiam  hie  ;  sed  in  A  nullum  erit 
asymptoticum  crus. 


2l8>  At  curva  Cluae  exhibet  in  fig.  25  legem  virium  pro  recta  CC'  transeunte  per  duo 
casus  prioris.  puncta  X,  Y,  est  admodum  diversa  a  priore.     Ea  facile  construitur  :   satis  est  pro  quovis 

ejus  puncto  d  assumere  in  fig.  I  duas  abscissae  aequales,  alteram  Yd  hujus  figurae,  alteram 
Xd  ejusdem,  &  sumere  hie  db  aequalem  [101]  summae,  vel  differentiae  binarum  ordinatarum 
pertinentium  ad  eas  abscissas,  prout  fuerint  ejusdem  directionis,  vel  contrariae,  &  earn 
ducere  ex  parte  attractiva,  vel  repulsiva,  prout  ambae  ordinatae  figurae  I,  vel  earum  major, 
attractiva  fuerit,  vel  repulsiva.  Habebitur  autem  asymptotus  bYc,  &  ultra  ipsam  crus 
asymptoticum  DE,  citra  ipsam  autem  crus  itidem  asymptoticum  dg  attractivum  respectu 
A,  cui  attractivum,  sed  directionis  mutatas  respectu  CC',  ut  in  fig.  superiore  diximus,  ad 
partes  oppositas  A  debet  esse  aliud  g'd',  habens  asymptotum  c'V  transeuntem  per  X ; 
ac  utrumque  crus  debet  continuari  usque  ad  A,  ubi  curva  secabit  axem.  Hoc  postremum 
patet  ex  eo,  quod  vires  oppositae  in  A  debeant  elidi ;  illud  autem  prius  ex  eo,  quod  si  a 
sit  prope  Y,  &  ad  ipsum  in  infinitum  accedat,  repulsio  ab  Y  crescat  in  infinitum,  vi,  quae 
provenit  ab  X,  manente  finita  ;  adeoque  tam  summa,  quam  differentia  debet  esse  vis 
repulsiva  respectu  Y,  &  proinde  attractiva  respectu  A,  quae  imminutis  in  infinitum  distantiis 
ab  Y  augebitur  in  infinitum.  Quare  ordinata  ag  in  accessu  ad  bYc  crescet  in  infinitum  ; 
unde  consequitur,  arcum  gd  fore  asymptoticum  respectu  Yc ;  &  eadem  erit  ratio  pro  a'g', 
&  arcu  g'd'  respectu  b'Xc'. 


Ejus   curvae    pro-  219.  Poterit  autem  etiam  arcus  curvae  interceptus  asymptotis  bYc,  b'Xc'  sive  cruribus 

at        mutata 
puncto- 

* 


mutata  ^S>  ^'g'  secare  alicubi  axem,  ut  exhibet  figura  26  ;  quin  immo  &  in  locis  pluribus,  si  nimirum 
distantia    puncto-  AY  sit  satis  major,  quam  AE  figurae  i,  ut  ab  Y  habeatur  alicubi  citra  A  attractio,  &  ab  X 


curva  casus*aiterius!  repulsio,  vel  ab  X  repulsio  major,  quam  repulsio  ab  Y.  Ceterum  sola  inspectione 
postremarum  duarum  figurarum  patebit,  quantum  discrimen  inducat  in  legem  virium,  vel 
curvam,  sola  distantia  punctorum  X,  Y.  Utraque  enim  figura  derivata  est  a  figura  I,  &  in 
fig.  25  assumpta  est  XY  sequalis  AE  figurae  I,  in  fig.  26  aequalis  AI,  ejusdem  quae  variatio 
usque  adeo  mutavit  figurse  genitae  ductum  ;  &  assumptis  aliis,  atque  aliis  distantiis  punc- 
torum X,  Y,  aliae,  atque  aliae  curvae  novae  provenirent,  quae  inter  se  collatae,  &  cum  illis, 
quae  habentur  in  recta  CAC'  perpendiculari  ad  XAY,  uti  est  in  fig.  24  ;  ac  multo  magis 
cum  iis,  quae  pertinentes  ad  alias  rectas  mente  concipi  possunt,  satis  confirmant  id,  quod 
supra  innui  de  tanta  multitudine,  &  varietate  legum  provenientium  a  sola  etiam  duo- 
rum  punctorum  agentium  in  tertium  dispositione  diversa  ;  ut  &  illud  itidem  patet  ex 
sola  etiam  harum  trium  curvarum  delineatione,  quanta  sit  ubique  conformitas  in  arcu  illo 
attractive  TpV,  ubique  conjuncta  cum  tanto  discrimine  in  arcu  se  circa  axem  contorquente. 


genera  hujus          220.  Verum  ex  tanto  discriminum  numero  unum  seligam  maxime  notatu    dignum, 
Usima!0  'g    &  maximo  nobis  usui  futurum  inferius.     Sit  in  fig.  2jC — 'AC  axis  idem,  ac  in  fig.  i,  &  quin- 

que  arcus  consequenter  accept!  alicubi  GHI,  IKL,  LMN,  NOP,  PQR  sint  aequales 
prorsus  inter  se,  ac  similes.  Ponantur  autem  bina  puncta  B',  B  hinc,  &  inde  ab  A  in  fig.  28 
[102]  ad  intervallum  aequale  dimidiae  amplitudini  unius  e  quinque  iis  arcubus,  uti  uni 
GI,  vel  IL ;  in  fig.  29  ad  intervallum  aequale  integrae  ipsi  amplitudini ;  in  fig.  30  ad 
intervallum  aequale  duplae  ;  sint  autem  puncta  L,  N  in  omnibus  hisce  figuris  eadem,  & 
quaeratur,  quae  futura  sit  vis  in  quovis  puncto  g  in  intervallo  LN  in  hisce  tribus  posi- 
tionibus  punctorum  B',  B. 


A  THEORY  OF  NATURAL  PHILOSOPHY 


169 


1 7o 


PHILOSOPHISE  NATURALIS  THEORIA 


A  THEORY  OF  NATURAL  PHILOSOPHY  171 

side  to  the  repulsive  side,  &  vice  versa.  For  along  the  whole  portion  CA,  the  force  of 
attraction  towards  A  has  the  direction  CC',  whilst  for  the  portion  AC',  the  force  of  attraction 
also  towards  A  has  the  direction  C'C.  Secondly,  it  will  be  clear,/  seen  that  the  force  at 
A  will  be  nothing  ;  for  there  indeed  the  forces,  being  equal  &  opposite,  cancel  one  another, 
&  so  the  curve  cuts  the  axis  there  ;  &  although  the  distances  AX,  AY  would  be  very  small, 
&  thus  the  repulsions  due  to  each  of  the  two  points  would  be  Immensely  great,  nevertheless, 
close  to  A,  the  resultants  would  be  very  small,  on  account  of  the  inclinations  of  the  two 
forces  to  XY  being  extremely  great  &  oppositely  inclined.  Also  if  AY,  AX  were  not  greater 
than  AE  in  Fig.  i,  the  last  arc  would  be  repulsive  ;  &  attractive,  if  they  were  greater  than 
AE,  but  not  greater  than  AG,  &  so  on  ;  for  the  forces  at  very  small  distances  from  A  must 
have  their  directions  the  same  as  that  required  in  Fig.  I  for  abscissae  that  are  slightly  greater 
than  YA.  The  final  branches  TpV,  T'p'V  will  plainly  be  attractive  ;  &,  if  in  Fig.  i  they 
were  asymptotic,  they  would  also  be  asymptotic  in  this  case  ;  but  there  will  not  be  an 
asymptotic  branch  at  A. 

218.  But  the  curve,  in  Fig.  25,  which  expresses  the  law  of  forces  for  the  straight  line  Construction    fo- 
CC',  when  it  passes  through  the  points  X,Y,  is  quite  different  from  the  one  just  considered.  j£|  the^aw^'tte 
It  is  easily  constructed  ;    it  is  sufficient,  for  any  point  d  upon  it,  to  take,  in  Fig.  i,  two  first  case, 
abscissae,  one  equal  to  Yd,  &  the  other  equal  to  Xd ;  &  then,  for  Fig.  25,  to  take  dh  equal 

to  the  sum  or  the  difference  of  the  two  ordinates  corresponding  to  these  abscissas,  according 
as  they  are  in  the  same  direction  or  in  opposite  directions ;  &,  according  as  each  ordinate, 
or  the  greater  of  the  two,  in  Fig.  i,  is  attractive  or  repulsive,  to  draw  dh  on  the  attractive 
or  repulsive  side  of  CC'.  Moreover  there  will  be  obtained  an  atymptote  bYc ;  on  the 
far  side  of  this  there  will  be  an  asymptotic  branch  DE,  &  on  the  near  side  of  it  there  will 
also  be  an  asymptotic  branch  dg,  which  will  be  attractive  with  respect  to  A  ;  &  with  respect 
to  this  part,  there  must  be  another  branch  g'd',  which  is  attractive  but,  since  the  direction 
with  regard  to  CC'  is  altered,  as  we  mentioned  in  the  case  of  the  preceding  figure,  falling 
on  the  opposite  side  of  CC' ;  this  has  an  asymptote  c'b'  passing  through  X.  Also  each 
branch  must  be  continuous  up  to  the  point  A,  where  it  cuts  thVaxis.  This  last  fact  is 
evident  from  the  consideration  that  the  equal  &  opposite  forces  at  A  must  cancel  one  another  ; 
&  the  former  is  clear  from  the  fact  that,  if  a  is  very  near  to  Y,  &  approaches  indefinitely 
near  to  it,  the  repulsion  due  to  Y  increases  indefinitely,  whilst  the  force  due  to  X  remains 
finite.  Thus,  both  the  sum  &  the  difference  must  be  repulsive  with  respect  to  Y,  &  therefore 
attractive  with  respect  to  A  ;  &  this,  as  the  distance  from  Y  is  diminished  indefinitely,  will 
increase  indefinitely.  Hence  the  ordinate  ag,  when  approaching  bYc,  increases  indefinitely  : 
&  it  thus  follows  that  the  arc  gd  will  be  asymptotic  with  respect  to  Yc ;  &  the  reasoning 
will  be  the  same  for  a'g',  &  the  arc  g'd',  with  respect  to  b'Xc', 

219.  Again,  it  is  even  possible  that  the  arc  intercepted  between  the  asymptotes  bYc,  The  properties  of 
b'Xc',  i.e.,  between  the  branches  dg,  d'g',  to  cut  the  axis  somewhere,  as  is  shown  in  Fig.  26  ;  encescor^espo'n'dSg 
nay  rather,  it  may  cut  it  in  more  places  than  one,  for  instance,  if  AY  is  sufficiently  greater  to    changed     dis- 
than  AE  in  Fig.  i  ;  so  that,  at  some  place  on  the  near  side  of  A,  there  is  obtained  an  attraction  p^s  ^"clrnpari6 
from  the  point  Y  &  a  repulsion  from  the  point  X,  or  a  repulsion  from  X  greater  than  the  son  with  the  curve 
repulsion  fiom  Y.     Besides,  by  a  mere  inspection  of  the  last  two  figures,  it  will  be  evident  other  case. in  the 
how  great  a  difference  in  the  law  of  forces,  &  the  curve,  may  be  derived  from  the  mere 

distance  apart  of  the  points  X  &  Y.  For  both  figures  are  derived  from  Fig.  I,  &,  in  Fig.  25, 
XY  is  taken  equal  to  AE  in  Fig.  i ,  whilst,  in  Fig.  26,  it  is  taken  equal  to  AI  of  Fig.  i  ;  & 
this  variation  alone  has  changed  the  derived  figure  to  such  a  degree  as  is  shown.  If  other 
distances,  one  after  another,  are  taken  for  the  points  X  &  Y,  fresh  curves,  one  after  the 
other,  will  be  produced.  If  these  are  compared  with  one  another,  &  with  those  that  are 
obtained  for  a  straight  line  CAC'  perpendicular  to  XAY,  like  the  one  in  Fig  24,  nay,  far 
more,  if  they  are  compared  with  those,  referring  to  other  straight  lines,  that  can  be  imagined, 
will  sufficiently  confirm  what  has  been  said  above  with  regard  to  the  immense  number  & 
variety  of  the  laws  arising  from  a  mere  difference  of  disposition  of  the  two  points  that  act 
on  the  third.  Also,  from  the  drawing  of  merely  these  three  curves,  it  is  plainly  seen 
what  great  uniformity  there  is  in  all  cases  for  the  attractive  arc  TpV,  combined  always 
with  a  great  dissimilarity  for  the  arc  that  is  twisted  about  the  axis. 

_  220.  But  I  will  select,  from  this  great  number  of  different  cases,  one  which  is  worth  T^Tee    classes    of 
notice  in  a  high  degree,  which  also  will  be  of  the  greatest  service  to  us  later.     In  Fig.  27,  weu 
let  CAC'  be  the  same  axis  as  in  Fig.  i,  &  let  the  five  arcs,  GHI,  IKL,  LMN,  NOP,  PQR 
taken  consecutively  anywhere  along  it,  be  exactly  equal  &  like  one  another.    Moreover, 
in  Fig.  28,  let  the  two  points  B  &  B',  one  on  each  side  of  A,  be  taken  at  a  distance  equal 
to  half  the  width  of  one  of  these  five  arcs,  i.e.,  half  of  the  one  GL,  or  LI  ;  in  Fig.  29,  at 
3.  distance  equal  to  the  whole  of  this  width  ;    &,  in  Fig.  30,  at  a  distance  equal  to  double 
the  width  ;   also  let  the  points  L,N  be  the  same  in  all  these  figures.     It  is  required  to  find 
the  force  at  any  point  g  in  the  interval  LN,  for  these  three  positions  of  the  points  B  &  B'. 


I72 


PHILOSOPHISE  NATURALIS  THEORIA 


Determinatip  vis 
compositaa  in  iis- 
dem. 


221.  Si  in  Fig.  27  capiantur  hinc,  &  inde  ab  ipso  g  intervalla  sequalia  intervallis  AB', 
AB  reliquarum  trium  figurarum  ita,  ut  ge,  gi  respondeant  figurae  28  ;  gc,  gm  figures  29 ; 
ga,  go  figurae  30  ;  patet,  intervallum  ei  fore  aequale  amplitudini  LN,  adeoque  Le,  Ni 
aequales  fore  dempto  communi  Lz,  sed  puncta  e,  i  debere  cadere  sub  arcus  proximos 
directionum  contrariarum  ;  ob  arcuum  vero  aequalitatem  fore  aequalem  vim  ef  vi  contrariae 
il,  adeoque  in  fig.  28  vim  ab  utraque  compositam,  respondentem  puncto  g,  fore  nullam. 
At  quoniam  gc,  gm  integrae  amplitudini  aequantur ;  cadent  puncta  c,  m  sub  arcus  IKL, 
NOP,  conformes  etiam  directione  inter  se,  sed  directionis  contraries  respectu  arcus  LMN, 
eruntque  asquales  wzN,  cl  ipsi  gL,  adeoque  attractiones  mn,  cd,  &  repulsioni  gh  aequales, 
&  inter  se  ;  ac  idcirco  in  figura  29  habebitur  vis  attractiva  gh  composita  ex  iis  binis  dupla 
repulsivae  figurae  27.  Demum  cum  ga,  go  sint  sequales  duplae  amplitudini,  cadent  puncta 
a,  o  sub  arcus  GHI,  PQR  conformis  directionis  inter  se,  &  cum  arcu  LMN,  eruntque 
pariter  binae  repulsiones  ab,  op  aequales  repulsioni  gh,  &  inter  se.  Quare  vis  ex  iis  com- 
positae  pro  fig.  30  erit  repulsio  gh  dupla  repulsionis  gh  figurae  27,  &  aequalis  attraction! 
figurae  29. 


vhn'in  tractu*'0116  222<  ^^  igitur  jam  patet,  loci  geometric!  exprimentis  vim  compositam,  qua  bina 

tinuo    nuiiam,    in  puncta  B',  B  agunt  in  tertium,  partem,  quae  respondet  intervallo  eidem  LN,  fore  in  prima 
aha    attractionem,  e  tribus  eorum  positionibus  propositis  ipsum  axem  LN,  in  secunda  arcum  attractivum 

in  aha  repulsionem,    T  ,  ,,.,     .  .  ,  .  i  i  •  ,  r 

manente  distantia ;  LMN,  in  tertia  repulsivum,  utroque  reccdente  ab  axe  ubique  duplo  plus,  quam  in  fig. 
Physica  27  ;  ac  pro  quovis  situ  puncti  g  in  toto  intervallo  LN  in  primo  e  tribus  casibus  fore  prorsus 
nullam,  in  secundo  fore  attractionem,  in  tertio  repulsionem  aequalem  ei,  quam  bina  puncta 
B',  B  exercerent  in  tertium  punctum  situm  in  g,  si  collocarentur  simul  in  A,  licet  in  omnibus 
hisce  casibus  distantia  puncti  ejusdem  g  a  medio  systematis  eorundem  duorum  punctorum, 
sive  a  centre  particulae  constantis  iis  duobus  punctis  sit  omnino  eadem.  Possunt  autem 
in  omnibus  hisce  casibus  puncta  B',  B  esse  simul  in  arctissimis  limitibus  cohaesionis  inter 
se,-  adeoque  particulam  quandam  constantis  positionis  constituere.  Aequalitas  ejusmodi 
accurata  inter  arcus,  &  amplitudines,  ac  limitum  distantias  in  figura  I  non  dabitur  uspiam  ; 
cum  nullus  arcus  curvae  derivatae  utique  continuae,  deductae  nimirum  certa  lege  a  curva 
continua,  possit  congruere  accurate  cum  recta  ;  at  poterunt  ea  omnia  ad  sequalitatem 
accedere,  quantum  [103]  libuerit ;  poterunt  haec  ipsa  discrimina  haberi  ad  sensum  per 
tractus  continues  aliis  modis  multo  adhuc  pluribus,  immo  etiam  pluribus  in  immensum, 
ubi  non  duo  tantummodo  puncta,  sed  immensus  eorum  numerus  constituat  massulas, 
quae  in  se  agant,  &  ut  in  hoc  simplicissimo  exemplo  deprompto  e  solo  trium  punctorum 
systemate,  multo  magis  in  systematis  magis  compositas,  &  plures  idcirco  variationes  admit- 
tentibus,  in  eadem  centrorum  distantia,  pro  sola  varia  positione  punctorum  componentium 
massulas  ipsas  vel  a  se  mutuo  repelli,  vel  se  mutuo  attrahere,  vel  nihil  ad  sensum  agere  in 
se  invicem.  Quod  si  ita  res  habet,  nihil  jam  mirum  accidet,  quod  quaedam  substantial 
inter  se  commixtse  ingentem  acquirant  intestinarum  partium  motum  per  effervescentiam, 
&  fermentationem,  quas  deinde  cesset,  particulis  post  novam  commixtionem  respective 
quiescentibus ;  quod  ex  eodem  cibo  alia  per  secretionem  repellantur,  alia  in  succum 
nutrititium  convertantur,  ex  quo  ad  eandem  prseterfluente  distantiam  alia  aliis  partibus 
solidis  adhaereant,  &  per  alias  valvulas  transmittantur,  aliis  libere  progredientibus.  Sed 
adhuc  multa  supersunt  notatu  dignissima,  quae  pertinent  ad  ipsum  etiam  adeo  simplex 
trium  punctorum  systema. 


Alius  casus  vis  nul- 
lius  trium  puncto- 
rum positorum  in 
directum  e  x  dis- 
tantiis  limitum : 
tres  alii  in  quorum 
binis  vis  nulla  ex 
elisione  contrari- 
arum. 


223.  Jaceant  in  figura  31  tria  puncta  A,D,B,  in  directum  :  ea  poterunt  respective 
quiescere,  si  omnibus  mutuis  viribus  careant,  quod  fieret,  si  tres  distantiae  AD,  DB,  AB 
omnes  essent  distantiae  limitum  ;  sed  potest  haberi  etiam  quies  respectiva  per  elisionem 
contrariarum  virium.  Porro  virium  mutuarum  casus  diversi  tres  esse  poterunt  :  vel  enim 
punctum  medium  D  ab  utroque  extremorum  A,  B  attrahitur,  vel  ab  utroque  repellitur, 
vel  ab  altero  attrahitur,  ab  altero  repellitur.  In  hoc  postremo  casu,  patet,  non  haberi 
quietem  respectivam  ;  cum  debeat  punctum  medium  moveri  versus  extremum  attrahens 
recedendo  simul  ab  altero  extremo  repellente.  In  reliquis  binis  casibus  poterit  utique 


A  THEORY  OF  NATURAL  PHILOSOPHY 


173 


c' 


C 


B'A   B 


FIG.  28. 


C' 


B'      A         B 


FIG.  29. 


cV 

1 

I 

g 

C 

B' 

T 

B 

L         N 

FIG.  30. 


'74 


PHILOSOPHIC  NATURALIS  THEORIA 


H 


R     C 


C' 


FIG.  27. 


Ln.      ," 5 


B'A   B 


FIG.  28. 


B'      A        B 


B7 


FIG  .29. 


B        L         N 


FIG.  30. 


A  THEORY  OF  NATURAL  PHILOSOPHY  175 

221.  If,  in  Fig.  27,  we  take,  on  either  side  of  this  point  g,  intervals  that  are  equal  to  Determination    of 
the  intervals  AB',  AB  of  the  other  three  figures  ;    so  that  ge,  gi  correspond  to  Fig.  28  ; 

gc,  gm  to  Fig.  29  ;  &  ga,  go  to  Fig.  30  ;  then  it  is  plain  that  the  interval  ei  will  be  equal 
to  the  width  LN,  &  thus,  taking  away  the  common  part  Lz,  we  have  L£  &  Ni  equal  to  one 
another,  but  the  points  e  &  i  must  fall  under  successive  arcs  of  opposite  directions.  Now, 
on  account  of  the  equality  of  the  arcs,  the  force  ef  will  be  equal  to  the  opposite  force  il  ; 
thus,  in  Fig.  28,  the  force  compounded  from  the  two,  corresponding  to  the  point  g,  will 
be  nothing.  Again,  in  Fig.  29,  since  gc,  gm  are  each  equal  to  the  whole  width  of  an  arc, 
the  points  c  &  m  fall  under  arcs  IKL,  NOP,  which  lie  in  the  same  direction  as  one  another, 
but  in  the  opposite  direction  to  the  arc  LMN.  Hence,  mN,  c\  will  be  equal  to  gL  ;  & 
thus  the  attractions  mn,  cd  will  be  equal  to  the  repulsion  gb,  &  to  one  another.  Therefore, 
in  Fig.  29,  we  shall  have  an  attractive  force,  compounded  of  these  two,  which  is  double 
of  the  repulsive  force  in  Fig.  27.  •  Lastly,  in  Fig.  30,  since  ga,  go  are  equal  to  double  the 
width  of  an  arc,  the  points  a  &  o  will  fall  beneath  arcs  GHI,  PQR,  lying  in  the  same  direction 
as  one  another,  &  as  that  of  the  arc  LMN  as  well.  As  before,  the  two  repulsions,  ab,  op 
will  be  equal  to  the  repulsion  gb,  &  to  one  another.  Hence,  in  Fig.  30,  the  force  compounded 
from  the  two  of  them  will  be  a  repulsion  gh  which  is  double  of  the  repulsion  gh  in  Fig.  27, 
&  equal  to  the  attraction  in  Fig.  29. 

222.  Therefore,  from  the  preceding  article,  it  is   now  evident  that  the  part  of  the  in     one    arrange- 


geometrical   locus    representing   the    resultant   force,   with   which  two   points  B',  B  act       "  '* 


region 
upon  a  third,  corresponding  to  the  same  interval  LN,  will  be  the  axis  LN  itself  in  the  first  no  force  at  ail,  in 

of  the  three  stated  positions  of  the  points  ;  in  the  second  position  it  will  be  an  attractive  attraction"5*  ^a 

arc  LMN,  &  in  the  third  a  repulsive  arc  ;    each  of  these  will  recede  from  the  axis  at  all  third  a    repulsion 

points  along  it  to  twice  the  corresponding  distance  shown  in  Fig.  27.     So,  for  any  position  maining^constant  • 

of  the  point  g  in  the  whole  interval  LN,  the  force  will  be  nothing  at  all  in  the  first  of  the  this  result  is  of  the 

three  cases,  an  attraction  in  the  second,  &  a  repulsion  in  the  third.    This  latter  will  be  -physics.  l 

equal  to  that  which  the  two  points  B',  B  would  exert  on  the  third  point,  if  they  were  both 

situated  at  the  same  time  at  the  point  A.     And  yet,  in  all  these  three  cases,  the  distance 

of  the  point  g  under  consideration  remains  absolutely  the  same,  measured  from  the  centre 

of  the  system  of  the  same  two  points,  or  from  the  mean  centre  of  a  particle  formed  from 

them.     Moreover,  in  all  three  cases,  the  points  B',B  may  be  in  the  positions  defining  the 

strongest  limits  of  cohesion  with  regard  to  one  another,  &  so  constitute  a  particle  fixed 

in  position.     Now  we  never  can  have  such  accurate  equality  as  this  between  the  arcs,  the 

widths,  &  the  distances  of  the  limit-points  ;  for  no  arc  of  the  derived  curve,  which  is  every- 

where continuous  because  it  is  obtained  by  a  given  law  from  a  continuous  curve,  can  possibly 

coincide  accurately  with  a  straight  line  ;   but  there  could  be  an  approximation  to  equality 

for   all  of   them,  to   any   degree    desired.     The    same    distinctions   could    be    obtained, 

approximately  for  continuous  regions  in  very  many  more  different  ways,  nay  the  number 

of  ways  is  immeasurable  ;    in  which  the  number  of  points  constituting  the  little  masses 

is  not  two  only,  but  a  very  large  number,  acting  upon  one  another  ;  &,  as  in  this  very  simple 

case  derived  from  a  consideration  of  a  single  system  of  three  points,  so,  much  more  in  systems 

that  are  more  complex  &  on  that  account  admitting  of  more  variations,  corresponding  to 

a  single  variation  of  the  points  composing  the  masses,  whilst  the  distance  between  the 

masses  themselves  remains  the  same,  there  may  be  either  mutual  repulsion,  mutual  attraction, 

or  no  mutual  action  to    any  appreciable    extent.      But,  that  being  the  case,    there  is 

nothing  wonderful  in  the  fact  that  certain   substances,  when  mixed  together,  acquire  a 

huge  motion  of  their  inmost  parts,  as  in  effervescence  &  fermentation  ;  this  motion  ceasing 

&  the  particles  attaining  relative  rest  after  rearrangement.     There  is  nothing  wonderful 

in  the  fact  that  from  the  same  food  some  things  are  repelled  by  secretion,  whilst  others 

are  converted  into  nutritious  juices  ;    &  that  from  these  juices,  though  flowing  past  at 

exactly  the  same  distances,  some  things  adhere  to  some  solid  parts  &  some  to  others  ;  that 

some  are  transmitted  through  certain  little  passages,  some  through  others,  whilst  some 

pass  along  uninterruptedly.     However,  there  yet  remain  many  things  with  regard  to  this  ever 

so  simple  system  of  three  points  ;   &  these  are  well  worth  our  attention. 

223.  In  Fig.  31,  let  A,D,B  be  three  points  in  a  straight  line.     These  will  be  at  rest  Another     instance 

•  i  11  i  n      i  •  .,     ,        of  no  force  in  the 

with  regard  to  one  another  if  they  lack  all  mutual  forces  ;   &  this  would  be  the  case,  it  the  case  of  three  points 
three  distances  AD,  DB,  AB  were  all  distances  corresponding  to  limit-points.     In  addition,  sitV?;,tf.d    i"a 

i     •  111          i       •        i          •  ,...*-»  in  •        f  ft.          straight  line  at  the 

relative  rest  could  be  obtained  owing  to  elimination  of  equal  &  opposite  iorces.     .further,  distances     corre- 
there  will  be  three  different  cases  with  regard  to  the  mutual  forces.     For,  either  the  middle  spending  to  Hmit- 

.  -i       r    i  i  •  >         T>  ••  11     ii  i       r    V.  points.     Ihree 

point  D  is  attracted  by  each  of  the  outside  points  A  &  B,  or  it  is  repelled  by  each  of  them,  others,  in   two  of 
or  it  is  attracted  by  one  of  them  &  repelled  by  the  other.     In  the  last  case,  it  is  evident  ^^res^tenl'farce 
that  relative  rest  could  not  obtain  ;   for  the  middle  point  must  then  be  moved  towards  the  arises  from  an  eii- 
outside  point  that  is  attracting  it,  &  recede  from  the  other  outside  point  which  is  repelling 
it  at  the  same  time.     But  in  the  other  two  cases,  it  is  at  least  possible  that  there  may  be 


PHILOSOPHIC  NATURALIS  THEORIA 


In  eorum  altero 
nisus  ad  recuper- 
andam  positionem, 
in  altero  ad  magis 
ab  ea  recedendum, 
si  incipiant  inde 
removeri. 


res  haberi  :  nam  vires  attractive,  vel  repulsive,  quas  habet  medium  punctum,  possunt 
esse  aequales ;  turn  autem  extrema  puncta  debebunt  itidem  attrahi  a  medio  in  primo  casu, 
repelli  in  secundo  ;  quae  si  se  invicem  e  contrario  aeque  repellant  in  casu  primo,  attrahant 
in  secundo  ;  poterunt  mutuse  vires  elidi  omnes. 

224.  Adhuc  autem  ingens  est  discrimen  inter  hosce  binos  casus.  Si  nimirum  puncta 
ilia  a  directione  rectae  lineae  quidquam  removeantur,  ut  nimirum  medium  punctum  D 
distet  jam  non  nihil  a  recta  AB,  delatam  in  C,  in  secundo  casu  adhuc  magis  sponte  recedet 
inde,  &  in  primo  accedet  iterum  ;  vel  si  vi  aliqua  externa  urgeatur,  conabitur  recuperare 
positionem  priorem,  &  ipsi  urgenti  vi  resi- 
stet.  Nam  binae  repulsiones  CM,  CN  adhuc 
habebuntur  in  secundo  casu  in  ipso  primo 
recessu  a  D  (licet  ese  mutatis  jam  satis  distan- 
tiis  BD,  AD  inBC,  AC,  evadere  possint  at- 
tractiones)  &  vim  com-[i  opponent  direc- 
tam  per  CH  contrariam  directioni  tendenti 
ad  rectam  AB.  At  in  primo  casu  habebuntur 
attractiones  CL,  CK,  quae  component  vim 
CF  directam  versus  AB,  quo  casu  attractio 
AP  cum  repulsione  AR,  et  attractio  BV, 
cum  repulsione  BS  component  vires  AQ,  BT, 
quibus  puncta  A,  B  ibunt  obviam  puncto  C 
redeunti  ad  rectam  transituram  per  illud 

T"V  /"* 

FIG.  31. 


Theoria  generalior 
indicata  :  t  r  i  u  m 
punctorum  jacen- 
tium  in  directum  : 
vis  maxima  ad 
conservandam  dis- 
tantiam. 


M 


R     A 


B     S 


punctum  E,  quod  est  in  triente  rectae  DC, 
&  de  quo  supra  mentionem  fecimus  num.  205. 

225.  Haec  Theoria  generaliter  etiam  non  rectilineae  tantum,  sed  &  cuivis  position! 
trium  massarum  applicari  potest,  ac  applicabitur  infra,  ubi  etiam  generale  simplicissimum, 
ac  fcecundissimum  theorema  eruetur  pro  comparatione  virium  inter  se  :  sed  hie  interea 
evolvemus  nonnulla,  quae  pertinent  ad  simpliciorem  hunc  casum  trium  punctorum.  In- 
primis  fieri  utique  potest,  ut  ejusmodi  tria  puncta  positionem  ad  sensum  rectilineam 
retineant  cum  prioribus  distantiis,  utcunque  magna  fuerit  vis,  quae  ilia  dimovere  tentet, 
vel  utcunque  magna  velocitas  impressa  fuerit  ad  ea  e  suo  respectivo  statu  deturbanda. 
Nam  vires  ejusmodi  esse  possunt,  ut  tarn  in  eadem  directione  ipsius  rectas,  quam  in 
directione  ad  earn  perpendicular!,  adeoque  in  quavis  obliqua  etiam,  quae  in  eas  duas  resolvi 
cogitatione  potest,  validissimus  exurgat  conatus  ad  redeundum  ad  priorem  locum,  ubi  inde 
discesserint  puncta.  Contra  vim  impressam  in  directione  ejusdem  rectae  satis  est,  si  pro 
puncto  medio  attractio  plurimum  crescat,  aucta  distantia  ab  utrolibet  extreme,  &  plurimum 
decrescat  eadem  imminuta  ;  ac  pro  utrovis  puncto  extreme  satis  est,  si  repulsio  decrescat 
plurimum  aucta  distantia  ab  extreme,  &  attractio  plurimum  crescat,  aucta  distantia  a 
medio,  quod  secundum  utique  fiet,  cum,  ut  dictum  est,  debeat  attractio  medii  in  ipsum 
crescere,  aucta  distantia.  Si  haec  ita  se  habuerint,  ac  vice  versa  ;  differentia  virium  vi 
extrinsecae  resistet,  sive  ea  tenet  contrahere,  sive  distrahere  puncta,  &  si  aliquod  ex  iis 
velocitatem  in  ea  directione  acquisiverit  utcunque  magnam,  poterit  differentia  virium 
esse  tanta,  ut  extinguat  ejusmodi  respectivam  velocitatem  tempusculo,  quantum  libuerit, 
parvo,  &  post  percursum  spatiolum,  quantum  libuerit,  exiguum. 


Quid  ubi  vis  exter- 


&  virgae  flexiiis. 


226.  Quod  si  vis  urgeat  perpendiculariter,  ut  ex.gr.  punctum  medium  D  moveatur 
per  rectam  DC  perpendicularem  ad  AB  ;  turn  vires  CK,  CL  possunt  utique  esse  ita  validae, 
ut  vis  composita  CF  sit  post  recessum,  quantum  libuerit,  exiguum  satis  magna  ad  ejusmodi 
vim  elidendam,  vel  ad  extinguendam  velocitatem  impressam.  In  casu  vis,  quas  constanter 
urgeat,  &  punctum  D  versus  C,  &  puncta  A,  B  ad  partes  oppositas,  habebitur  inflexio  ;  ac 
in  casu  vis,  quae  agat  in  eadem  directione  rectae  jungentis  puncta,  habebitur  contractio, 
seu  distractio  ;  sed  vires  resistentes  ipsis  poterunt  esse  ita  validae,  ut  &  inflexio,  &  contractio, 
vel  distractio,  sint  prorsus  insensibiles ;  [105]  ac  si  actione  externa  velocitas  imprimatur 
punctis  ejusmodi,  quae  flexionem,  vel  contractionem,  aut  distractionem  inducat,  turn 
ipsa  puncta  permittantur  sibi  libera  ;  habebitur  oscillatio  quasdam,  angulo  jam  in  alteram 
plagam  obverso,  jam  in  alteram  oppositam,  ac  longitudine  ejus  veluti  virgae  constantis  iis 
tribus  punctis  jam  aucta,  jam  imminuta,  fieri  poterit ;  ut  oscillatio  ipsa  sensum  omnem 
effugiat,  quod  quidem  exhibebit  nobis  ideam  virgae,  quam  vocamus  rigidam,  &  solidam, 
contractionis  nimirum,  &  dilatationis  incapacem,  quas  proprietates  nulla  virga  in  Natura 

[The  reader  should  draw  a  more  general  figure  for  Art.  224  &  227,  taking  AD,  DB 
unequal  and  CD  not  at  right  angles  to  AB.] 


A  THEORY  OF  NATURAL  PHILOSOPHY  177 

relative  rest ;  for  the  attractive,  or  repulsive,  forces  which  are  acting  on  the  middle  point 
may  be  equal.  But  then,  in  these  cases,  the  outside  points  must  be  respectively  attracted, 
or  repelled  by  the  middle  point ;  &  if  they  are  equally  &  oppositely  repelled  by  one  another 
in  the  first  case,  &  attracted  by  one  another  in  the  second  case,  then  it  will  be  possible  for 
all  the  mutual  forces  to  cancel  one  another. 

224.  Further,  there  is  also  a  very  great  difference  between  these  two  cases.     For  in    one   of   these 
instance,  if  the  points  are  moved  a  small  distance  out  of  the  direct  straight  line,  so  that  endeavour6 towards 
the  middle  point  D,  say,  is  now  slightly  off  the  straight  line  AB,  being  transferred  to  C,  a  recovery  of  posi- 
then,  if  left  to  itself,  it  will  recede  still  further  from  it  in  the  first  case,  &  will  approach  t^warts^iurther 
it  once  more  in  the  second  case.     Or,  if  it  is  acted  on  by  some  external  force,  it  will  endeavour  recession    from  it, 
to  recover  its  position  &  will  resist  the  force  acting  on  it.     For  two  repulsions,  CM,  CN,  moved  o 

will  at  first  be  obtained  in  the  second  case,  at  the  first  instant  of  motion  from  the  position  position. 
D  ;  although  indeed  these  may  become  attractions  when  the  distances  BD,  AD  are 
sufficiently  altered  into  the  distances  BC,  AC.  These  will  give  a  resultant  force 
acting  along  CH  in  a  direction  away  from  the  straight  line  AB.  But  in  the  first 
case  we  shall  have  two  attractions  CL,  CK ;  &  these  will  give  a  force  directed 
towards  AB.  In  this  case,  the  attraction  AP  combined  with  the  repulsion  AR,  & 
the  attraction  BV  combined  with  the  repulsion  BY,  will  give  resultant  forces,  AQ,  BT, 
under  the  action  of  which  the  points  A,B  will  move  in  the  opposite  direction  to  that  of 
the  point  C,  as  it  returns  to  the  straight  line  passing  through  that  point  E,  which  is  a  third 
of  the  way  along  the  straight  line  DC,  of  which  mention  was  made  above  in  Art.  205. 

225.  This  Theory  can  also  be  applied  more  generally,  to  include  not  only  a  position  Enunciation    of  a 
of  the  three  points  in  a  straight  line  but  also  any  position  whatever.     This  application  more  general  theory 
will  be  made  in  what  follows,  where  also  a  general  theorem,  of  a  most  simple  &  fertile  nature  ly^g  ^  a  straight 
will  be  deduced  for  comparison  of  forces  with  one  another.     But  for  the  present  we  will  line  ;  possibility  of 
consider  certain  points  that  have  to  do  with  this  more  simple  case  of  three  points.     First  tendSg^conser^ 
of  all,  it  may  come  about  that  three  points  of  this  kind  may  maintain  a  position  practically  vation  of  distance, 
in  a  straight  line,  no  matter  how  great  the  force  tending  to  drive  them  from  it  may  be, 

or  no  matter  how  great  a  velocity  may  be  impressed  upon  them  for  the  purpose  of  disturbing 
them  from  their  relative  positions.  For  there  may  be  forces  of  such  a  kind  that  both  in 
the  direction  of  the  straight  line,  &  perpendicular  to  it,  &  hence  in  any  oblique  direction 
which  may  be  mentally  resolved  into  the  former,  there  may  be  produced  an  extremely 
strong  endeavour  towards  a  return  to  the  initial  position  as  soon  as  the  points  had  departed 
from  it.  To  counterbalance  the  force  impressed  in  the  direction  of  the  same  straight  line 
itself,  it  is  sufficient  if  the  attraction  for  the  middle  point  should  increase  by  a  large  amount 
when  the  distance  from  either  of  the  outside  points  is  increased,  &  should  be  decreased 
by  a  large  amount  if  this  distance  is  decreased.  For  either  of  the  outside  points  it  is  sufficient 
if  the  repulsion  should  greatly  decrease,  as  the  distance  is  increased,  from  the  outside  point, 
and  the  attraction  should  greatly  increase,  as  the  distance  is  increased,  from  the  middle 
point ;  &  this  second  requirement  will  be  met  in  every  case,  since,  as  has  been  said,  and 
attraction  on  it  of  the  middle  point  will  necessarily  increase  when  the  distance  is  increased. 
If  matters  should  turn  out  to  be  as  stated,  or  vice  versa,  then  the  difference  of  the  forces 
will  resist  the  external  force,  whether  it  tries  to  bring  the  points  together  or  to  drive  them 
apart ;  &  if  any  one  of  them  should  have  acquired  a  velocity  in  the  direction  of  the  straight 
line,  no  matter  how  great,  there  will  be  a  possibility  that  the  difference  of  the  forces  may 
be  so  great  that  it  will  destroy  any  relative  velocity  of  this  kind,  in  any  interval  of  time, 
no  matter  how  short  the  time  assigned  may  be ;  &  this,  after  passing  over  any  very  small 
assigned  space,  no  matter  how  small. 

226.  But  if  the  force  acts  perpendicularly,  so  that,  for  instance,  the  point  D  moves  what   happens    if 
along  the  line  DC  perpendicular  to  AB,  then  the  forces  CK,  CL,  can  in  any  case  be  so  ^es  n^tTct  ah^g 
strong  that  the  resultant  force  CF  may  become,  after  a  recession  of  any  desired  degree  the  straight  line ; 
of  smallness,  large  enough  to  eliminate  any  force  of  this  kind,  or  to  destroy  any  impressed 

velocity.  In  the  case  of  a  force  continually  urging  the  point  D  towards  C,  &  the  points 
A  &  B  in  the  opposite  direction,  there  will  be  some  bending  ;  &  in  the  case  of  a  force  acting 
in  the  same  direction  as  the  straight  line  joining  the  two  points,  there  will  be  some  contraction 
or  distraction.  But  the  forces  resisting  them  may  be  so  strong  that  the  bending,  the 
contraction,  or  the  distraction  will  be  altogether  inappreciable.  If  by  external  action  a 
velocity  is  impressed  on  points  of  this  kind,  &  if  this  induces  bending,  contraction  or  distraction, 
&  if  the  points  are  then  left  to  themselves,  there  will  be  produced  an  oscillation,  in  which 
the  angle  will  jut  out  first  on  one  side  &  then  on  the  other  side  ;  &  the  length  of,  so  to 
speak,  the  rod  consisting  of  the  three  points  will  be  at  one  time  increased  &  at  another 
decreased  ;  &  it  may  possibly  be  the  case  that  the  oscillation  will  be  totally  unappreciable  ; 
&  this  indeed  will  give  us  the  idea  of  a  rod,  such  as  we  call  rigid  &  solid,  incapable  of 
being  contracted  or  dilated  ;  these  properties  are  possessed  by  no  rod  in  Nature  perfectly 

N 


I78 


PHILOSOPHLE  NATURALIS  THEORIA 


habet  accurata  tales,  sed  tantummodo  ad  sensum.  Quod  si  vires  sint  aliquanto  debiliores, 
turn  vero  &  inflexio  ex  vi  externa  mediocri,  &  oscillatio,  ac  tremor  erunt  majores,  &  jam 
hinc  ex  simplicissimo  trium  punctorum  systemate  habebitur  species  quaedam  satis  idonea 
ad  sistendum  animo  discrimen,  quod  in  Natura  observatur  quotidie  oculis.  inter  virgas 
rigidas,  ac  eas,  quae  sunt  flexiles,  &  ex  elasticitate  trementes. 

Systemate    inflexo  227.  Ibidem  si  binse  vires,  ut   AQ,  BT   fuerint    perpendiculares  ad  AB,  vel  etiam 

vlsr  ^ncti^medii  utcunque  parallels  inter  se,  tertia  quoque  erit  parallela  illis,  &  aequalis  earum  summae, 
contraria  extremis,  sed  directionis  contrariae.  Ducta  enim  CD  parallela  iis,  turn  ad  illam  KI  parallela  BA, 
"  '  erit  ob  CK>  yB  sequales,  triangulum  CIK  aequale  simili  BTV,  sive  TBS,  adeoque  CI  squalls 

BT,  IK  aequalis  BS,  sive  AR,  vel  QP.  Quare  si  sumpta  IF  aequali  AQ  ducatur  KF  ;  erit 
triangulum  FIK  aequale  AQP,  ac  proinde  FK  aequalis,  &  parallela  AP,  sive  LC,  &  CLFK 
parallelogrammum,  ac  CF,  diameter  ipsius,  exprimet  vim  puncti  C  utique  parallelam 
viribus  AQ,  BT,  &  asqualem  earum  summae,  sed  directionis  contrariae.  Quoniam  vero 
est  SB  ad  BT,  ut  BD  ad  DC  ;  ac  AQ  ad  AR,  ut  DC  ad  DA  ;  erit  ex  aequalitate  perturbata 
AQ  ad  BT,  ut  BD  ad  DA,  nimirum  vires  in  A,  &  B  in  ratione  reciproca  distantiarum  AD, 
DB  a  recta  CD  ducta  per  C  secundum  directionem  virium. 


& 

summae. 


Postremum  theo- 
rema  generate,  ubi 
etiam  tria  puncta 
non  jaceant  in  di- 
rectum. 


Equilibrium  trium 
punctorum  non  in 
directum  jacentium 
impossible  sine  vi 
externa,  nisi  sint 
in  distantiis  limi- 
tum  :  cum  iis  qui 
nisus  ad  retinen- 
dam  formam  syste- 
matis. 


228.  Ea,  quas  hoc  postremo  numero  demonstravimus,  aeque  pertinent  ad  actiones 
mutuas  trium  punctorum  habentium  positionem  mutuam  quamcunque,  etiam  si  a  rectilinea 
recedat  quantumlibet  ;    nam  demonstratio  generalis  est  :  sed  ad  massas  utcunque  inaequales, 
&  in  se  agentes  viribus  etiam  divergentibus,  multo  generalius  traduci  possunt,  ac  traducentur 
inferius,  &  ad  aequilibrii  leges,  &  vectem,  &  centra  oscillationis  ac  percussionis  nos  deducent. 
Sed  interea  pergemus  alia  nonnulla  persequi  pertinentia  itidem  ad  puncta   tria,  quae  in 
directum  non  jaceant. 

229.  Si  tria  puncta  non  jaceant  in  directum,  turn  vero  sine  externis  viribus  non  poterunt 
esse  in  aequilibrio  ;    nisi  omnes  tres  distantiae,  quae  latera  trianguli  constituunt,  sint  dis- 
tantiae   limitum   figurae    i.     Cum  enim  vires  illae  mutuae  non  habeant    [106]  directiones 
oppositas  ;    sive  unica  vis  ab  altero  e  reliquis  binis  punctis  agat  in  tertium  punctum,  sive 
ambae  ;    haberi  debebit  in  illo  tertio  puncto  motus,  vel  in  recta,  quae  jungit  ipsum  cum 
puncto  agente,  vel  in  diagonali  parallelogrammi,  cujus  latera  binas  illas  exprimant  vires. 
Quamobrem  si  assumantur  in  figura  I  tres  distantiae  limitum  ejusmodi,  ut  nulla  ex  iis  sit 
major  reliquis  binis  simul  sumptis,  &  ex  ipsis  constituatur  triangulum,  ac  in  singulis  angu- 
lorum  cuspidibus  singula  materiae  puncta  collocentur  ;  habebitur  systema  trium  punctorum 
quiescens,  cujus  punctis  singulis  si  imprimantur  velocitates  aequales,  &  parallelae  ;  habebitur 
systema   progrediens    quidem,   sed   respective   quiescens  ;     adeoque  istud   etiam  systema 
habebit  ibi  suum  quemdam  limitem,  sed  horum  quoque  limitum  duo  genera  erunt  :    ii, 
qui  orientur  ab  omnibus  tribus  limitibus  cohaesionis,  erunt  ejusmodi,  ut  mutata  positione, 
conentur  ipsam  recuperare,  cum  debeant  conari  recuperare  distantias  :    ii  vero,  in  quibus 
etiam  una  e  tribus  distantiis  fuerit  distantia  limitis  non  cohaesionis,  erunt  ejusmodi,  ut 
mutata  positione  :  ab  ipsa  etiam  sponte  magis  discedat  systema  punctorum  eorundem.     Sed 
consideremus  jam  casus  quosdam  peculiares,  &  elegantes,  &  utiles,  qui  hue  pertinent. 

Eiegans  theoria          230.  Sint  in  fig.  32  tria  puncta  A,E,B  ita  collocata,  ut  tres  distantise  AB,  AE,  BE  sint 
ratto  eiupsTs  binis  distantiae  limitum  cohaesionis,  &  postremae  duae 
aiiis    occupantibus  sint  aequales.      Focis  A,   B  concipiatur  ellipsis 

transiens  per  E,  cujus  axis  transversus  sit  FO, 

conjugatus  EH,  centrum  D  :  sit  in  fig.   I   AN 

aequalis  semiaxi  transverse  hujus  DO,  sive  BE, 

vel  AE,  ac  sit  DB  hie  minor,  quam  in  fig.   I 

amplitude  proximorum  arcuum  LN,  NP,  &  sint 

in  eadem  fig.  i   arcus  ipsi  NM,  NO  similes,  & 

aequales  ita,  ut  ordinatae  uy,  zt,  aeque  distantes 

ab  N,  sint  inter  se  aequales.     Inprimis  si  punctum 

materiae  sit  hie  in  E  ;  nullum  ibi  habebit  vim, 

cum  AE,  BE  sint  aequales  distantiae  AN  limitis 

N  figurae   I  ;    ac   eadem  est  ratio  pro    puncto 

collocate  in  H.     Quod  si  fuerit  in  O,  itidem 

erit  in  aequilibrio.     Si  enim  assumantur  in  fig.  I 

Az,  AM  aequales  hisce  BO,  AO  ;    erunt  Nz, 


foco  :    vis  nulla  in 
verticibus  axium. 


illius  aequales  DB,  DA  hujus,  adeoque  &  inter  se.  Quare  &  vires  illius  zt,  uy  erunt  aequales 
inter  se,  quae  cum  pariter  oppositae  directionis  sint,  se  mutuo  elident  ;  ac  eadem  ratio  est 
pro  collocatione  in  F.  Attrahetur  hie  utique  A,  &  repelletur  B  ab  O  ;  sed  si  limes,  qui 
respondet  distantiae  AB,  sit  satis  validus  ;  ipsa  puncta  nihil  ad  sensum  discedent  a  focis 


A  THEORY  OF  NATURAL  PHILOSOPHY  179 

accurately,  but  only  approximately.  But  if  the  forces  are  somewhat  more  feeble,  then 
indeed,  under  the  action  of  a  moderate  external  force,  the  bending,  the  oscillation  &  the 
vibration  will  all  be  greater  ;  &  from  this  extremely  simple  system  of  three  points  we  now 
obtain  several  kinds  of  cases  that  are  adapted  to  giving  us  a  mental  conception  of  the  differences, 
that  meet  our  eyes  every  day  in  Nature,  between  rigid  rods  &  those  that  are  flexible  & 
elastically  tremulous. 

227.  At  the  same  time,  if  the  two  forces,  represented  by  AQ,  BT,  were  perpendicular  In  *  system  dis- 

A  T>  n   i  i_       •  ^i.     ^L-    j  f  u     f      i  n   i    torted  by  parallel 

to  AB,  or  parallel  to  one  another  in  any  manner,  then  the  third  force  would  also  be  parallel  forces  the  force  on 
to  them,  equal  to  their  sum,  but  in  the  opposition  direction.     For,  if  CD  is  drawn  parallel  the  middle  point  is 
to  the  forces,  &  KI  parallel  to  BA  to  meet  CD  in  I,  then,  since   CK  &  VB  are  equal  to  direction  t°  thaTof 
one  another,  the  triangle  CIK  will  be  equal  to  the  similar  triangle  BTV,  or  to  the  triangle  the  outside  forces, 
TBS  ;  &  therefore  CI  will  be  equal  to  BT,  IK  to  BS  or  AR  or  QP.     Hence  if  IF  is  taken  sumT* 
equal  to  AQ  &  KF  is  drawn,  then  the  triangle  FIK  will  be  equal  to  AQP,  &  thus  FK 
will  be  equal  £  parallel  to  AP  or  LC,  CLFK  will  be  a-  parallelogram,  &  its  diagonal  CF 
will  represent  the  force  for  the  point  C,  in  every  case  parallel  to  the  forces  AQ,  BT,  & 
equal  to  their  sum,  but    opposite  in    direction.      But,  because   SB  :   BT  :  :  BD  :  DC,  & 
AQ  :  AR  :  :  DC  :  DA  ;  hence,  ex  cequali  we  have  AQ  :  BT  :  :  BD  :  DA,  that  is  to  say,  the 
forces  on  A  &  B  are  in  the  inverse  ratio  of  the  distances  AD  &  DB,  drawn  from  the  straight 
line  CD  in  the  direction  of  the  forces. 

228.  What  has  been  proved  in  the  last  article  applies  equally  to  the  mutual  actions  The  last  theorem  in 
of  three  points  having  any  relative  positions  whatever,  even  if  it  departs  from  a  rectilinear  fhe  tiTree^point^do 
position  to  any  extent  you  may  please.     For  the  demonstration  is  general  ;   &,  further,  the  not  He  in  a  straight 
results  can  be  deduced  much  more  generally  for  masses  that  are  in  every  manner  unequal,  line- 

&  that  act  upon  one  another  even  with  diverging  forces  ;  &  they  will  be  thus  deduced 
later  ;  &  these  will  lead  us  to  the  laws  of  equilibrium,  the  lever,  &  the  centres  of  oscillation 
&  percussion.  But  meanwhile  we  will  go  straight  on  with  our  consideration  of  some 
matters  relating  in  the  same  manner  to  three  points,  which  do  not  lie  in  a  straight  line. 

229.  If  the  three  points  do  not  lie  in  a  straight  line,  then  indeed  without  the  presence  Equilibrium    of 
of  an  external  force  they  cannot  be  in  equilibrium  ;   unless  all  three  distances,  which  form  ^o^no*0^  in^a 
the  sides  of  the  triangle,  are  those  corresponding  to  the  limit-points  in    Fig.  I.     For,  since  straight  line  isim- 
the  mutual  forces  do  not  have  opposite  directions,  either  a  single  force  from  one  of  the  F^^^^^1^0^ 

.  i        i  •    i  i  TT  i  iri  presence  01  an 

remaining  two  points  acts  on  the  third,  or  two  such  forces.     Hence  there  must  be  for  that  external    force, 
third  point  some  motion,  either  in  the  direction  of  the  straight  line  joining  it  to  the  acting  "?Bless3l.  thHisl^« 

'  o  j  o  o    are      at     distances 

point,  or  along  the  diagonal  of  the  parallelogram  whose  sides  represent  those  two  forces,  corresponding    to 


Therefore,  if  in  Fig.  i  we  take  three  limit-distances  of  such  a  kind,  that  no  one  of  them  is  8  ' 


in    his 
greater  than  the  other  two  taken  together,  &  if  from  them  a  triangle  is  formed  &  at  each  case,   to  'conserve 

vertical  angle  a  material  point  is  situated,  then  we  shall  have  a  system  of  three  points  at  rest,  sygte^"11    of   the 

If  to  each  point  of  the  system  there  is  given  a  velocity,  and  these  are  all  equal  &  parallel  to  one 

another,  we  shall  have  a  system  which  moves  indeed,  but  which  is  relatively  at  rest.     Thus 

also  that  system  will  have  a  certain  limit  of  its  own  ;  moreover,  of  such  limits  there  are  also 

two  kinds.     Namely,  those  that  arise  from  all  three  limit-points  being  those  of  cohesion 

which  will  be  such  that,  if  the  relative  position  is  altered,  they  will  strive  to  recover  it  ; 

for  they  are  bound  to  try  to  restore  the  distances.     Secondly,  those  in  which  one  of  the 

three  distances  corresponds  to  a  limit-point  of  non-cohesion,  which  will  be  such  that,  if 

the  relative  position  is  altered,  the  system  will  of  its  own  accord  depart  still  more  from  it. 

However,  let  us  now  consider  certain  special  cases,  that  are  both  elegant  &  useful,  for  which 

this  is  the  appropriate  place. 

230.  In  Fig.  32,  let  the  three  points  A,E,B  be  so  placed  that  the  three  distances  AB,  An  elegant  theory 
AE,  BE  correspond  to  limit-points  of  cohesion,  &  let  the  two  last  be  equal  to  one  another.  i^the'periineter  of 
Suppose  that  an  ellipse,  whose  foci  are  A  &  B,  passes  through  E  ;  let  the  transverse  axis  of  an  ellipse,  each  of 
this  be  FO,  &  the  conjugate  axis  EH,  &  the  centre  D.  In  Fig.  i,  let  AN  be  equal  to  behVpiaced  irTa 
the  transverse  semiaxis  DO  of  Fig.  32,  that  is  equal  to  BE  or  AE  ;  also  in  the  latter  figure  focus  ;  no  force  at 
let  DB  be  less  than  the  width  of  the  successive  arcs  LN,  NP  of  Fig.  i  ;  also,  in  Fig.  i,  let  en 
the  arcs  NM,  NO  be  similar  &  equal,  so  that  the  ordinates  uy,  zt,  which  are  equidistant 
from  N,  are  equal  to  one  another.  Then,  first  of  all,  if  in  Fig.  32,  the  point  of  matter 
is  situated  at  E,  there  will  be  no  force  upon  it  ;  for  AE,  BE  are  equal  to  the  distance  AN 
of  the  limit-point  N  in  Fig.  i  ;  &  the  argument  is  the  same  for  a  point  situated  at  H. 
Further,  if  it  is  at  O,  it  will  in  like  manner  be  in  equilibrium.  For,  if  in  Fig.  i  we  take 
Az,  Au  equal  to  AO,  BO  of  Fig.  32,  then  Nz,  NM  of  the  former  figure  will  be  equal  to  DB, 
DA  of  the  latter  ;  &  thus  equal  also  to  one  another.  Hence  also  the  forces  in  that  figure, 
zt  &  uy,  will  be  equal  to  one  another  ;  &  since  they  are  likewise  opposite  in  direction,  they 
will  cancel  one  another  ;  &  the  argument  is  the  same  for  a  point  situated  at  F.  Here 
in  every  case  A  is  attracted  &  B  is  repelled  from  O  ;  but  if  the  limit-point,  which  corresponds 
to  the  distance  AB  is  strong  enough,  the  points  will  not  depart  to  any  appreciable  extent 


i8o 


PHILOSOPHISE  NATURALIS  THEORIA 


In  reliquis  puncti 
perimetri  vis  direc- 
ta  per  ipsam  peri- 
metrum versus  ver- 
tices axis  conju- 
gati. 


Analogia  verticum 
binorum  axium 
cum  limitibus  cur- 
vae virium. 


Quando  limites 
contrario  m  o  d  o 
positi  :  casus  ele- 
gantissimi  alterna- 
tionis  p  1  u  r  i  u  m 
limitum  in  peri- 
metro ellipseos. 


N 


ellipseos,  in  quibus  fuerant  collocata,  vel  si  debeant  discedere  ob  limitem  minus  validum, 
considerari  poterunt  per  externam  vim  ibidem  immota,  ut  contemplari  liceat  solam 
relationem  tertii  puncti  ad  ilia  duo. 

231.  Manet   igitur   immotum,  ac   sine   vi, 
punctum  collocatum  tarn  in  verticibus  axis  con- 
jugati    ejus    ellipseos,  quam  in    verticibus  axis 
transversi ;   &  si  ponatur  in  quovis   puncto  C 
[107]  perimetri  ejus  ellipseos,  turn  ob  AC,  CB 
simul  aequales  in  ellipsi  axi  transverse,  sive  duplo 
semiaxi  DO  ;  erit  AC  tanto  longior,  quam  ipsa 
DO,  quanto  BC  brevior  ;  adeoque  si  jam  in  fig. 
I  sint  AM,  Az  aequales  hisce  AC,   BC  ;   habe- 
buntur  ibi  utique  uy,  zt  itidem  aequales  inter  se. 
Quare  hie    attractio  CL   sequabitur   repulsioni 
CM,  &  LIMC  erit  rhombus,  in  quo  inclinatio  1C 
secabit  bifariam  angulum  LCM  ;  ac  proinde  si 
ea  utrinque  producatur  in  P,  &  Q  ;  angulus  ACP, 
qui  est  idem,  ac  LCI,  erit  aequalis  angulo  BCQ, 
qui  est  ad  verticem  oppositus  angulo  ICM.    Quse 
cum  in  ellipsi  sit  notissima  proprietas  tangentis 

relatae  ad  focos ;  erit  ipsa  PQ  tangens.  Quamobrem  dirigetur  vis  puncti  C  in  latus  secundum 
tangentem,  sive  secundum  directionem  arcus  elliptici,  atque  id,  ubicunque  fuerit  punctum 
in  perimetro  ipsa,  versus  verticem  propiorem  axis  conjugati,  &  sibi  relictum  ibit  per  ipsam 
perimetrum  versus  eum  verticem,  nisi  quatenus  ob  vim  centrifugam  motum  non  nihil 
adhuc  magis  incurvabit. 

232.  Quamobrem  hie  jam  licebit  contemplari  in  hac  curva  perimetro  vicissitudinem 
limitum  prorsus  analogorum  limitibus  cohaesionis,  &  non  cohaesionis,  qui  habentur  in  axe 
rectilineo  curvae  primigeniae  figures  I.     Erunt  limites  quidam  in  E,  in  F,  in  H,  in  O,  in 
quibus  nimirum  vis  erit  nulla,  cum  in  omnibus  punctis  C  intermediis  sit  aliqua.     Sed  in 

E,  &  H  erunt  ejusmodi,  ut  si  utravis  ex  parte  punctum  dimoveatur,  per  ipsam  perimetrum, 
debeat  redire  versus  ipsos  ejusmodi  limites,  sicut  ibi  accidit  in  limitibus  cohaesionis ;   at  in 

F,  &  O  erit  ejusmodi,  ut  in  utramvis  partem,  quantum  libuerit,  parvum  inde  punctum 
dimotum  fuerit,  sponte  debeat  inde  magis  usque  recedere,  prorsus  ut  ibi  accidit  in  limitibus 
non  cohaesionis. 

233.  Contrarium  accideret,  si  DO  aequaretur  distantiae  limitis  non  cohaesionis  :    turn 
enim  distantia  BC  minor  haberet  attractionem  CK,  distantia  major  AC  repulsionem  CN, 
&  vis  composita  per  diagonalem  CG  rhombi  CNGK  haberet  itidem  directionem  tangentis 
ellipseos  ;    &  in  verticibus  quidem  axis  utriusque  haberetur  limes    quidam,  sed  punctum 
in  perimetro  collocatum  tenderet  versus  vertices  axis  transversi,  non  versus  vertices  axis 
conjugati,  &  hi  referrent  limites  cohaesionis,  illi  e  contrario  limites  non  cohaesionis.     Sed 
adhuc  major  analogia  in  perimetro  harum  ellipsium  habebitur  cum  axe  curvae  primigeniae 
figurae  I  ;   si  fuerit  DO  asqualis  distantiae  limitis  cohaesionis  AN  illius,  &  DB  in  hac  major, 
quam  in  fig.  i  amplitude  NL,  NP ;    multo  vero  magis,  si  ipsa  hujus  DB  superet  plures 
ejusmodi  amplitudines,  ac  arcuum  aequalitas  maneat  hinc,  &  inde  per  totum  ejusmodi 
spatium.     Ubi  enim  AC  hujus  figurae  fiet  aequalis  abscissae  AP  illius,  etiam  BC  hujus  fiet 
pariter  aequalis  AL  illius.     Quare  in  ejusmodi  loco  habebitur  limes,  &  ante  ejusmodi  locum 
versus  A  distantia  [108]  longior  AC  habebit    repulsionem,  &  BC  brevior   attractionem, 
ac  rhombus  erit  KGNC,  &  vis  dirigetur  versus  O.     Quod  si  alicubi  ante  in  loco  adhuc 
propriore  O  distantiae  AC,  BC  aequarentur  abscissis   AR,  AI  figurae  i  ;    ibi   iterum  esset 
limes ;    sed  ante  eum  locum  rediret  iterum  repulsio  pro  minore  distantia,  attractio  pro 
majore,  &  iterum  rhombi  diameter  jaceret  versus  verticem  axis  conjugati  E.     Generaliter 
autem  ubi  semiaxis  transversus  aequatur  distantiae  cujuspiam  limitis  cohaesionis,  &  distantia 
punctorum  a  centre  ellipseos,  sive  ejus  eccentricitas  est  major,  quam  intervallum  dicti 
limitis  a  pluribus  sibi  proximis  hinc,  &  inde,  ac  maneat  aequalitas  arcuum,  habebuntur  in 
singulis  quadrantibus    perimetri    ellipeos  tot    limites,  quot  limites  transibit  eccentricitas 
hinc  translata  in  axem  figurae  I,  a  limite  illo  nominato,  qui  terminet  in  fig.  i  semiaxem 
transversum   hujus  ellipseos ;     ac   praetererea  habebuntur  limites  in  verticibus  amborum 
ellipseos  axium  ;   eritque  incipiendo  ab  utrovis  vertice  axis  conjugati  in  gyrum  per  ipsam 
perimetrum  is  limes  primus  cohaesionis,  turn  illi  proximus  esset  non  cohaesionis,  deinde 


A  THEORY  OF  NATURAL  PHILOSOPHY  181 

from  the  foci  of  the  ellipse,  in  which  they  were  originally  situated  ;  or,  if  they  are  forced 
to  depart  therefrom  owing  to  the  insufficient  strength  of  the  limit-point,  they  may  be 
considered  to  be  kept  immovable  in  the  same  place  by  means  of  an  external  force,  so  that 
we  may  consider  the  relation  of  the  third  point  to  those  two  alone. 

2i>i.  A  point,  then,  which  is  situated  at  one  of  the  vertices  of  the  conjugate  axis  of  At  remaining  points 

,         ,.J.  ,  '  ,  •  r     i_  •  •  •       i          o  j          -L       of     tne     perimeter 

the  ellipse  or  at  one  of  the  vertices  of  the  transverse  axis  remains  motionless  &  under  the  the  force  directed 
action  of  no  force.     If  it  is  placed  at  any  point  C  in  the  perimeter  of  the  ellipse,  then,  since  alons the  perimeter 

A  ^i     /~.T>  i  •        i         IT  i  i  •  Till  .is  towards  the  ver- 

AC,  CB  taken  together  are  m  the  ellipse  equal  to  the  transverse  axis,  or  double  the  semi-  tices  of  the  conju- 

axis  DO,  AC  will  be  as  much  longer  than  DO  as  BC  is  shorter.     Hence,  if  in  Fig.  i  AM,  8ate  axis- 

Az  are  equal  to  these  lines  AC,  BC,  we  shall  have  in  every  case,  in  Fig.  I,  uy,  zt  also  equal 

to  one  another.     Therefore,  in  Fig.  32,  the  attraction  CL  will  be  equal  to  the  repulsion 

CM,  &  LIMC  will  be  a  rhombus,  in  which  the  inclination  1C  will  bisect  the  angle  LCM. 

Hence  if  it  is  produced  on  either  side  to  P  &  Q,  the  angle  ACP,  which  is  the  same  as  the 

angle  LCI  will  be  equal  to  the  angle  BCQ,  which  is  vertically  opposite  to  the  angle  ICM. 

Now  this  is  a  well-known  property  with  respect  to  the  tangent  referred  to  the  foci  in  the 

case  of  an  ellipse  ;  &  therefore  PQ  is  the  tangent.     Hence  the  force  on  the  point  C  is  directed 

laterally  along  the  tangent,  i.e.,  in  the  direction  of  the  arc  of  the  ellipse  ;   &  this  is  true, 

no  matter  where  the  point  is  situated  on  the  perimeter,  &  the  force  is  towards  the  nearest 

vertex  of  the  conjugate  axis ;    if  left  to  itself,  the  point  will  travel  along  the  perimeter 

towards  that  vertex,  except  in  so  far  as  its  motion  is  disturbed  somewhat  in  addition,  owing 

to  centrifugal  force. 

232.  Hence  we  can  consider  in  this  curved  perimeter  the  alternation  of  limit-points  Analogy    between 
as  being  perfectly  analogous  to  those  of  cohesion  &  non-cohesion,  which  were  obtained  in  two  ^xes63*    the 
the  rectilinear  axis  of  the  primary  curve  of  Fig.  I.     There  will  be  certain  limit-points  at  limit-points  of  the 
E,  F,  H,  O,  in  which  there  is  no  force,  whilst  in  all  intermediate  points  such  as  C  there  c 

will  be  some  force.  But  at  E  &  H  they  will  be  such  that,  if  the  point  is  moved  towards 
either  side  along  the  perimeter,  it  must  return  towards  such  limit-points,  just  as  it  has  to 
do  in  the  case  of  limit-points  of  cohesion  in  Fig.  I.  But  at  F  &  O,  the  limit-point  would 
be  such  that,  if  the  point  is  moved  therefrom  to  either  side  by  any  amount,  no  matter 
how  small,  it  must  of  its  own  accord  depart  still  further  from  it ;  exactly  as  it  fell  out  in 
Fig.  i  for  the  limit-points  of  non-cohesion. 

233.  Just  the  contrary  would  happen,  if  DO  were  equal  to  the  distance  corresponding  when    the    limit 
to  a  limit-point  of  non-cohesion.     For  then  the  smaller  distance  BC  would  have  an  Pomif  are  disposed 

ATT-  i  T  A  ^i  i  •         /^XT         i  i  r  -i  ,       in     the    opposite 

attraction  CK,  &  the  greater  distance  AC  a  repulsion  CJN  ;    the  resultant  force  along  the  way ;  most  elegant 
diagonal  CG  of  the  rhombus  CNGK  would  in  the  same  way  have  its  direction  along  the  instances  of  aiter- 

,,,.  .  .  r     •  -i  •       i  111  .,..9  nation    of    several 

tangent  to  the  ellipse,  &  at  the  vertices  of  either  axis  there  would  be  certain  limit-points ;  limit-points  in  the 

but  a  point  situated  in  the  perimeter  would  tend  towards  the  vertices  of  the  transverse  g^™6*"  of  the 

axis,  &  not  towards  the  vertices  of  the  conjugate  axis ;    &  the  latter  are  of  the  nature  of 

limit-points  of  cohesion  &  the  former  of  non-cohesion.     However,  a  still  greater  analogy 

in  the  case  of  the  perimeter  of  these  ellipses  with  the  axis  of  the  primary  curve  of  Fig.  i 

would  be  obtained,  if  DO  were  taken  equal  to  the  distance  corresponding  to  the  limit-point 

of  cohesion  AN  in  that  figure,  &  in  the  present  figure  DB  were  taken  greater  than  the 

width  of  NL,  NP  in  Fig.  i  ;  much  more  so,  if  DB  were  greater  than  several  of  these  widths, 

&  the  equality  between  the  areas  on  one  side  &  the  other  held  good  throughout  the  whole 

of  the  space  taken.     For  where  AC  in  the  present  figure  becomes  equal  to  the  abscissa  AP 

of  the  former,  BC  in  the  present  figure  will  likewise  become  equal  to  AL  in  the  former. 

Hence  at  a  position  of  this  kind  there  will  be  a  limit-point  ;   &  before  a  position  of  this 

kind,  towards  O,  the  longer  distance  AC  will  have  a  repulsion  &  the  shorter  distance  BC 

an  attraction,  KGNC  will  be  a  rhombus,  &  the  force  will  be  directed  towards  O.     But  if 

at  some  position,  on  the  side  of  O,  &  still  nearer  to  O,  the  distances  AC,  BC  were  equal 

to  the  abscissae  AR,  AI  of  Fig.  I,  then  again  there  would  be  a  limit-point  ;    but  before 

that    position  there  would  return  once  more  a  repulsion  for  the  smaller  distance  &  an 

attraction  for  the  greater,  &  once  more  the  diagonal  of  the  rhombus  would  lie  in  the  direction 

of  E,  the  vertex  of  the  conjugate  axis.     Moreover,  in  general,  whenever  the  transverse 

semiaxis  is  equal  to  the  distance  corresponding  to  any  limit-point  of  cohesion,  &  the  distance 

of  the  points  from  the  centre  of  the  ellipse,  i.e.,  its  eccentricity,  is  greater  than  the  interval 

between  the  said  limit-point  &  several  successive  limit-points  on  either  side  of  it,  &  the 

equality  of  the  arcs  holds  good,  then  for  each  quadrant  of  the  perimeter  of  the  ellipse  there 

will  be  as  many  limit-points  as  the  number  of  limit-points  in  the  axis  of  Fig.  I  that  the 

eccentricity  will  cover  when  transferred  to  it  from  the  present  figure,  measured  from  that 

limit-point  mentioned  as  terminating  in  Fig.  I  the  transverse  semiaxis  of  the  ellipse  of  the 

present  figure  ;    in  addition  there  will  be  limit-points  at  the  vertices  of  both  axes  of  the 

ellipse.     Beginning  at  either  vertex -of  the  conjugate  axis,  &  going  round  the  perimeter, 

the  first  limit-point  will  be  one  of  cohesion,  then  the  next  to  it  one  of  non-cohesion,  then 


PHILOSOPHISE  NATURALIS  THEORIA 


alter  cohaesionis,  &  ita  porro,  donee  redeatur  ad  primum,  ex  quo  incceptus  fuerit  gyrus, 
vi  in  transitu  per  quemvis  ex  ejusmodi  limitibus  mutante  directionem  in  oppositam.  Quod 
si  semiaxis  hujus  ellipseos  aequetur  distantiae  limitis  non  cohsesionis  figurae  i  ;  res  ecdem 
ordine  pergit  cum  hoc  solo  discrimine,  quod  primus  limes,  qui  habetur  in  vertice  semiaxis 
conjugati  sit  limes  non  cohaesionis,  turn  eundo  in  gyrum  ipsi  proximus  sit  cohsesionis  limes, 
deinde  iterum  non  cohaesionis,  &  ita  porro. 

Perimetn   piunum  2, .    Verum  est  adhuc  alia  quaedam  analogia  cum  iis  limitibus  ;    si  considerentur 

elhpsium     aequiva-  JT  ..    .  ....      ,      .  °       ..  '..... 

lentes  limitibus.  plures  ellipses  nsdem  illis  iocis,  quarum  semiaxes  ordine  suo  aequentur  distantns,  in  altera 
cujuspiam  e  limitibus  cohaesionis  figuras  I,  in  altera  limitis  non  cohaesionis  ipsi  proximi, 
&  ita  porro  alternatim,  communis  autem  ilia  eccentricitas  sit  adhuc  etiam  minor  quavis 
amplitudine  arcuum  interceptorum  limitibus  illis  figurse  I,  ut  nimirum  singulae  ellipsium 
perimetri  habeant  quaternos  tantummodo  limites  in  quatuor  verticibus  axium.  Ipsae 
ejusmodi  perimetri  totae  erunt  quidam  veluti  limites  relate  ad  accessum,  &  recessum  a 
centro.  Punctum  collocatum  in  quavis  perimetro  habebit  determinationem  ad  motum 
secundum  directionem  perimetri  ejusdem  ;  at  collocatum  inter  binas  perimetros  diriget 
semper  viam  suam  ita,  ut  tendat  versus  perimetrum  definitam  per  limitem  cohaesionis 
figurae  I,  &  recedat  a  perimetro  definita  per  limitem  non  cohaesionis ;  ac  proinde  punctum 
a  perimetro  primi  generis  dimotum  conabitur  ad  illam  redire  ;  &  dimotum  a  perimetro 
secundi  generis,  sponte  illam  adhuc  magis  fugiet,  ac  recedet. 

Demonstrate.  235.  Sint  enim  in  fig.   33.  ellipsium  FEOH,  F'E'O'H',  F"E"O"H"  semiaxes  DO, 

D'O',  D"O"  aequales  primus  di-[iO9]-stantiae  AL  limitis  non  cohaesionis  figurae  i  ;  secundus 
distantiae  AN  limitis  cohaesionis ;  tertius  distantiae  AP  limitis  iterum  non  cohaesionis,  & 
primo  quidem  collocetur  C  aliquanto  ultra  perimetrum  mediam  F'E'O'H'  :  erunt  AC, 
BC  majores,  quam  si  essent  in  perimetro,  adeoque  in  fig.  I  factis  AM,  Az  majoribus,  quam 
essent  prius,  decrescet  repulsio  zt,  crescet  attractio  uy  ;  ac  proinde  hie  in  parallelogram  mo 
LCMI  erit  attractio  CL  major,  quam  repulsio  CM,  &  idcirco  accedet  directio  diagonalis 
CI  magis  ad  CL,  quam  ad  CM,  &  inflectetur  introrsum  versus  perimetrum  mediam. 
Contra  vero  si  C'  sit  intra  perimetrum  mediam,  factis  BC',  AC'  minoribus,  quam  si  essent 
in  perimetro  media  ;  crescet  repulsio  C'M',  &  decrescet  attractio  C'L',  adeoque  directio 
C'l'  accedet  magis  ad  priorem  C'M',  quam  ad  posteriorem  C'L',  &  vis  dirigetur  extrorsum 
versus  eandem  mediam  perimetrum.  Contrarium  autem  accideret  ob  rationem  omnino 
similem  in  vicinia  primae  vel  tertias  perimetri  :  atque  inde  patet,  quod  fuerat  propositum. 


blematum  s  e  g  e  s, 
sed  minus  utilis  : 
immensa  combina- 
tionum  varietas. 


Alias  curvas  eiiip-  236.  Quoniam  arcus  hinc,  &  inde  a  quovis  limite  non  sunt  prorsus  aequales ;  quanquam, 

das13;  ampfa^pro-  ut  suPra  observavimus  num.  184,  exigui  arcus  ordinatas  ad  sensum  aequales  hinc,  &  inde 
habere  debeant ;  curva,  per  cujus  tangentem  perpetuo  dirigatur  vis,  licet  in  exigua  eccen- 
tricitate  debeat  esse  ad  sensum  ellipsis,  tamen  nee  in  iis  erit  ellipsis  accurate,  nee  in 
eccentricitatibus  majoribus  ad  ellipses  multum  accedet.  Erunt  tamen  semper  aliquae 
curvae,  quae  determinent  continuam  directionem  virium,  &  curvse  etiam,  quae  trajectoriam 
describendam  definiant,  habita  quoque  ratione  vis  centifugae  :  atque  hie  quidem  uberrima 
seges  succrescit  problematum  Geometrise,  &  Analysi  exercendae  aptissimorum  ;  sed  omnem 
ego  quidem  ejusmodi  perquisitionem  omittam,  cujus  nimirum  ad  Theoriae  applicationem 
usus  mihi  idoneus  occurrit  nullus ;  &  quae  hue  usque  vidimus,  abunde  sunt  ad  ostendendam 
elegantem  sane  analogiam  alternationis  in  directione  virium  agentium  in  latus,  cum  virium 
primigeniis  simplicibus,  ac  harum  limitum  cum  illarum  limitibus,  &  ad  ingerendam  animo 
semper  magis  casuum,  &  combinationum  diversarum  ubertatem  tantam  in  solo  etiam 
trium  punctorum  systemate  simplicissimo  ;  unde  conjectare  liceat,  quid  futurum  sit,  ubi 
immensus  quidam  punctorum  numerus  coalescat  in  massulas  constituentes  omnem  hanc 
usque  adeo  inter  se  diversorum  corporum  multitudinem  sane  immensam. 


Conversio  t  o  t  i  u  s 
systematis  illaesi : 
impulsu  per  peri- 
metrum ellip  sees 
oscil latio:  idea 
liquationis,  &  con- 
glaciationis. 


237.  At  praeterea  est  &  alius  insignis,  ac  magis  determinatus  fructus,  quern  ex  ejusmodi 
contemplationibus  capere  possumus,  usui  futurus  etiam  in  applicatione  Theoriae  ad 
Physicam.  Si  nimirum  duo  puncta  A,  &  B  sint  in  distantia  limitis  cohaesionis  satis  validi, 
&  punctum  tertium  collocatum  in  vertice  axis  conjugati  in  E  distantiam  a  reliquis  habeat, 
quam  habet  limes  itidem  cohaesionis  satis  validus ;  poterit  sane  [no]  vis,  qua  ipsum 
retinetur  in  eo  vertice,  esse  admodum  ingens  pro  utcunque  exigua  dimotione  ab  eo  loco, 


A  THEORY  OF  NATURAL  PHILOSOPHY 


183 


FIG.  33. 


i84 


PHILOSOPHISE  NATURALIS  THEORIA 


FIG.  33. 


A  THEORY  OF  NATURAL  PHILOSOPHY  185 

another  of  cohesion,  &  so  on,  until  we  arrive  at  the  first  of  them,  from  which  the  circuit 
was  commenced  ;  &  the  force  changes  direction  as  we  pass  through  each  of  the  limit-points 
of  this  kind  to  the  exactly  opposite  direction.  But  if  the  semiaxis  of  this  ellipse  is  equal 
to  the  distance  corresponding  to  a  limit-point  of  non-cohesion  in  Fig.  i,  the  whole  matter 
goes  on  as  before,  with  this  difference  only,  namely,  that  the  first  limit-point  at  the  vertex 
of  the  conjugate  semiaxis  becomes  one  of  non-cohesion  ;  then,  as  we  go  round,  the  next 
to  it  is  one  of  cohesion,  then  again  one  of  non-cohesion,  &  so  on. 

234.  Now  there  is  yet  another  analogy  with  these  limit-points.     Let  us  consider  a  The  perimeters  of 
number  of  ellipses  having  the  same  foci,  of  which  the  semiaxes  are  in  order  equal  to  the  sev<rral  ellipses 

,.  ,....,-,.  ,  -        .       .         £  equivalent  to  limit- 

distances  corresponding  to  limit-points  in  .tig.  I,  namely  to  one  of  cohesion  for  one,  to  points. 

that  of  non-cohesion  next  to  it  for  the  second,  &  so  on  alternately  ;  also  suppose  that  the 
eccentricity  is  still  smaller  than  any  width  of  the  arcs  between  the  limit-points  of  Fig.  I, 
so  that  each  of  the  elliptic  perimeters  has  only  four  limit-points,  one  at  each  of  the  four 
vertices  of  the  axes.  The  whole  set  of  such  perimeters  will  be  somewhat  of  the  nature 
of  limit-points  as  regards  approach  to,  or  recession  from  the  centre.  A  point  situated  in 
any  one  of  the  perimeters  will  have  a  propensity  for  motion  along  that  perimeter.  If  it 
is  situated  between  two  perimeters,  it  will  always  direct  its  force  in  such  a  way  that  it  will 
tend  towards  a  perimeter  corresponding  to  a  limit-point  of  cohesion  in  Fig.  i,  &  will 
recede  from  a  perimeter  corresponding  to  a  limit-point  of  non-cohesion.  Hence,  if  a  point 
is  disturbed  out  of  a  position  on  a  perimeter  of  the  first  kind,  it  will  endeavour  to  return 
to  it ;  but  if  disturbed  from  a  position  on  a  perimeter  of  the  second  kind,  it  will  of  its 
own  accord  try  to  get  away  from  it  still  further,  &  will  recede  from  it. 

235.  In  Fig.  33,  of  the  semiaxes  DO,  DO',  DO"  of  the  ellipses  FEOH,  F'E'O'H',  Demonstration. 
F"E"O"H".  let  the  first  be  equal  to  the  distance  corresponding  to  AL,  a  limit-point  of 
non-cohesion  in  Fig.  I,  the  second  to  AN,  one  of  cohesion,  the  third  to  AP,  one  of  non- 
cohesion.     In  the  first  place,  let  the  point  C  be  situated  somewhere  outside  the  middle 
perimeter  F'E'O'H'  ;  then  AC,  BC  will  be  greater  than  if  they  were  drawn  to  the  perimeter. 

Hence,  in  Fig.  I,  since  AM,  Az  would  be  made  greater  than  they  were  formerly,  the  repulsion 
zt  would  decrease,  &  the  attraction  uy  would  increase.  Therefore,  in  Fig.  33,  in  the 
parallelogram  LCMI,  the  attraction  CL  will  be  greater  than  the  repulsion  CM,  &  so  the 
direction  of  the  diagonal  CI  will  approach  more  nearly  to  CL  than  to  CM,  &  will  be  turned 
inwards  towards  the  middle  perimeter.  On  the  other  hand,  if  C'  is  within  the  middle 
perimeter,  BC',  AC'  are  made  smaller  than  if  they  were  drawn  to  the  middle  perimeter  ; 
the  repulsion  C'M'  will  increase,  &  the  attraction  C'L'  will  decrease,  &  thus  the  direction 
of  CT  will  approach  more  nearly  to  the  former,  C'M',  than  to  the  latter,  C'L' ;  &  the 
force  will  be  directed  outwards  towards  the  middle  perimeter.  Exactly  the  opposite  would 
happen  in  the  neighbourhood  of  the  first  or  third  perimeter,  &  the  reasoning  would  be 
similar.  Hence,  the  theorem  enunciated  is  evidently  true. 

236.  Now,  since  the  arcs  on  either  side  of  any  chosen  limit-point  are  not  exactly  equal,  other    curves     to 
&  yet,  as  has  been  mentioned  above  in  Art.  184,  very  small  arcs  on  either  side  are  bound  b,f.  substltuted  for 

i  •  IT  i  11  i  •  ellipses ;   an    ample 

to  nave  approximately  equal  ordmates ;    the  curve,  along  the  tangent  to  which  the  force  crop  of   theorems, 

is  continually  directed,  although  for  small  eccentricity  it  must  be  practically  an  ellipse,  b"ga"ot  °v^|eth  USoi 

yet  will  neither  be  an  ellipse  accurately  in  this  case,  nor  approach  very  much  to  the  form  combinations. 

of  an  ellipse  for  larger  eccentricity.     Nevertheless,  there  will  always  be  certain  curves 

determining  the  continuous  direction  of  the  force,  &  also  curves  determining  the  path 

described  when  account  is  taken  of  the  centrifugal  force.     Here  indeed  there  will  spring 

up  a  most  bountiful  crop  of  problems  well-adapted  for  the  employment  of  geometry  & 

analysis.     But  I  am  going  to  omit  all  discussion  of  that  kind  ;   for  I  can  find  no  fit  use  for 

them  in  the  application  of  my  Theory.     Also  those  which  we  have  already  seen  are  quite 

suitable  enough  to  exhibit  the  truly  elegant  analogy  between  the  alternation  in  direction 

of  forces  acting  in  a  lateral  direction  &  the  simple  primary  forces,  between  the  limit-points 

of  the  former  &  those  of  the  latter ;    also  for  impressing  on  the  mind  more  &  more  the 

great  wealth  of  cases  &  different  combinations  to  be  met  with  even  in  the  single  very  simple 

system  of    three  points.     From  this  it  may  be  conjectured  what  will  happen  when  an 

immeasurable  number  of  points  coalesce  into  small  masses,  from  which  are  formed  all  that 

truly  immense  multitude  of  bodies  so  far  differing  from  one  another. 

237.  In  addition  to  the  above,  there  is  another  noteworthy  &  more  determinate  result  Rotation   of   the 
to  be  derived  from  considerations  of  this  kind,  &  one  that  will  be  of  service  in  the  application  ™**  °' e    sy.,s,t  ®  m 

t    1      »-r>i  T.I       •  i-i       •  •!•    -i  A   n   T>  T  T          mtact   )    oscillation 

oi  the  1  neory  to  rnysics.    r  or  instance,  it  the  two  points  A  &  is  are  at  a  distance  corresponding  along  the  perimeter 
to  a  limit-point  of  cohesion  that  is   sufficiently  strong,  &  the  third  point  situated  at  the  of  the  ellipse  due  to 

T-,     £    ,  .  .     .  , .  '  1-1  an     impulse ;     the 

vertex  r,  oi  the  conjugate  axis  is  at  a  distance  from  the  other  two  which  corresponds  to  idea  of  liquefaction 
a  limit-point  of  cohesion  that  is  also  sufficiently  strong,  then  the  force  retaining  the  point  &  congelation, 
at  that  vertex  might  be  great  enough,  for  any  slight  disturbance  from  that  position,  to 
prevent  it  from  being  moved  any  further,  unless  through  the  action  of  a  huge  external 


1 86  PHILOSOPHIC  NATURALIS  THEORIA 

ut  sine  ingenti  externa  vi  inde  magis  dimoveri  non  possit.  Turn  quidem  si  quis  impediat 
motum  puncti  B,  &  circa  ipsum  circumducat  punctum  A,  ut  in  fig.  34  abeat  in  A' ;  abibit 
utique  &  E  versus  E',  ut  servetur  forma  trianguli  AEB,  quam  necessario  requirit  conver- 
satio  distantiarum,  sive  laterum  inducta  a  limitum  validitate,  & 
in  qua  sola  poterit  respective  quiescere  systema,  ac  habebitur 
idea  quaedam  soliditatis  cujus  &  supra  injecta  est  mentio.  At 
si  stantibus  in  fig.  32  punctis  A,  B  per  quaspiam  vires  externas, 
quae  eorum  motum  impediant,  vis  aliqua  exerceatur  in  E  ad 
ipsum  a  sua  positione  deturbandum ;  donee  ea  fuerit  medio- 
cris,  dimovebit  illud  non  nihil ;  turn,  ilia  cessante,  ipsum  se  resti- 
tuet,  &  oscillabit  hinc,  &  inde  ab  illo  vertice  per  perimetrum 
curvae  cujusdam  proximse  arcui  elliptico.  Quo  major  fuerit  vis 
externa  dimovens,  eo  major  oscillatio  net ;  sed  si  non  fuerit 
tanta,  ut  punctum  a  vertice  axis  conjugati  recedens  deveniat  ad 
verticem  axis  transversi  ;  semper  retro  cursus  reflectetur,  &  de- 
scribetur  minus,  quam  semiellipsis.  Verum  si  vis  externa  coegerit 
percurrere  totum  quadrantem,  &  transilire  ultra  verticem  axis 
transversi ;  turn  verogyrabit  punctum  circumquaque  per  totam  FIG.  34. 

perimetrum  motu  continue,  quern  a  vertice   axis  conjugati  ad 

verticem  transversi  retardabit,  turn  ab  hoc  ad  verticem  conjugati  accelerabit,  &  ita  porro, 
nee  sistetur  periodicus  conversionis  motus,  nisi  exteriorum  punctorum  impedimentis 
occurrentibus,  quae  sensim  celeritatem  imminuant,  &  post  ipsos  ejusmodi  motus  periodicos 
per  totum  ambitum  reducant  meras  oscillationes,  quas  contrahant,  &  pristinam  debitam 
positionem  restituant,  in  qua  una  haberi  potest  quies  respectiva.  An  non  ejusmodi  aliquid 
accidit,  ubi  solida  corpora,  quorum  partes  certam  positionem  servant  ad  se  invicem,  ingenti 
agitatione  accepta  ab  igneis  particulis  liquescunt,  turn  iterum  refrigescentes,  agitatione 
sensim  cessante  per  vires,  quibus  igneae  particulae  emittuntur,  &  evolant,  positionem  prio- 
rem  recuperant,  ac  tenacissime  iterum  servant,  &  tuentur  ?  Sed  haec  de  trium  punctorum 
systemate  hucusque  dicta  sint  satis. 
Systema  punctorum  238.  Quatuor,  &  multo  magis  plurium,  punctorum  systemata  multo  plures  nobis 

quatuor,  in   eodem  .     .J  ,  ..  .      .  ,',...,  .r  -p, 

piano  cum  distan-  vanationes  objicerent  ;    si  rite  ad  examen  vocarentur  ;    sed  de  us  id  unum  innuam.     H,a 
tiis    hmitum,  suao  quidem  in  piano  eodem  possunt  positionem  mutuam  tueri  tenacissime  ;    si  singulorum 

forma;  tenax.  f.  .  r    ,.       .  .......  .  ,.  ,  ,. 

distantiae  a  reliquis  sequentur  distantns  hmitum  satis  validorum  tigurae  I  :  neque  emm 
in  eodem  piano  positionem  respectivam  mutare  possunt,  aut  aliquod  ex  iis  exire  e  piano 
ducto  per  reliqua  tria,  nisi  mutet  distantiam  ab  aliquo  e  reliquis,  cum  datis  trium  punctorum 
distantiis  mutuis  detur  triangulum,  quod  constituere  debent,  turn  datis  distantiis  quarti 
a  duobus  detur  itidem  ejus  positio  respectu  eorum  in  eodem  piano,  &  detur  distantia  ab 
eorum  tertio,  quae,  si  id  punctum  exeat  e  [in]  priore  piano,  sed  retineat  ab  iis  duobus 
distantiam  priorem,  mutari  utique  debet,  ut  facili  negotio  demonstrari  potest. 

Alia  ratio  system-  239.  Quin  immo  in  ipsa  ellipsi   considerari  possunt  puncta  quatuor,  duo  in  focis,  & 

quatuor  ^^eodem  a^a  ^uo  nmc>  &  'm<^e  a  vertice  axis  conjugati  in  ea  distantia  a  se  invicem,  ut  vi  mutua 

piano    cum     idea  repulsiva  sibi  invicem  elidant  vim,  qua  juxta  praecedentem  Theoriam  urgentur  in  ipsum 

flexiiis  •"Systema  verticem  ;    quo  quidem  pacto  rectangulum  quoddam  terminabunt,  ut  exhibet  fig.  35,  in 

eorundem     forms  punctis  A,  B,  C,  D.     Atque  inde  si  supra  angulos  quadratae  basis  assurgant  series  ejusmodi 

nes^rifmrticufa-  punctorum  exhibentium  series  continuas  rectangulorum,  habebitur  quaedam  adhuc  magis 

rum  pyramidaiium.  praecisa  idea  virgae  solidae,  in  qua  si  basis  ima  inclinetur  ;    statim  omnia  superiora  puncta 

movebuntur  in  latus,  ut  rectangulorum  illorum  positionem  retineant,&  celeritas  conversionis 

erit  major,  vel  minor,  prout  major  fuerit,  vel  minor  vis  ilia  in  latus,  quae  ubi  fuerit  aliquanto 

languidior,  multo  serius  progredietur  vertex,  quam  fundum,    . 

&  inflectetur  virga,  quae    inflexio    in    omni   virgarum   genere 

apparet  adhuc  multo  magis  manifesta,  si  celeritas  conversionis        C  O 

fuerit  ingens.  Sed  extra  idem  planum  possunt  quatuor  puncta 
collocata  ita,  ut  positionem  suam  validissime  tueantur,  etiam 
ope  unicae  distantiae  limitis  unici  satis  validi.  Potest  enim  fieri 
pyramis  regularis,  cujus  latera  singula  triangularia  habeant 

ejusmodi  distantiam.     Turn  ea  pyramis  constituet  particulam        •  • 

quandam  suae  figurae  tenacissimam,  quae  in  puncta,  vel  pyra-        /\  3 

mides  ejusmodi  aliquanto  remotiores  ita  poterit  agere,  ut  ejus  FlG  35 

puncta  respectivum  situm  nihil  ad  sensum  mutent.     Ex  quatuor 

ejusmodi  particulis  in  aliam  majorem  pyramidem  dispositis  fieri  poterit  particula  secundi 
ordinis  aliquanto  minus  tenax  ob  majorem  distantiam  particularum  primi  earn  componen- 


A  THEORY  OF  NATURAL   PHILOSOPHY  187 

force.  In  that  case,  if  the  motion  of  the  point  B  were  prevented,  &  the  point  A  were  set 
in  motion  round  B,  so  that  in  Fig.  34  it  moved  to  A',  then  the  point  E  would  move  off 
to  E'  as  well,  so  as  to  conserve  the  form  of  the  triangle  AEB,  as  is  required  by  the  conservation 
of  the  sides  or  distances  which  is  induced  by  the  strength  of  the  limits ;  &  the  system  can 
be  relatively  at  rest  in  this  form  only  ;  thus  we  get  an  idea  of  a  certain  solidity,  of  which 
casual  mention  has"  already  been  made  above.  But  if,  in  Fig.  2,  whilst  the  points  A,B, 
are  kept  stationary  by  means  of  an  external  force  preventing  their  motion,  some  force  is 
exerted  on  the  point  at  E  to  disturb  it  from  its  position,  then,  as  long  as  the  force  is  only 
moderate,  it  will  move  the  point  a  little  ;  &  afterwards,  when  the  force  ceases,  the  point 
will  recover  its  position,  &  will  then  oscillate  on  each  side  of  the  vertex  along  a  perimeter 
of  the  curve  that  closely  approximates  to  an  elliptic  arc.  The  greater  the  external  force 
producing  the  motion,  the  greater  the  oscillation  will  be  ;  but  if  it  is  not  so  great  as  to  make 
the  point  recede  from  the  vertex  of  the  conjugate  axis  until  it  reaches  the  vertex  of  the 
transverse  axis,  its  path  will  always  be  retraced,  &  the  arc  described  will  be  less  than  a 
semi-ellipse.  But  if  the  external  force  should  compel  the  point  to  traverse  a  whole  quadrant 
&  pass  through  the  vertex  of  the  transverse  axis,  then  indeed  the  point  will  make  a  complete 
circuit  of  the  whole  perimeter  with  a  continuous  motion  ;  this  will  be  retarded  from  the 
vertex  of  the  conjugate  axis  to  that  of  the  transverse  axis,  then  accelerated  from  there 
onwards  to  the  vertex  of  the  conjugate  axis,  &  so  on  ;  there  will  not  be  any  periodic  reversal 
of  motion,  unless  there  are  impediments  met  with  from  external  points  that  appreciably 
diminish  the  speed  ;  in  which  case,  following  on  such  periodic  motions  round  the  whole  circuit, 
there  will  be  a  return  to  mere  oscillations ;  &  these  will  be  shortened,  &  the  original  position 
restored,  the  only  one  in  which  there  can  possibly  be  relative  rest.  Probably  something 
of  this  sort  takes  place,  when  solid  bodies  whose  parts  maintain  a  definite  position  with 
regard  to  one  another,  if  subjected  to  the  enormous  agitation  produced  by  fiery  particles, 
liquefy ;  &  once  more  freezing,  as  the  agitation  practically  ceases  on  account  of  forces  due 
to  the  action  of  which  the  fiery  particles  are  driven  out  &  fly  off,  recover  their  initial  position 
&  again  keep  &  preserve  it  most  tenaciously.  But  let  us  be  content  with  what  has  been  said 
above  with  regard  to  a  system  of  three  points  for  the  present. 

238.  Systems  of  four,  &  much  more  so  for  more,  points  would  yield  us  many  more  varia-  A  system  of  four 
tions,  if  they  were  examined  carefully  one  after  the  other  ;  but  I  will  only  mention  one  thing  jj^^ces^orre* 
about  such  systems.     It  is  possible  that  such  systems,  in  one  plane,  may  conserve  their  rela-  spending  to  Hmit- 
tive  positions  very  tenaciously,  if  the  distances  of  each  from  the  rest  are  equal  to  the  dis-  ^ves^tslform.0011 
tances  in  Fig.  i  corresponding  to  limit-points  of  sufficient  strength.     For  neither  can  they 

change  their  relative  position  in  the  plane,  nor  can  any  one  of  them  leave  the  plane  drawn 
through  the  other  three  ;  since,  if  the  distances  of  three  points  from  one  another  is  given, 
then  we  are  given  the  triangle  which  they  must  form  ;  &  then  being  given  the  distances 
of  the  fourth  point  from  two  of  these,  we  are  also  given  the  position  of  this  fourth  point 
from  them,  &  therefore  also  the  distance  from  the  third  of  them.  If  the  point  should 
depart  from  the  plane  mentioned,  &  yet  preserve  its  former  distances  from  the  two  points 
the  distance  from  the  third  point  must  be  changed  in  any  case,  as  can  be  easily  proved. 

239.  Again,  we  may  consider  the  case  of  four  points  in  the  ellipse,  two  being  at  the  A  further  consider- 
foci,  &  the  other  two  on  either  side  of  a  vertex  of  the  conjugate  axis  at  such  a  distance  from  a{io"ou°f  ^i^f6,™ 
one  another,  that  the  mutual  repulsive  force  between  them  will  cancel  the  force  with  which  connection    with 
they  are  urged  towards  that  vertex,  according  to  the  preceding  theorem.     Thus,  they  are  the  idea  of  rigid  & 

f  i  .      T->-  i  •          A   T,  V-i  T-V     flexible     rods;    a. 

at  the  vertices  of  a  rectangle,  as  is  shown  in  rig.  35,  where  they  occupy  the  points  A,B,U,D.  system     of    four 
Hence,  if  we  have  a  series  of  points  of  this  kind  to  stand  above  the  four  angles  of  the  quadratic  P°mts  m  the  fo.rm 

r  i     11     i       •      r  i  •         ^         •   •         of      a      pyramid; 

base,  so  as  to  represent  continuous  series  of  rectangles,  we  shall  obtain  from  this  supposition  different    arrange- 
a  more  precise  idea  than  hitherto  has  been  possible  of  a  solid  rod,  in  which,  if  the  lowest  ments  °f  particular 

r     r    •  •      •      v        i        n  •  T        i  1-1  pyramids. 

set  or  points  is  inclined,  all  the  points  above  are  immediately  moved  sideways,  but 
so  that  they  retain  the  positions  in  their  rectangles ;  &  the  speed  of  rotation  will  be  greater 
or  less  according  as  the  force  sideways  was  greater  or  less ;  even  where  this  force  is  somewhat 
feeble,  the  top  will  move  considerably  later  than  the  base  &  the  rod  will  be  bent ;  &  the 
amount  of  bending  in  every  kind  of  rod  will  be  still  more  apparent  if  the  speed  of  rotation  is 
very  great.  Again,  four  points  not  in  the  same  plane  can  be  so  situated  that  they  preserve 
their  relative  position  very  tenaciously ;  &  that  too,  when  we  make  use  of  but  a  single 
distance  corresponding  to  a  limit-point  of  sufficient  strength.  For  they  can  form  a  regular 
pyramid,  of  which  each  of  the  sides  of  the  triangles  is  of  a  length  equal  to  this  distance. 
Then  this  pyramid  will  constitute  a  particle  that  is  most  tenacious  as  regards  its  form ; 
&  this  will  be  able  to  act  upon  points,  or  pyramids  of  the  same  kind,  that  are  more  remote, 
in  such  a  manner  that  its  points  do  not  alter  their  relative  position  in  the  slightest  degree 
for  all  practical  purposes.  From  four  particles  of  this  kind,  arranged  to  form  a  larger 
pyramid,  we  can  obtain  a  particle  of  the  second  order,  somewhat  less  tenacious  of  form  on 
account  of  the  greater  distance  between  the  particles  of  the  first  order  that  compose  it ; 


1 88  PHILOSOPHIC   NATURALIS  THEORIA 

tium,  qua  fit,  ut  vires  in  easdem  ab  externis  punctis  impressae  multo  magis  inaequales  inter  se 
sint,^quam  fuerint  in  punctis  constituentibus  particulas  ordinis  primi ;  ac  eodem  pacto  ex 
his  secundi  ordinis  particulis  fieri  possunt  particulse  ordinis  tertii  adhuc  minus  tenaces  figurae 
suae,  atque  ita  porro,  donee  ad  eas  deventum  sit  multo  majores,  sed  adhuc  multo  magis 
mobiles,  atque  variabiles,  ex  quibus  pendent  chemica;  operationes,  &  ex  quibus  haec  ipsa 
crassiora  corpora  componuntur,  ubi  id  ipsum  accideret,  quod  Newtonus  in  postrema  Optics 
questione  proposuit  de  particulis  suis  primigeneis,  &  elementaribus,  alias  diversorum  ordi- 
num  particulas  efformantibus.  Sed  de  particularibus  hisce  systematis  determinati  punc- 
torum  numeri  jam  satis,  ac  ad  massas  potius  generaliter  considerandas  faciemus  gradum. 

Transitus     ad  240.  In  massis  primum  nobis  se  offerunt  considerandas  elegantissimse  sane,  ac  £  foccund- 

massas  :    quid  cen-    •  m        **t*     '  •  •  •  •  j  T->I 

trum      gravitatis :  issimae,  &  utilissimae  propnetates  centn  gravitatis,  quse  quidem  e  nostra    1  heona  sponte 
theoremata  hk  de  propemodum  fluunt,  aut  saltern  eius  ope  evidentissime  demonstrantur.     Porro  centrum 

eo  demonstrando.      L       *••.      •  .  •  •..,     .  .  .  .  .  ,          . 

gravitatis  a  gravium  aequihbno  nomen  accepit  suum,  a  quo  etiam  ejus  consideratio  ortum 

duxit ;  sed  id  quidem  a  gravi-[ii2]-tate  non  pendet,  sed  ad  massam  potius  pertinet. 
Quamobrem  ejus  definitionem  proferam  ab  ipsa  gravitate  nihil  omnino  pendentem,  quan- 
quam  &  nomen  retinebo,  &  innuam,  unde  originem  duxerit ;  turn  demonstrabo  accuratissime, 
in  quavis  massa  haberi  aliquod  gravitatis  centrum,  idque  unicum,  quod  quidem  passim 
omittere  solent,  &  perperam ;  deinde  ad  ejus  proprietatem  praecipuam  exponendam 
gradum  faciam,  demonstrando  celeberrimum  theorema  a  Newtono  propositum,  centrum 
gravitatis  commune  massarum,  sive  mihi  punctorum  quotcunque,  &  utcunque  disposi- 
torum,  quorum  singula  moveantur  sola  inertiae  vi  motibus  quibuscunque,  qui  in  singulis 
punctis  uniformes  sint,  in  diversis  utcunque  diversi,  vel  quiescere,  vel  moveri  uniformiter 
in  directum  :  turn  vero  mutuas  actiones  quascunque  inter  puncta  quaelibet,  vel  omnia 
simul,  nihil  omnino  turbare  centri  communis  gravitatis  statum  quiescendi  vel  movendi 
uniformiter  in  directum,  unde  nobis  &  actionis,  ac  reactionis  aequalitas  in  massis  quibusque, 
&  principia  collisiones  corporum  definientia,  &  alia  plurima  sponte  provenient.  Sed 
aggrediamur  ad  rem  ipsam. 

Definitio    centri  241.  Centrum    igitur    commune    gravitatis    punctorum    quotcunque.    &     utcunque 

gravitatis    non    j-          •  n   f       -j  j     •    j  i 

pendens    ab    idea  dispositorum,  appellabo  id  punctum,  per  quod  si  ducatur  planum  quodcunque  ;    summa 

gravitatis  :  ejus  distantiarum  perpendicularium  ab  eo  piano  punctorum  omnium  jacentium  ex  altera 
idea8communi.C  *  ejusdem  parte,  sequatur  summa  distantiarum  ex  altera.  Id  quidem  extenditur  ad  quas- 
cunque, &  quotcunque  massas ;  nam  eorum  singulae  punctis  utique  constant,  &  omnes 
simul  sunt  quaedam  punctorum  diversorum  congeries.  Nomen  traxit  ab  aequilibrio 
gravium,  &  natura  vectis,  de  quibus  agemus  infra  :  ex  iis  habetur  illud,  singula 
pondera  ita  connexa  per  virgas  inflexiles,  ut  moveri  non  possint,  nisi  motu  circa  aliquem 
horizontalem  axem,  exerere  ad  conversionem  vim  proportionalem  sibi,  &  distantiae  perpen- 
diculari  a  piano  verticali  ducto  per  axem  ipsum  ;  unde  fit,  ut  ubi  ejusmodi  vires,  vel,  ut 
ea  vocant,  momenta  virium  hinc,  &  inde  asqualia  fuerint,  habeatur  aequilibrium.  Porro 
ipsa  pondera  in  nostris  gravibus,  in  quibus  gravitatem  concipimus,  ac  etiam  ad  sensum 
experimur,  proportionalem  in  singulis  quantitati  materiae,  &  agentem  directionibus  inter 
se  parallelis,  proportionalia  sunt  massis,  adeoque  punctorum  eas  constituentium  numero  ; 
quam  ob  rem  idem  est,  ea  pondera  in  distantias  dncere,  ac  assumere  summam  omnium 
distantiarum  omnium  punctorum  ab  eodem  piano.  Quod  si  igitur  respectu  aggregati 
cujuscunque  punctorum  materiae  quotcunque,  &  quomodocunque  dispositorum  sit  aliquod 
punctum  spatii  ejusmodi,  ut,  ducto  per  ipsum  quovis  piano,  summa  distantiarum  ab  illo 
punctorum  jacentium  ex  parte  altera  aequetur  summse  distantiarum  jacentium  ex  altera  ; 
concipiantur  autem  singula  ea  puncta  animata  viribus  aequalibus,  &  parallelis,  cujusmodi  sunt 
vires,  quas  in  nostris  gravibus  concipimus ;  illud  utique  consequitur,  [113]  suspense  utcunque 
ex  ejusmodi  puncto,  quale  definivimus  gravitatis  centrum,  omni  eo  systemate,  cujus 
systematis  puncta  viribus  quibuscunque,  vel  conceptis  virgis  inflexibilus,  &  gravitate 
carentibus,  positionem  mutuam,  &  respectivum  statum,  ac  distantias  omnino  servent,  id 
systema  fore  in  aequilibrio  ;  atque  illud  ipsum  requiri,  ut  in  aequilibrio  sit.  Si  enim  vel 
unicum  planum  ductum  per  id  punctum  sit  ejusmodi,  ut  summae  illae  distantiarum  non 
sint  aequales  hinc,  &  inde  ;  converse  systemate  omni  ita,  ut  illud  planum  evadat  verticale, 
jam  non  essent  aequales  inter  se  summae  momentorum  hinc,  &  inde,  &  altera  pars  alteri 
prseponderaret.  Verum  haec  quidem,  uti  supra  monui,  fuit  occasio  quaedam  nominis 
imponendi  ;  at  ipsum  punctum  ea  lege  determinatum  longe  ulterius  extenditur,  quam 


A  THEORY  OF  NATURAL  PHILOSOPHY  189 

for  from  this  fact  it  comes  about  that  the  forces  impressed  upon  these  from  external  points 
are  much  more  unequal  to  one  another. than  they  would  be  for  the  points  constituting 
particles  of  the  first  order.  In  the  same  manner,  from  these  particles  of  the  second  order 
we  might  obtain  particles  of  the  third  order,  still  less  tenacious  of  form,  &  so  on  ;  until 
at  last  we  reach  those  which  are  much  greater,  still  more  mobile,  &  variable  particles,  which 
are  concerned  in  chemical  operations ;  &  to  those  from  which  are  formed  the  denser  bodies, 
with  regard  to  which  we  get  the  very  thing  set  forth  by  Newton,  in  his  last  question  in 
Optics,  with  respect  to  his  primary  elemental  particles,  that  form  other  particles  of  different 
orders.  We  have  now,  however,  said  enough  concerning  these  systems  of  a  definite  number 
of  points,  &  we  will  proceed  to  consider  masses  rather  more  generally. 

240.  In  dealing  with  masses,  the  first  matters  that  present  themselves  for  our  considera-  Passing   on  to 
tion  are  certain  really  very  elegant,  as  well  as  most  fertile  &  useful  properties  of  the  centre  of  "ntr?  of  "gravity^ 
gravity.     These  indeed  come  forth  almost  spontaneously  from  my  Theory,  or  at  least  are  Theorems     to    be 
demonstrated  most  clearly  by  means  of  it.     Further,  the  centre  of  gravity  derived  its  name 

from  the  equilibrium  of  heavy  (gravis)  bodies,  &  the  first  results  in  connection  with  the  former 
were  developed  by  means  of  the  latter ;  but  in  reality  it  does  not  depend  on  gravity,  but  rather  is 
related  to  masses.  On  this  account,  I  give  a  definition  of  it,  which  in  no  way  depends  on 
gravity,  although  I  will  retain  the  name,  &  will  mention  whence  it  derived  its  origin.  Then 
I  will  prove  with  the  utmost  rigour  that  in  every  body  there  is  a  centre  of  gravity,  &  one 
only  (a  thing  which  is  usually  omitted  by  everybody,  quite  unjustifiably).  Then  I  will 
proceed  to  expound  its  chief  property,  by  proving  the  well-known  theorem  enunciated  by 
Newton  ;  that  the  centre  of  gravity  of  masses,  or,  in  my  view,  of  any  number  of  points  in 
any  positions,  each  of  which  is  moved  in  any  manner  by  the  force  of  inertia  alone,  this 
force  being  uniform  for  the  separate  points  but  maybe  non-uniform  to  any  extent  for 
different  points,  will  be  either  at  rest  or  will  move  uniformly  in  a  straight  line.  Finally, 
I  will  show  that  any  mutual  action  whatever  between  the  points,  or  all  of  them  taken 
together,  will  in  no  way  disturb  the  state  of  rest  or  of  uniform  motion  in  a  straight  line  of 
the  centre  of  gravity.  From  which  the  equality  of  action  &  reaction  in  all  bodies,  &  the 
principles  governing  the  collision  of  solids,  &  very  many  other  things  will  arise  of  them- 
selves. However  let  us  set  to  work  on  the  matter  itself. 

241.  Accordingly,  I  will  call  the  common  centre  of  gravity  of  any  number  of  points,  Definition    of  the 
situated  in  any  positions  whatever,  that  point  which  is  such  that,  if  through  it  any  plane  in^tndent^fTn7 
is  drawn,  the  sum  of  the  perpendicular  distances  from  the  plane  of  all  the  points  lying  on  idea  of  gravitation ; 
one  side  of  it  is  equal  to  the  sum  of  the  distances  of  all  the  points  on  the  other  side  of  it.  ^  dtfaiSorT 
The  definition  applies  also  to  masses,  of  any  sort  or  number  whatever  ;    for  each  of  the  the  usual  idea, 
latter  is  made  up  of  points,  &  all  of  them  taken  together  are  certain  groups  of  different 

points.  The  name  is  taken  from  the  equilibrium  of  weights  (gravis),  &  from  the  principle 
of  the  lever,  with  which  we  shall  deal  later.  Hence  we  obtain  the  principle  that  each  of 
the  weights,  connected  together  by  rigid  rods  in  such  a  manner  that  the  only  motion  possible 
to  them  is  one  round  a  horizontal  axis,  will  exert  a  turning  force  proportional  to  itself  & 
to  its  perpendicular  distance  from  a  vertical  plane  drawn  through  this  axis.  From  which 
it  comes  about  that,  when  the  forces  of  this  sort  (or,  as  they  are  called,  the  moments  of  the 
forces)  are  equal  to  one  another  on  this  side  &  on  that,  then  there  is  equilibrium.  Further, 
the  weights  in  our  heavy  bodies,  in  which  we  conceive  the  existence  of  gravity  (&  indeed 
find  by  experience  that  there  is  such  a  thing)  proportional  in  each  to  the  quantity  of  matter, 
&  acting  in  directions  parallel  to  one  another,  are  proportional  to  the  masses,  &  thus  to 
the  number  of  points  that  go  to  form  them.  Therefore,  the  product  of  the  weights  into 
the  distances  comes  to  the  same  thing  as  the  sum  of  all  the  distances  of  all  the  points  from 
the  plane.  If  then,  for  an  aggregate  of  points  of  matter,  of  any  sort  &  number  whatever, 
situated  in  any  way,  there  is  a  point  of  space  of  such  a  nature  that,  for  any  plane  drawn 
through  it,  the  sum  of  the  distances  from  it  of  all  points  lying  on  one  side  of  it  is  equal  to  the 
sum  of  the  distances  of  all  the  points  lying  on  the  other  side  of  it ;  if  moreover  each  of  the 
points  is  supposed  to  be  endowed  with  a  force,  &  these  forces  are  all  equal  &  parallel  to  one 
another,  &  of  such  a  kind  as  we  conceive  the  forces  in  our  weights  to  be  ;  then  it  follows 
directly  that,  if  the  whole  of  this  system  is  suspended  in  any  way  from  a  point  of  the  sort 
we  have  defined  the  centre  of  gravity  to  be,  the  points  of  the  system,  on  account  of  certain 
assumed  forces  or  rigid  weightless  rods,  preserving  their  mutual  position,  their  relative 
state  &  their  distances  absolutely  unchanged,  then  the  system  will  be  in  equilibrium.  Such  a 
point  is  to  be  found,  in  order  that  the  system  may  be  in  equilibrium.  For,  if  any  one  plane 
can  be  drawn  through  the  point,  such  that  the  sum  of  the  distances  on  the  one  side  are 
not  equal  to  those  on  the  other  side,  &  thewhole  system  is  turned  so  that  this  plane  becomes 
vertical,  then  the  sums  of  the  moments  will  not  be  equal  to  one  another  on  each 
side,  but  one  part  will  outweigh  the  other  part.  This  indeed,  as  I  said  above,  was  the  idea 
that  gave  rise  to  the  term  centre  of  gravity  ;  but  the  point  determined  by  this  rule  has 


190 


PHILOSOPHIC  NATURALIS  THEORIA 


Corollarium  g  e  it- 
erate pertinens  ad 
summas  distanti- 
arum  omnium 
punctorum  massse 
a  piano  transeunte 
per  centrum  gravi- 
tatis  xquales  utrin- 
que. 


Bi.n  a  theoremata 
per  tinentia  ad 
planum  parallel  urn 
piano  distantiarum 
aequalium  cum 
eorum  demonstra- 
tiouibus. 


Com  pie  me  n  turn 
demonstrationis  ut 
e  x  t  e  n  d[a  t  u  r  ad 
omnes  casus. 


ad  solas  massas  animatas  viribus  asqualibus,  &  parallelis,  cujusmodi  concipiuntur  a  nobis 
in  nostris  gravibus,  licet  ne  in  ipsis  quidem  accurate  sint  tales.  Quamobrem  assumpta 
superiore  definitione,  quae  a  gravitatis,  &  sequilibrii  natura  non  pendet,  progrediar  ad 
deducenda  inde  corollaria  quaaedam,  quae  nos  ad  ejus  proprietates  demonstrandas  deducant. 

242.  Primo  quidem  si  aliquod  fuerit  ejusmodi  planum,  ut  binae  summae  distantiarum 
perpendicularium    punctorum    omnium    hinc    &    inde  acceptorum   aequenter  inter  se  : 
aequabuntur  &  summae  distantiarum  acceptarum  secundum  quancunque  aliam  directionem 
datam,  &  communem  pro  omnibus.     Erit  enim  quaevis  distantia  perpendicularis  ad  quanvis 
in  dato  angulo  inclinatam  semper  in  eadem  ratione,  ut  patet.     Quare  &  sunimae  illarum 
ad  harum  summas  erunt  in  eadem  ratione,  ac  asqualitas  summarum  alterius  binarii  utriuslibet 
secum  trahet   aequalitatem   alterius.     Quare   in   sequentibus,  ubi   distantias   nominavero, 
nisi    exprimam    perpendiculares,    intelligam    generaliter    distantias    acceptas    in    quavis 
directione  data. 

243.  Quod  si  assumatur  planum  aliud  quodcunque  parallelum  piano  habenti  aequales 
hinc,  &  inde  distantiarum  summas ;    summa  distantiarum  omnium  punctorum  jacentium 
ex  parte  altera  superabit  summam  jacentium  ex  altera,  excessu  aequali  distantiae  planorum 
acceptae  secundum  directionem  eandem  ductae  in  nwmerum  punctorum  :    &  vice  versa  si 
duo  plana    parallela  sint,  ac  is  excessus  alterius  summas  supra  summam  alterius  in  altero 
ex  iis  aequetur  eorum  distantiae  ductae  in  numerum  punctorum  ;   planum  alterum  habebit 
oppositarum  distantiarum  summas  aequales.     Id  quidem  facile  concipitur  ;    si  concipiatur, 
planum    distantiarum    aequalium  moveri  versus    illud    alterum    planum    motu  parallelo 
secundum  earn  directionem,  secundum  quam  sumuntur  distantiae.     In  eo  motu  distantiae 
singulse  ex  altera  parte  crescunt,  ex  altera  decrescunt  continue  tantum,  quantum  promo- 
vetur  planum,  &  si  aliqua  distantia  evanescit  interea  ;  jam  eadem  deinde  incipit  tantundem 
ex  parte  contraria  crescere.     Quare  patet  excessum  omnium  citeriorum  [114]  distantiarum 
supra  omnes  ulteriores  aequari  progressui  plani  toties  sumpto,  quot  puncta  habentur,  & 
in  regressu  destruitur  e  contrario,  quidquid  in  ejusmodi  progressu  est  factum,  atque  idcirco 
ad  aequalitatem  reditur.      Verum  ut   demonstratio 

quam  accuratissima  evadat,  exprimat  in  fig.  36  recta 
AB  planum  distantiarum  aequalium,  &  CD  planum 
ipsi  parallelum,  ac  omnia  puncti  distribui  poterunt 
in  classes  tres,  in  quorum  prima  sint  omnia  jacentia 
citra  utrumque  planum,  ut  punctum  E  ;  in  secunda 
omnia  puncta  jacentia  inter  utrumque,  ut  F,  in  tertia 
omnia  puncta  adhuc  jacentia  ultra  utrumque,  ut  G. 
Rectae  autem  per  ipsa  ductae  in  directione  data 
quacunque,  occurrant  rectae  AB  in  M,  'H,  K,  & 
rectae  CD  in  N,  I,  L  ;  ac  sit  quaedam  reacta  direc- 
tionis  ejusdem  ipsis  AB,  CD  occurrens  in  O,  P. 
Patet,  ipsam  OP  fore  aequalem  ipsis  MN,  HI,  KL. 
Dicatur  jam  summa  omnium  punctorum  E  primae 
classis  E,  &  distantiarum  omnium  EM  summa  e ; 
punctorum  F  secundae  classis  F,  &  distantiarum  / ; 
punctorum  G  tertiae  classis  summa  G,  &  distantiarum 
earundem  g ;  distantia  vero  OP  dicatur  O.  Patet,  sum- 
mam omnium  MN  fore  E  X  O ;  summam  omnium 
HI  fore  F  X  O  ;  summam  omnium  KL  fore  G  X  O  ;  erit  autem  quaevis  EN  —  EM  +MN  ; 
quaevis  FI  =  HI  —  FH  ;  quaevis  GL  =  KG  —  KL.  Quare  summa  omnium  EN  erit 
e  +  E  xO  ;  summa  omnium  FI  =  F  x  O  —  /,  &  summa  omnium  GL  =  g  —  G  X  O  ; 
adeoque  summa  omnium  distantiarum  punctorum  jacentium  citra  planum  CD,  primae  nimi- 
rum,  ac  secundae  classis,  erit  e  +  E  xO+F  X  O  —  /,  &  summa  omnium  jacentium  ultra, 
nimirum  classis  tertiae,  erit  g  —  G  X  O.  Quare  excessus  prioris  summae  supra  secundam 
erit  e  +  E  X  O  -f  F  xO  —  /  —  g+  G  xO;  adeoque  si  prius  fuerit  e  =  f  +  g  ; 
delete  e  —  f—  g,  totus  excessus  erit  E  x  O  +  F  X  O  +  G  X  O,  sive  (E  +  F  +  G)  X  O, 
summa  omnium  punctorum  ducta  in  distantiam  planorum  ;  &  vice  versa  si  is  excessus 
respectu  secundi  plani  CD  fuerit  aequalis  huic  summas  ductae  in  distantiam  O,  oportebit 
e  —  f  —  /aequetur  nihilo,  adeoque  sit  e  =  f  -\-  g,  nimirum  respectu  primi  plani  AB  summas 
distantiarum  hinc,  &  inde  aequales  esse. 

244.  Si  aliqua  puncta  sint  in  altero  ex  iis  planis,  ea  superioribus  formulis  contineri 
possunt,  concepta  zero  singulorum  distantia  a  piano,  in  quo  jacent ;  sed  &  ii  casus  involvi 
facile  possent,  concipiendo  alias  binas  punctorum  classes ;  quorum  priora  sint  in  priore 
piano  AB,  posteriora  in  posteriore  CD,  quae  quidem  nihil  rem  turbant  :  nam  prioris  classis 


FIG.  36. 


A  THEORY  OF  NATURAL  PHILOSOPHY  191 

a  far  wider  application  than  to  the  single  cases  of  mass  endowed  with  equal  &  parallel  forces 
such  as  we  have  assumed  to  exist  in  our  heavy  bodies ;  &  indeed  such  do  not  exist  accurately 
even  in  the  latter.  Hence,  taking  the  definition  given  above,  which  is  independent  of 
gravity  &  the  nature  of  equilibrium  of  weights,  I  will  proceed  to  deduce  from  it  certain 
corollaries,  which  will  enable  us  to  demonstrate  the  properties  of  the  centre  of  gravity. 

242.  First  of  all,  then,  if  there  should  be  any  plane  such  that  the  two  sums  of  the  General     corollary 
perpendicular  distances  of  all  the  points  on  either  side  of  it  taken  together  are  equal  to  Of  ^h^d^ances'of 
one  another,  then  the  sums  of  the  distances  taken  together  in  any  other  given  direction,  ail  the  points  of  a 
that  is  the  same  for  all  of  them,  will  also  be  equal  to  one  another.     For,  any  perpendicular  ^slng™ •hrVu'g'h 
distance  will  evidently  be  in  the  same  ratio  to  the  corresponding  distance  inclined  at  a  the  centre  of  grav- 
given  angle.     Hence  the  sums  of  the  former  distances  will  bear  the  same  ratio  to  the  sums  e^he/sicfeoTit!  ° 
of  the  latter  distances  ;    &  therefore  the  equality  of  the  sums  in  either  of  the  two  cases 

will  involve  the  equality  of  the  sums  for  the  other  also.  Consequently,  in  what  follows, 
whenever  I  speak  of  distances,  I  intend  in  general  distances  in  any  given  direction,  unless 
I  expressly  say  that  they  are  perpendicular  distances. 

243.  If  now  we  take  any  other  plane  parallel  to  the  plane  for  which  the  sums  of  the  Two.  theorems 
distances  on  either  side  are  equal,  then  the  sum  of  the  distances  of  all  the  points  lying  pearaiief    °o  P  the 
on  the  one  side  of  it  will  exceed  the  sum  for  those  lying  on  the  other  side  by  an  amount  P|ane     o  f    equal 
equal  to  the  distance  between  the  two  planes  measured  in  the  like  direction  multiplied  demonstrations, 
by  the  number  of  all  the  points.     Conversely,  if  there  are  two  parallel  planes,  &  if  the 

excess  of  the  sum  of  the  distances  from  one  of  them  over  the  sum  of  the  distances  from 
the  other  is  equal  to  the  distance  between  the  planes  multiplied  by  the  number  of  the 
points,  then  the  second  plane  will  have  the  sums  of  the  opposite  distances  equal  to  one 
another.  This  is  easily  seen  to  be  true  ;  for,  if  the  plane  of  equal  distances  is  assumed  to 
be  moved  towards  the  other  plane  by  a  parallel  motion  in  the  direction  in  which  the  distances 
the  measured,  then  as  the  plane  is  moved  each  of  the  distances  on  the  one  side  increase, 
&  those  on  the  other  side  decrease  by  just  the  amount  through  which  the  plane  is  moved  ; 
&  should  any  distance  vanish  in  the  meantime,  there  will  be  an  increase  on  the  other  side 
of  just  the  same  amount.  Thus,  it  is  evident  that  the  excess  of  all  the  distances  on  the 
near  side  above  the  sum  of  all  the  distances  on  the  far  side  will  be  equal  to  the  distance 
through  which  the  plane  has  been  moved,  taken  as  many  times  as  there  are  points.  On 
the  other  hand,  when  the  plane  is  moved  back  again,  this  excess  is  destroyed,  namely  exactly 
the  amount  that  was  produced  as  the  plane  moved  forward,  &  consequently  equality  will 
be  restored.  But  to  give  a  more  rigorous  demonstration,  let  the  straight  line  AB,  in  Fig.  36, 
represent  the  plane  of  equal  distances,  &  let  CD  represent  a  plane  parallel  to  it.  Then 
all  the  points  can  be  grouped  into  three  classes ;  let  the  first  of  these  be  that  in  which  we 
have  every  point  that  lies  on  the  near  side  of  both  the  planes,  as  E  ;  let  the  second  be  that 
in  which  every  point  lies  between  the  two  planes,  as  F ;  &  the  third,  every  point  lying 
on  the  far  side  of  both  planes,  as  G.  Let  straight  lines,  drawn  in  any  given  direction  whatever, 
through  the  points  meet  AB  in  M,  H,  K,  &  the  straight  line  CD  in  N,  I,  L  ;  also  let  any 
straight  line,  drawn  in  the  same  direction,  meet  AB,  CD  in  O  &  P.  Then  it  is  clear  that 
OP  will  be  equal  to  MN,  HI,  or  KL.  Now,  let  us  denote  the  sum  of  all  the  points  of  the 
first  class,  like  E,  by  the  letter  E,  &  the  sum  of  all  the  distances  like  EM  by  the  letter  e  ; 
&  those  of  the  second  class  by  the  letters  F  &  / ;  those  of  the  third  class  by  G  &  g ;  & 
the  distance  OP  by  O.  Then  it  is  evident  that  the  sum  of  all  the  MN's  will  be  E  X  O  ; 
the  sum  of  all  the  Hi's  will  be  F  X  O  ;  the  sum  of  all  the  KL's  will  be  G  X  O  ;  also 
in  every  case,  EN  =  EM  +  MN,  FI  =  HI  —  FH,  &  GL  =  KG  —  KL.  Hence  the  sum 
of  the  EN's  will  be  e  +  E  X  O,  the  sum  of  the  FI's  will  be  F  X  O—  /,  &  the  sum  of  the 
GL's  will  be  g  —  G  X  O.  Hence,  the  sum  of  all  the  distances  of  the  points  lying  on  the 
near  side  of  the  plane  CD,  that  is  to  say,  those  belonging  to  the  first  &  second  classes,  will 
be  equal  to<?+ExO  +  Fx  O—  / ;  &  the  sum  of  all  those  lying  on  the  far  side,  that 
is,  of  the  third  class,  will  be  equal  to  g  —  G  X  O.  Hence,  the  excess  of  the  former  over 
the  latter  will  be  equal  to  ^+ExO+FxO  —  /  —  g+GxO.  Therefore,  if  at 
first  we  had  e  =  /  +  g,  then,  on  omitting  e  —  f  —  g,  we  have  the  total  excess  equal  to 
ExO+FxO  +  GxO,or(E+F-fG)xO,  i.e.,  the  sum  of  all  the  points  multiplied 
by  the  distance  between  the  planes.  Conversely,  if  the  excess  with  respect  to  the  second 
plane  CD  were  equal  to  this  sum  multiplied  by  the  distance  O,  it  must  be  that  e  —  f  —  g 
is  equal  to  nothing,  &  thus  e  =  f  -\-  g;  in  other  words  the  sum  of  the  distances  with  respect 
to  the  first  plane  AB  must  be  equal  on  one  side  &  the  other. 

244.  If  any  of  the  points  should  be  in  one  or  other  of  the  two  planes,  these  may  also  C°^lestio"s°tf0  ^e 
be  included  in  the  foregoing  formulae,  if  we  suppose  that  the  distance  for  each  of  them  ^°^e'  au  apos°ib"e 
is  zero  distance  from  the  plane  in  which  they  lie.     Then  these  cases  may  also  be  included  cases. 

by  considering  that  there  are  two  fresh  classes  of  points ;  namely,  first  those  lying  in  the 
first  plane  AB,  &  secondly  those  lying  in  the  second  plane  CD  ;  &  these  classes  will  in 


192 


PHILOSOPHIC  NATURALIS  THEORIA 


Theoremata  pro 
piano  posito  ultra 
omnia  pun  eta: 
eorum  extensio  ad 
qua; vis  plana. 


Cuivis  piano  in- 
veniri  posse  paral- 
lelum  planum  dis- 
tantiarum  aequa- 
lium. 


Thoorema  prseci- 
puum  si  tria  plana 
distantiarum  aequa- 
lium  habeant  uni- 
cum  punctum 
commune  ;  rcliqua 
ornnia  por  id  tran- 
seuntia  erunt  ejus- 
modi. 


Demonstratio  ejus- 
dem. 


distantiae  a  priore  piano  erunt  omnes  simul  zero,  &  a  posteriore  sequabuntur  distantiae  O 
ductae  in  eorum  numerum,  quae  summa  accedit  priori  summae  punctorum  jacentium  citra  ; 
posterioris  autem  classis  distantiaa  a  priore  erant  prius  simul  aequales  summaa  ipsorum 
ductae  itidem  in  O,  &  deinde  fiunt  nihil ;  adeoque  [115]  summse  distantiarum  punctorum 
jacentium  ultra,  demitur  horum  posteriorum  punctorum  summa  itidem  ducta  in  O,  & 
proinde  excessui  summse  citeriorum  supra  summam  ulteriorum  accedit  summa  omnium 
punctorum  harum  duarum  classium  ducta  in  eandem  O. 

245.  Quod  si  planum  parallelum  piano  distantiarum  aequalium  jaceat   ultra   omnia 
puncta  ;  jam  habebitur  hoc  theorema  :  Summa  omnium  distantiarum  punctorum  omnium  ab 
eo  piano  cequabitur  distantly  planorum  ducta  in  omnium  punctorum  summam,  &  si  fuerint  duo 
plana  parallela  ejusmodi,  ut  alterum  jaceat  ultra  omnia  puncta,  fcsf  summa  omnium   distanti- 
arum ab  ipso  cequetur  distantice  planorum  ductce  in  omnium  punctorum  numerum  ;  alterum  illud 
planum  erit  planum  distantiarum  cequalium.     Id  sane  patet  ex  eo,  quod  jam  secunda  sum- 
ma pertinens  ad  puncta  ulteriora,  quae  nulla  sunt,  evanescat,  &  excessus  totus  sit  sola  prior 
summa.    Quin  immo  idem  theorema  habebit  locum  pro  quovis  piano  habente  etiam  ulteriora 
puncta,  si  citeriorum  distantiae  habeantur  pro  positivis,  &  ulteriorum  pro  negativis  ;  cum 
nimirum  summa  constans  positivis,  &  negativis  sit  ipse  excessus  positivorum  supra  negativa  ; 
quo  quidem  pacto  licebit  considerare  planum  distantiarum  aequalium,  ut  planum,   in  quo 
summa  omnium  distantiarum  sit  nulla,  negativis  nimirum  distantiis  elidentibus  positivas. 

246.  Hinc  autem  facile  jam  patet,  data  cuivis  piano  haberi  aliquod  planum  parallelum, 
quod  sit  planum  distantiarum  cequalium  ;    quin  immo  data  positione  punctorum,  &  piano  illo 
ipso,  facile  id  alterum  definitur.     Satis  est  ducere  a  singulis  punctis  datis  rectas  in  data 
directione  ad  planum  datum,  quae  dabuntur ;    turn  a  summa  omnium,  quae  jacent  ex  parte 
altera,  demere  summam  omnium,  si  quae  sunt,  jacentium  ex  opposita,  ac  residuum  dividere 
per  numerum  punctorum.     Ad  earn  distantiam  ducto  piano  priori  parallelo,  id  erit  planum 
quaesitum   distantiarum   aequalium.     Patet    autem   admodum   facile   &   illud    ex    eadem 
demonstratione,  &  ex  solutione  superioris  problematis,  dato  cuivis  piano  non  nisi  unicum 
esse  posse  planum  distantiarum  aequalium,  quod  quidem  per  se  satis  patet. 

247.  Hisce  accuratissime  demonstratis,  atque  explicatis,  progrediar  ad  demonstrandum 
haberi  aliquod  gravitatis  centrum  in  quavis  punctorum  congerie,  utcunque  dispersorum, 
&  in  quotcunque  massas  ubicunque  sitas 

coalescentium.  Id  net  ope  sequentis 
theorematis ;  si  per  quoddam  punctum  tran- 
seant  tria  plana  distantiarum  cequalium  se 
non  in  eadem  communi  aliqua  recta  secan- 
tia  ;  omnia  alia  plana  transeuntia  per  illud 
idem  punctum  erunt  itidem  distantiarum 
esqualium  plana.  Sit  enimin  fig.  37,  ejus- 
modi punctum  C,  per  quod  transeant  tria 
plana  GABH,  XABY,  ECDF,  qua;  om- 
nia sint  plana  distantiarum  aequalium, 
ac  sit  quodvis  aliud  planum  KICL  tran- 
[i  i6]-siens  itidem  per  C,  ac  secans  pri- 
mum  ex  iis  recta  CI  quacunque  ;  opor- 
tet  ostendere,  hoc  quoque  fore  planum 
distantiarum  aequalium,  si  ilia  priora 
ejusmodi  sint.  Concipiaturquodcunque 
punctum  P  ;  &  per  ipsum  P  concipiatur 
tria  plana  parallela  planis  DCEF,  ABYX, 
GABH,  quorum  sibi  priora  duo  mutuo 
occurrant  in  recta  PM,  postrema  duo 
in  recta  PV,  primum  cum  tertio  in 
recta  PO  :  ac  primum  occurrat  piano 
GABH  in  MN,  secundum  vero  eidem 
in  MS,  piano  DCEF  in  QR,  ac  piano  CIKL  in  SV,  ducaturque  ST  parallela  rectis  QR,  MP, 
quas,  utpote  parallelorum  planorum  intersectiones,  patet  fore  itidem  parallelas  inter  se,  uti 
&  MN,  PO,  DC  inter  se,  ac  MS,  PTV,  BA  inter  se. 

248.  Jam  vero  summa  omnium  dis  antiarum  a  piano  KICL  secundum  datam  direc- 
tionem  BA  erit  summa  omnium  PV,  quae  resolvitur  in  tres  summas,  omnium  PR,  omnium 
RT,  omnium  TV,  sive  eae,  ut  figura  exhibet  in  unam  colligendss  sint,  sive,  quod  in  aliis 
plani  novi  inclinationibus  posset  accidere,  una  ex  iis  demenda  a  reliquis  binis,  ut  habeatur 
omnium  PV  summa.     Porro  quaevis  PR  est  distantia  a  piano  DCEF  secundum  eandem 
earn   directionem  ;  quaevis  RT  est  aequalis  QS  sibi  respondenti,  quae  ob  datas  directiones 
laterum  trianguli  SCQ  est  ad  CQ,  aequalem  MN,  sive  PO,  distantiae  a  piano  XABY  secundum 


A  THEORY  OF  NATURAL  PHILOSOPHY  193 

no  way  cause  any  difficulty.  For  the  distances  of  the  points  of  the  first  class  from  the  first 
plane,  all  together,  will  be  zero,  &  their  distances  from  the  second  plane  will,  all  together, 
be  equal  to  the  distance  O  multiplied  by  the  number  of  them  ;  &  this  sum  is  to  be  added 
to  the  former  sum  for  the  points  lying  on  the  near  side.  Again,  the  distances  of  the  points 
of  the  second  class  from  the  first  plane  were,  all  together,  at  first  equal  to  the  distance 
O  multiplied  by  their  number,  &  then  are  nothing  for  the  second  plane.  Hence  from  the 
sum  of  the  distances  of  the  points  lying  on  the  far  side,  we  have  to  take  away  the  sum  of 
these  last  points  also  multiplied  by  the  distance  O  ;  &  thus,  to  the  excess  of  the  sum  of  the 
points  on  the  near  side  over  the  sum  of  the  points  on  the  far  side  we  have  to  add  the  sum 
of  all  the  points  in  these  two  classes  multiplied  by  the  same  distance  O. 

245.  Now,  if  the  plane  parallel  to  the  plane  of  equal  distances  should  lie  on  the  far  Theorems    for   a 
side  of  all  the  points   then  the  following  theorem  is  obtained.     The  sum  of  all  the  distances  P1,3;"6  lvins  beyond 

/      M     i  •         i  7  •        i  -77  7  77-  7  7         i  ,   .    , .    ,   ,       a  1 1  » t  h  e     points ; 

of  all  the  -points  from  this  plane  will  be  equal  to  the  distance  between  the  planes  multiplied  by  extension  of  these 

the  sum  of  all  the  points  ;   y  if  there  were  two  parallel  planes,  such  that  one  of  them  lies  beyond  theorems  to ^any 

all  the  points,  £ff  if  the  sum  of  all  the  distances  from  this  plane  is  equal  to  the  distance  between 

the  planes  multiplied  by  the  number  of  points,  then  the  other  plane  will  be  the  plane  of  equal 

distances.     This  is  perfectly  clear  from  the  fact  that  in  this  case  the  second  sum  relating 

to  the  points  that  lie  beyond  the  planes  vanishes,  for  there  are  no  such  points,  &  the  whole 

excess  corresponds  to  the  first  sum  alone.     Further,  the  same  theorem  holds  good  for  any 

plane  even  if  there  are  points  beyond  it,  if  the  distances  of  points  on  the  near  side  of  it 

are  reckoned  as  positive  &  those  on  the  far  side  as  negative  ;   for  the  sum  formed  from  the 

positives  &  the  negatives  is  nothing  else  but  the  excess  of  the  positives  over  the  negatives. 

In  precisely  the  same  manner,  we  may  consider  the  plane  of  equal  distances  to  be  a  plane 

for  which  the  sum  of  all  the  distances  is  nothing,  that  is  to  say,  the  positive  distances  cancel 

the  negative  distances. 

246.  From  the  foregoing  theorem  it  is  now  clear  that  for  any  given  plane  there  exists  Given   any  plane, 
another  plane  parallel  to  it,  which  is  a  plane  of  equal  distances  ;  further,  if  we  are  given  the  a  ^fane"  of  equal 
position  of  the  points,  y  also  the  plane  is  given,  then  the  parallel  plane  is  easily  determined,  distances,  parallel 
It  is  sufficient  to  draw  from  each  of  the  points  straight  lines  in  a  given  direction  to  the 

given  plane,  &  then  these  are  all  given  ;  then  from  the  sum  of  all  of  them  that  lie  on  the 
one  side  to  take  away  the  sum  of  all  those  that  lie  on  the  other  side,  if  any  such  there  are  ; 
&  lastly  to  divide  the  remainder  by  the  number  of  the  points.  If  a  plane  is  drawn  parallel  to 
the  first  plane,  &  at  a  distance  from  it  equal  to  the  result  thus  found,  then  this  plane  will 
be  a  plane  of  equal  distances,  as  was  required.  Moreover  it  can  be  seen  quite  clearly, 
&  that  too  from  the  very  demonstration  just  given,  that  to  any  given  plane  there  can  cor- 
respond but  one  single  plane  of  equal  distances ;  indeed  this  is  sufficiently  self-evident 
without  proof. 

247.  Now  that  the  foregoing  theorems  have  received  rigorous  demonstrations    &  The    important 
explanation,  I  will  proceed  to  prove  that  there  is  a  centre  of  gravity  for  any  set  of  points,  three^i'anes'  o'f 
no  matter  how  they  are  dispersed  or  what  the  number  of  masses  may  be  into  which  they  equal     distances 
coalesce,  or  where  these  masses  may  be  situated.     The  proof  follows  from  the  theorem  : —  p^ft(  ^heiT^ny 
//  through  any  point  there  pass  three  planes  of  equal  distances  that  do  not  all  cut  one  another  in  other  plane  through 
some  common  line  then  all  other  planes  passing  through  this  same  point  will  also  be  planes  of  equal  \^  ^^e  nature!  ° 
distances.     In  Fig.  37,  let  C  be  a  point  of  this  sort,  &  through  it  suppose  that  three  planes, 

GABH,  XABY,  ECDF,  pass ;  also  suppose  that  all  the  planes  are  planes  of  equal  distances. 
Let  KICL  be  any  other  plane  passing  through  C  also,  &  cutting  the  first  of  the  three  planes 
in  any  straight  line  CI ;  we  have  to  prove  that  this  latter  plane  is  a  plane  of  equal  dis- 
tances, if  the  first  three  are  such  planes.  Take  any  point  P ;  &  through  P  suppose  three 
planes  to  be  drawn  parallel  to  the  planes  DCEF,  ABYX,  GABH  ;  let  the  first  two  of 
these  meet  one  another  in  the  straight  line  PM,  the  last  two  in  the  straight  line  PV,  &  the 
first  &  third  in  the  straight  line  PO.  Also  let  the  first  meet  the  plane  GABH  in  the 
straight  line  MN,  the  second  meet  this  same  plane  in  MS,  &  the  plane  DCEF  in  QR, 
the  plane  CIKL  in  SV,  &  let  ST  be  drawn  parallel  to  the  straight  lines  QR  &  MP,  which, 
since  they  are  intersections  with  parallel  planes,  are  parallel  to  one  another  ;  similarly  MN, 
PO,  DC  are  parallel  to  one  another,  as  also  are  MS,  PTV  &  BA  parallel  to  one  another. 

248.  Now,  the  sum  of  all  the  distances  from  the  plane  KICL,  in  the  given  direction  Proof  of  the  theo- 
BA,  will  be  equal  to  the  sum  of  all  the  PV's ;   &  this  can  be  resolved  into  the  three  sums,  r 

that  of  all  the  PR's,  that  of  all  the  RT's,  &  that  of  all  the  TV's ;  whether  these,  as  are 
shown  in  the  figure,  have  to  be  all  collected  into  one  whole,  or,  as  may  happen  for  other 
inclinations  of  a  fresh  plane,  whether  one  of  the  sums  has  to  be  taken  away  from  the  other 
two,  to  give  the  sum  of  all  the  PV's.  Now  each  PR  is  the  distance  of  a  point  P  from  the 
plane  DCEF,  measured  in  the  given  direction  ;  &  eachRT  is  equal  to  the  QS  that  corresponds 
to  it,  which,  on  account  of  the  given  directions  of  the  sides  of  the  triangle  SCQ  bears  a 
given  ratio  to  CQ  ,  the  latter  being  equal  to  MN  or  PO,  the  distance  of  P  from  the  plane 


194  PHILOSOPHISE  NATURALIS  THEORIA 

datam  directionem  DC,  in  ratione  data  ;  &  quaevis  VT  est  itidem  in  ratione  data  ad  TS 
aequalem  PM,  distantiae  a  piano  GABH  secundum  datam  directionem  EC  ;  ac  idcirco 
etiam  nulla  ex  ipsis  PR,  RT,  TV  poterit  evanescere,  vel  directione  mutata  abire  e  positiva 
in  negativam,  aut  vice  versa,  mutato  situ  puncti  P,  nisi  sua  sibi  respondens  ipsius  puncti 
P  distantia  ex  iis  PR,  PO,  PM  evanescat  simul,  aut  directionem  mutet.  Quamobrem  & 
summa  omnium  positivarum  vel  PR,  vel  RT,  vel  TV  ad  summam  omnium  positivarum 
vel  PR,  vel  PO,  vel  PM,  &  summa  omnium  negativarum  prioris  directionis  ad  summam 
omnium  negativarum  posterioris  sibi  respondentis,  erit  itidem  in  ratione  data  ;  ac  proinde 
si  omnes  positivae  directionum  PR,  PO,  PM  a  suis  negativis  destruuntur  in  illis  tribus 
aequalium  distantiarum  planis,  etiam  omnes  positivae  PR,  RT,  TV  a  suis  negativis  destru- 
entur,  adeoque  &  omnes  PV  positivae  a  suis  negativis.  Quamobrem  planum  LCIK  erit 
planum  distantiarum  aequalium.  Q.E.D. 
[Haberi  semper  2A.Q.  Demonstrato  hoc  theoremate  iam  sponte  illud  consequitur,  in  quavis  punclorum 

aliquod     pravitatis  •  J  7-  77-  ?•:;•_.• 

centrum,  atque  id  fongene,  adeoque  massarum  utcunque  dispersarum  summa,  haben  semper  aiiquod  gravitatis 

esse  unicum.]  centrum,  atque  id  esse  unicum,  quod  quidem  data  omnium  -punctorum  positione  facile  determin- 

abitur.  Nam  assumpto  puncto  quovis  ad  arbitrium  ubicunque,  ut  puncto  P,  poterunt  duci 
per  ipsum  tria  plana  quaecunque,  ut  OPM,  RPM,  RPO.  Turn  singulis  poterunt  per 
num.  246  inveniri  plana  parallela,  [117]  quae  sint  plana  distantiarum  sequalium,  quorum 
priora  duo  si  sint  DCEF,  XABY,  se  secabunt  in  aliqua  recta  CE  parallela  illorum  inter- 
section! MP  ;  tertium  autem  GABH  ipsam  CE  debebit  alicubi  secare  in  C  ;  cum  planum 
RPO  secet  PM  in  P  :  nam  ex  hac  sectione  constat,  hanc  rectam  non  esse  parallelam  huic 
piano,  adeoque  nee  ilia  illi  erit,  sed  in  ipsum  alicubi  incurret.  Transibunt  igitur  per 
punctum  C  tria  plana  distantiarum  aequalium,  adeoque  per  num.  247  &  aliud  quodvis 
planum  transiens  per  punctum  idem  C  erit  planum  aequalium  distantiarum  pro  quavis 
directione,  &  idcirco  etiam  pro  distantiis  perpendicularibus ;  ac  ipsum  punctum  C  juxta 
definitionem  num.  241,  erit  commune  gravitatis  centrum  omnium  massarum,  sive  omnis 
congeriei  punctorum,  quod  quidem  esse  unicum,  facile  deducitur  ex  definitione,  &  hac 
ipsa  demonstratione  ;  nam  si  duo  essent,  possent  utique  per  ipsa  duci  duo  plana  parallela 
directionis  cujusvis,  &  eorum  utrumque  esset  planum  distantiarum  aequalium,  quod  est 
contra  id,  quod  num.  246  demonstravimus. 

^nm^nlaberiseml  25O-  Demonstr<indum  necessario  fuit,  haberi  aliquod  gravitatis   centrum,  atque  id 

per  centrum  gravi-  esse  unicum  ;  &  perperam  id  quidem  a  Mechanicis  passim  omittitur  ;  si  enim  id  non 
ubique  adesset,  &  non  esset  unicum,  in  paralogismum  incurrerent  quamplurimae  Mechanic- 
orum  ipsorum  demonstrationes,  qui  ubi  in  piano  duas  invenerunt  rectas,  &  in  solidis  tria 
plana  determinantia  aequilibrium,  in  ipsa  intersectione  constituunt  gravitatis  centrum,  & 
supponunt  omnes  alias  rectas,  vel  omnia  alia  plana,  quae  per  id  punctum  ducantur,  eandem 
aequilibrii  proprietatem  habere,  quod  utique  fuerat  non  supponendum,  sed  demonstrandum. 
Et  quidem  facile  est  similis  paralogismi  exemplum  praebere  in  alio  quodam,  quod  magni- 
tudinis  centrum  appellare  liceret,  per  quod  nimirum  figura  sectione  quavis  secaretur  in 
duas  partes  asquales  inter  se,  sicut  per  centrum  gravitatis  secta,  secatur  in  binas  partes 
aequilibratas  in  hypothesi  gravitatis  constantis,  &  certam  directionem  habentis  piano 
secanti  parallelam. 

mapiltudinis^noii  2SI-  Erraret  sane,  qui  ita  defmiret  centrum  magnitudinis,  turn  determinaret  id  ipsum 

semper  haberi.  in  datis  figuris  eadem  ilia  methodo,  quae  pro  centri 
gravitatis  adhibetur.  Is  ex.  gr.  pro  triangulo  ABG 
in  fig.  38  sic  ratiocinationem  institueret.  Secetur 
AG  bifariam  in  D,  ducaturque  BD,  quse  utique 
ipsum  triangulum  secabit  in  duas  partes  aequales. 
Deinde,  secta  AB  itidem  bifariam  in  E,  ducatur  GE, 

quam  itidem  constat,  debere  secare  triangulum  in  /   C"^^   \C*          \ 

partes  aequales  duas.     In  earum  igitur  concursu  C       •" '  '"•^AVx 

habebitur  centrum  magnitudinis.  Hoc  invento  si 
progrederetur  ulterius,  &  haberet  pro  aequalibus 
partes,  quae  alia  sectione  quacunque  facta  per  C 
obtinentur ;  erraret  pessime.  Nam  ducta  ED,  jam 
constat,  fore  ipsam  ED  parallelam  BG,  &  ejus  dimi- 
diam  ;  adeoque  similia  fore  triangula  [118]  ECD, 
BCG,  &  CD  dimidiam  CB.  Quare  si  per  C  ducatur  FH  parallela  AG  ;  triangulum  FBH, 
erit  ad  ABG,  ut  quadratum  BC  ad  quadratum  BD,  seu  ut  4  ad  9,  adeoque  segmentum 
FBH  ad  residuum  FAGH  est  ut  4  ad  5,  &  non  in  ratione  sequalitatis. 

Ubi  haec  primo  252.  Nimirum   quaecunque   punctorum,   &    massarum   congeries,    adeoque   &   figura 

demonstrata  •       •  •    •  •    p    •  e  • 

quaevis,  in  qua  concipiatur  punctorum  numerus  auctus  in  innnitum,  donee  ngura  ipsa 
evadat  continua,  habet  suum  gravitatis  centrum  ;  centrum  magnitudinis  infinites  earum 
non  habent  ;  &  illud  primum,  quod  hie  accuratissime  demonstravi,  demonstraveram  jam 


A  THEORY  OF  NATURAL  PHILOSOPHY  195 

XABY,  measured  in  the  given  direction  DC ;  lastly,  VT  is  also  in  a  given  ratio  to  TS,  the 
latter  being  equal  to  PM,  the  distance  of  the  point  P  from  the  plane  GABH,  measured  in 
the  given  direction  EC.  Hence,  none  of  the  distances  PR,  RT,  TV  can  vanish  or,  having 
changed  their  directions,  pass  from  positive  to  negative,  or  vice  versa,  by  a  change  in  the 
position  of  the  point  P,  unless  that  one  of  the  distances  PR,  PO,  PM,  of  the  point  P,  which 
corresponds  to  it  vanishes  or  changes  its  direction  at  the  same  time.  Therefore  also  the 
sum  of  all  the  positives,  whether  PR,  or  RT,  or  TV  to  the  sum  of  all  the  positives,  PR, 
or  PO,  or  PM,  &  the  sum  of  all  the  negatives  for  the  first  direction  to  the  sum  of  all  the 
negatives  for  the  second  direction  which  corresponds  to  it,  will  also  be  in  a  given  ratio. 
Thus,  finally,  if  all  the  positives  out  of  the  direction  PR,  PO,  PM  are  cancelled  by  the 
corresponding  negatives  in  the  case  of  the  three  planes  of  equal  distances ;  then  also  all 
the  positive  PR's,  RT's,  TV's  are  cancelled  by  their  corresponding  negatives,  &  therefore 
also  all  the  positive  PV's  are  cancelled  by  their  corresponding  negatives.  Consequently, 
the  plane  LCIK  will  be  a  plane  of  equal  distances.  Q.E.D. 

249.  Now  that  we  have  demonstrated  the  above  theorem,  it    follows    immediately  There    is    always 
from  it  that,  for  any  group  of  -points,  tj  therefore  also  for  a  set  of  masses  scattered  in  any  manner,  ty^^^ty1^ 
there  exists  a  centre  of  gravity,  W  there  is  but  one  ;  W  this  can  be  easily  determined  when  the 

position  of  each  of  the  points  is  given.  For  if  a  point  is  taken  at  random  anywhere,  like  the 
point  P  there  could  be  drawn  through  it  any  three  planes,  OPM,  RPM,  RPO.  Then 
corresponding  to  each  of  these  there  could  be  found,  by  Art.  245,  a  parallel  plane,  such 
that  these  planes  were  planes  of  equal  distances.  If  the  first  two  of  these  are  DCEF  & 
XABY,  they  will  cut  one  another  in  some  straight  line  CE  parallel  to  their  intersection 
MP  ;  also  the  third  plane  GABH  must  cut  this  straight  line  CE  somewhere  in  C  ;  for 
the  plane  RPO  will  cut  PM  in  P,  &  from  this  fact  it  follows  that  the  latter  line  is  not  parallel 
to  the  latter  plane,  &  therefore  the  former  line  is  not  parallel  to  the  former  plane,  but  will 
cut  it  somewhere.  Hence  three  planes  of  equal  distances  will  pass  through  the  point  C, 
&  therefore,  by  Art.  247,  any  other  plane  passing  through  this  point  C  will  also  be  a  plane 
of  equal  distances  for  any  direction,  &  thus  also  for  perpendicular  distances.  Hence,  according 
to  the  definition  of  Art.  241,  the  point  C  will  be  the  common  centre  of  gravity  of  all  the 
masses,  or  of  the  whole  group  of  points.  That  there  is  only  one  can  be  easily  derived  from 
the  definition  &  the  demonstration  given  ;  for,  if  there  were  two,  there  could  in  every 
case  be  drawn  through  them  two  parallel  planes  in  any  direction,  &  each  of  these  would 
be  a  plane  of  equal  distances ;  which  is  contrary  to  what  we  have  proved  in  Art.  246. 

250.  It  was  absolutely  necessary  to  prove  that  there  always  exists  a  centre  of  gravity,  The    need    for 
&  that  there  is  only  one  in  every  case  ;  &  this  proof  is  everywhere  omitted  by  Mechanicians,  ^centre^f  gray6 
quite  unjustifiably.     For,  if  there  were  not  one  in  every  case,  or  if  it  were  not  unique,  ity  in  every  case, 
very  many  of  the  proofs  given  by  these  Mechanicians  would  result  in  fallacious  argument. 

Where,  for  instance,  they  find  two  straight  lines,  in  the  case  of  a  plane,  &  in  the  case  of  solids 
three  planes,  determining  equilibrium,  &  suppose  that  all  other  lines,  &  all  other  planes, 
which  are  drawn  through  the  point  to  have  the  same  property  of  equilibrium ;  this  in 
every  case  ought  not  to  be  a  matter  of  supposition,  but  of  proof.  Indeed  it  is  easy  to  give 
a  similar  example  of  fallacious  argument  in  the  case  of  something  else,  which  we  may  call 
the  centre  of  magnitude  ;  for  instance,  where  a  figure  is  cut,  by  any  section,  into  two  parts 
equal  to  one  another ;  just  as  when  the  section  passes  through  the  centre  of  gravity  it  is 
cut  into  two  parts  that  balance  one  another,  on  the  hypothesis  of  uniform  gravitation 
acting  in  a  fixed  direction  parallel  to  the  cutting  plane. 

251.  He  would  indeed  be  much  at  fault,  who  would  so  define  the  centre  of  magnitude  For  there  is  not 
&  then  proceed  to  determine  it  in  given  figures  by  the  same  method  as  that  used  for  the  m^aitude061 
centre  of  gravity.     For  example,  the  reasoning  he  would  use  for  the  triangle  ABG,  in  Fig.  38, 

would  be  as  follows.  Let  AB  be  bisected  in  D,  &  through  D  draw  BD  ;  this  will  certainly 
divide  the  triangle  into  two  equal  parts.  Then,  having  bisected  AB  also  in  E,  draw  GE  ; 
it  is  true  that  this  also  divides  the  triangle  into  two  equal  parts.  Hence  their  point  of 
intersection  C  will  be  the  centre  of  magnitude.  If  then,  having  found  this,  he  proceeded 
further,  &  said  that  those  parts  were  equal,  which  were  obtained  by  any  other  section  made 
through  C  ;  he  would  be  very  much  in  error.  For,  if  ED  is  drawn,  it  is  well  known  that 
we  now  have  ED  parallel  to  BG  &  equal  to  half  of  it ;  &  therefore  the  triangles  BCD,  BCG 
would  be  similar,  &  CD  half  of  CB.  Hence,  if  FH  is  drawn  through  C  parallel  to  AG, 
the  triangle  FBH  will  be  to  the  triangle  ABG,  as  the  square  on  BC  is  to  the  square  on  BD, 
or  as  4  is  to  9 ;  &  thus  the  segment  FBH  is  to  the  remainder  FAGH  as  4  is  to  5,  &  not 
in  a  ratio  of  equality. 

252.  Thus,  any  group  of  points  or  masses,  &  therefore  any  figure  in  which  the  number  Where  the  first 
of  points  is  supposed  to  be  indefinitely  increased  until  the  figure  becomes  continuous,  *™°    c 
possesses  a  centre  of  gravity ;    but  there  are  an  infinite  number  of  them  which  have  not 

got  a  centre  of    magnitude.     The  first  of   these,  of  which  I  have  here  given  a  rigorous 


196 


PHILOSOPHISE  NATURALIS  THEORIA 


olim  methodo  aliquanto  contractiore  in  dissertatione  De  Centra  Gravitatis ;  hujus  vero 
secundi  exemplum  hie  patet,  ac  in  dissertatione  De  Centra  Magnitudinis,  priori  illi  addita 
in  secunda  ejusdem  impressione,  determinavi  generaliter,  in  quibus  figuris  centrum 
magnitudinis  habeatur,  in  quis  desk  ;  sed  ea  ad  rem  praesentem  non  pertinent. 


Inde  ubi  sit  cen- 
t  r  u  m  commune 
massarum  duarum. 


Inde  &  communis 
methodus  pro  quot- 
cunque  massis. 


Inde  &  theorema, 
ope  cujus  investi- 
gatur  id  in  figuris 
continuis. 


253.  Ex  hac  general!  determinatione  centri  gravitatis  facile  colligitur  illud,  centrum 
commune  binarum  massarum  jacere  in  directum  cum  centris  gravitatis  singularum,  & 
horum  distantias  ab  eodem  esse  reciproce,  ut  ipsas  massas.     Sint  enim  binae  massae,  quarum 
centra  gravitatis  sint  in  fig.  39  in  A,  &  B.     Si  per  rectam  AB  ducatur  planum  quodvis,  id 
debet  esse  planum  distantiarum  sequalium  re- 

spectu  utriuslibet.  Quare  etiam  respectu 
summae  omnium  punctorum  ad  utrumque 
simul  pertinentium  distantiae  omnes  hinc,  & 
inde  acceptae  sequantur  inter  se  ;  ac  proinde  id 
etiam  respectu  summae  debet  esse  planum  dis- 
tantiarum aequalium,  &  centrum  commune  FlG  3g 
debet  esse  in  quovis  ex  ejusmodi  planis,  ade- 

oque  in  intersectione  duorum  quorumcunque  ex  iis,  nimirum  in  ipsa  recta  AB.  Sit 
id  in  C,  &  si  jam  concipiatur  per  C  planum  quodvis  secans  ipsam  AB  ;  erit  summa  omnium 
distantiarum  ab  eo  piano  secundum  directionem  AB  punctorum  pertinentium  ad  massam 
A,  si  a  positivis  demantur  negativae,  aequalis  per  num.  243  numero  punctorum  massae  A 
ducto  in  AC,  &  summa  pertinentium  ad  B  numero  punctorum  in  B  ducto  in  BC  ;  quae 
producta  aequari  debent  inter  se,  cum  omnium  distantiarum  summae  positivae  a  negativis 
elidi  debeant  respectu  centri  gravitatis  C.  Erit  igitur  AC  ad  CB,  ut  numerus  punctorum 
in  B  ad  numerum  in  A,  nimirum  in  ratione  massarum  reciproca. 

254.  Hinc  autem  facile  deducitur  communis  methodus  inveniendi  centrum  gravitatis 
commune  plurium  massarum.     Conjunguntur  prius  centra  duarum,  &?  eorum  distantia  dividitur 
in  ratione  reciproca  ipsarum.     Turn  harum  commune  centrum  sic  inventum  conjungitur  cum 
centra  tertics,  tsf  dividitur  distantia  in  ratione  reciproca  summa  massarum  priorum  ad  massam 
tertiam,    &    ita    porro.     Quin  immo    possunt  seorsum  inveniri  centra  gravitatis  binarum 
quarumvis,  ternarum,  denarum  quocunque  [119]  ordine,  turn  binaria  conjungi  cum  ternariis, 
denariis,  aliisque,  ordine  itidem  quocunque,  &  semper  eadem  methodo  devenitur  ad  centrum 
commune  gravitatis  masses  totius.     Id  patet,  quia  quotcunque  massae  considerari  possunt 
pro  massa  unica,  cum  agatur  de  numero  punctorum  massae  tantummodo,  &  de  summa 
distantiarum  punctorum  omnium  ;    summae  massarum  constituunt   massam,  &  summae 
distantiarum  summam  per  solam  conjunctionem  ipsarum.     Quoniam  autem  ex  generali 
demonstratione  superius  facta  devenitur  semper  ad  centrum  gravitatis,  atque  id  centrum 
est  unicum  ;    quocunque  ordine  res  peragatur,  ad  illud  utique  unicum  devenitur. 

255.  Inde  vero  illud  consequitur,  quod  est  itidem  commune,  si  plurium  massarum 
centra  gravitatis  sint  in  eadem  aliqua  recta,  fore  etiam  in  eadem  centrum  gravitatis  summce 
omnium  ;    quod  viam  sternit  ad  investiganda  gravitatis  centra  etiam  in  pluribus  figuris 
continuis.     Sic  in  fig.  38  centrum  commune  gravitatis  totius  trianguli  est  in  illo  puncto, 
quod  a  recta  ducta  a  vertice  anguli  cujusvis  ad  mediam  basim  oppositam  relinquit  trientem 
versus  basim  ipsam.     Nam  omnium  rectarum  basi  parallelarum,  quae  omnes  a  recta  BD 
secantur  bifariam,  ut  FH,  centra  gravitatis  sunt  in  eadem  recta,  adeoque  &  areae  ab  iis 
contextae  centrum  gravitatis  est  tarn  in  recta  BD,  quam  in  recta  GE  ob  eandem  rationem, 
nempe  in  illo  puncto  C.     Eadem  methodus  applicatur  aliis  figuris  solidis,  ut  pyramidibus  ; 
at  id,  ut  &  reliqua  omnia  pertinentia  ad  inventionem  centri  gravitatis  in  diversis  curvis 
lineis,  superficiebus,  solidis,  hinc  profluentia,  sed  meae    Theoriae    communia    jam    cum 
vulgaribus  elementis,  hie  omittam,  &  solum  illud   iterum  innuam,  ea  rite  procedere,  ubi 
jam  semel  demonstratum  fuerit,  haberi  in  massis  omnibus  aliquod  gravitatis  centrum,  & 
esse  unicum,  ex  quo  nimirum  hie  &  illud  fluit,  areas  FAGH,  FBH  licet  inaequales,  habere 
tamen  aequales  summas  distantiarum  omnium  suorum  punctorum  ab  eadem  recta  FH. 


Difficuitas  demon-  2c6.  In  communi  methodo  alio  modo  se  res  habet.     Posteaquam  inventum  est  in 

strationis    in   com-    „  .  •      A      n    T.     •  •  T\/~>     o 

muni  methodo.  fig.  40  centrum  gravitatis  commune  massis  A,  &  B,  juncta  pro  tertia  massa  DC,  &  secta 
in  F  in  ratione  massarum  D,  &  A  +  B  reciproca,  habetur  F  pro  centro  communi  omnium 
trium.  Si  prius  inventum  esset  centrum  commune  E  massarum  D,  B,  &  juncta  AE,  ea 
secta  fuisset  in  F  in  ratione  reciproca  massarum  A,  &  B  +  D ;  haberetur  itidem  illud 


A  THEORY  OF  NATURAL  PHILOSOPHY  197 

demonstration,  I  proved  some  time  ago  in  a  somewhat  shorter  manner  in  my  dissertation 
De  Centra  Gravitatis ;  &  a  case  of  the  second  is  here  clearly  shown  ;  &  in  the  dissertation 
De  Centra  Magnitudinis,  which  was  added  as  a  supplement  to  the  former  in  the  second 
edition,  I  determined  in  general  the  figures  in  which  there  existed  a  centre  of  magnitude 
&  those  in  which  there  was  none  ;  but  such  things  have  no  Hearing  on  the  matter  now 
in  question. 

253.  From  this  general  determination  of  the  centre  of  gravity  it  is  readily  deduced  that  Hence  to  determine 
the  common  centre  of  two  masses  lies  in  the  straight  line  joining  the  centres  of  each  of  the 

masses,  &  that  the  distances  of  the  masses  from  this  point  will  be  reciprocally  proportional  two  masses. 

to  the  masses  themselves.     For  suppose  we  have  two  masses,  &  that  their  centres  of  gravity 

are,  in  Fig.  39,  at  A  &  B.     If  through  the  straight  line  AB  any  plane  is  drawn,  it  must  be 

a  plane  of  equal  distances  for  either  of  the  masses.     Therefore  also,  with  regard  to  the 

sum  of  the  points  of  both  masses  taken  together,  all  the  distances  taken  on  one  side  &  on 

the  other  side  will  be  equal  to  one  another.     Hence  also  with  regard  to  this  sum  it  must 

be  a  plane  of  equal  distances ;  the  common  centre  must  lie  in  any  one  of  these  planes,  & 

therefore  in  the  line  of  intersection  of  any  two  of  them,  that  is  to  say,  in  the  straight  line 

AB.    Suppose  it  is  at  C  ;  &  suppose  that  any  plane  is  drawn  through  C  to  cut  AB.     Then 

the  sum  of  all  the  distances  from  this  plane  in  the  direction  AB  of  all  the  points  belonging 

to  the  mass  A,  the  negatives  being  taken  from  the  positives,  will  by  Art.  243  be  equal  to 

the  number  of  points  in  the  mass  A  multiplied  by  AC ;   &  the  sum  of  those  belonging  to 

the  mass  B  to  the  number  of  points  in  the  mass  B  multiplied  by  BC.     These  products 

must  be  equal  to  each  other,  since  the  positives  in  the  sum  of  all  the  distances  must  be 

cancelled  by  the  negatives  with  regard  to  the  centre  of  gravity  C.     Hence  AC  is  to  CB 

as  the  number  in  B  is  to  the  number  of  points  in  A,  i.e.,  in  the  reciprocal  ratio  of  the  masses. 

254.  Further,  from  the  foregoing  theorem  can  be  readily  deduced  the  usual  method  Hence,  the    usual 
of  finding  the  common  centre  of  gravity  of  several  masses.     First  of  all  the  centres  of  two  of  ™mtec  Offmassesy 
them  are  joined,  &  the  distance  between  them  is  divided  in  the  reciprocal  ratio  of  the  masses. 

Then  the  common  centre  of  these  two  masses,  thus  found,  is  joined  to  the  centre  of  a  third,  &  the 
distance  is  divided  in  the  reciprocal  ratio  of  the  sum  of  the  first  two  masses  to  the  third  mass  ; 
W  so  on.  Indeed,  we  may  find  the  centres  of  gravities  of  any  groups  of  two,  three,  or  ten,  in 
any  order,  &  then  the  groups  of  two  may  be  joined  to  the  threes,  the  tens,  or  what  not,  also  in 
any  order  whatever  ;  &  in  every  case,  in  precisely  the  same  manner,  we  shall  arrive  at  the 
common  centre  of  gravity  of  the  whole  mass.  This  is  evidently  the  case,  for  the  reason  that 
any  number  of  masses  can  be  reckoned  as  a  single  mass,  since  it  is  only  a  question  of  the 
number  of  points  in  the  mass  &  the  sum  of  the  distances  of  all  the  points ;  the  sum  of  the 
masses  constitute  a  mass,  &  the  sums  of  the  distances  a  sum  of  distances,  merely  by  taking 
them  as  a  whole.  Moreover,  since,  by  the  general  demonstration  given  above,  a  centre  of 
gravity  is  always  obtained,  &  since  this  centre  is  unique,  it  follows  that,  no  matter  in  what 
order  the  operations  are  performed,  the  same  centre  is  arrived  at  in  every  case. 

255.  From  the  above  we  have  a  theorem,  which  is  also  well  known,  namely  : — //  the  Hence,  a  theorem, 
centres  of  gravity  of  several  masses  all  lie  in  one  &  the  same  straight  line,  then  the  centre  of  Which  "the  "centre 
gravity  of  the  whole  set  will  also  lie  in  the  same  straight  line.     This  indicates  a  method  for  of  gravity  for  con- 
investigating  the  centres  of  gravity  also  in  the  case  of  many  continuous  figures.     Thus, 

in  Fig.  38,  the  centre  of  gravity  of  the  whole  triangle  is  at  that  point,  which  cuts  off,  from 
the  straight  line  drawn  through  the  vertex  of  any  angle  to  the  middle  point  of  the  base 
opposite  to  it,  one-third  of  its  length  on  the  side  nearest  to  the  base.  For,  the  centre  of 
gravity  of  every  line  drawn  parallel  to  the  base,  such  as  FH,  since  each  of  them  is  bisected 
by  BD,  lies  in  this  latter  straight  line.  Hence  the  centre  of  gravity  of  the  area  formed 
from  them  lies  in  this  straight  line  BD  ;  as  it  also  does  in  GE  for  a  similar  reason  ;  that 
is  to  say,  it  is  at  the  point  C.  The  same  method  can  be  applied  to  some  solid  figures,  such 
as  pyramids.  But  I  omit  all  this  here,  just  as  I  do  all  the  other  matters  relating  to  the 
finding  of  the  centre  of  gravity  for  diverse  curved  lines,  surfaces  &  solids,  to  be  derived  from 
what  has  been  proved,  but  in  which  my  theory  is  in  agreement  with  the  usual  fundamental 
principles ;  I  will  only  remark  once  again  that  these  all  will  follow  in  due  course  when  once 
it  has  been  shown  that  for  all  masses  there  exists  a  centre  of  gravity,  &  that  there  is  only 
one ;  and  from  this  indeed  there  follows  also  the  theorem  that,  although  the  areas 
FAGH,  FBH  are  unequal,  yet  the  sums  of  the  distances  from  the  straight  line  FH  of  all 
the  points  forming  them  are  equal  to  one  another. 

256.  In  the  ordinary  method  it  is  quite  another  thing.      Afterthat,  in  Fig.  40,    the  The  difficulty  of 
common  centre  of  gravity  of  the  masses  A  &  B  has  been  found,  for  the  third  mass,  whose  ^ary' method. 
centre  is  D,  join  DC  and  divide  it  at  F  in  the  reciprocal  ratio  of   D  to  A  +  B,  then 

F  is  obtained  as  the  common  centre  for  all  three  masses.  If,  first  of  all,  the  common 
centre  E  of  the  masses  D  &  B  had  been  found,  &  AE  were  joined,  &  the  latter 
divided  at  F  in  the  reciprocal  ratio  of  the  masses  A  &  B  +  D  ;  then  the  point  of  section, 


198 


PHILOSOPHISE  NATURALIS  THEORIA 


Similis  difficultas 
in  summa,  &  mul- 
tiplicatione  plurium 
numerorum,  &  in 
vi  composita  ex 
pluribus :  methodus 
componendi  simul 
omnes. 


sectionis  punctum  pro  centre  gravitatis.  Nisi 
generaliter  demonstratum  fuisset,  haberi  sem- 
per aliquod,  &  esse  unicum  gravitatis  cen- 
trum ;  oporteret  hie  iterum  demonstrare  id 
novum  sectionis  punctum  fore  idem,  ac  illud 
prius  :  sed  per  singulos  casus  ire,  res  infi- 
nita  esset,  cum  diversae  rationes  conjungendi 
massas  eodem  redeant,  quo  diversi  ordines 
litterarum  conjungendarum  in  voces,  de  qua- 
rum  multitudine  immensa  in  exiguo  etiam  ter- 
minorum  numero  mentionem  fecimus  num.  1 14. 
[120]  257.  Atque  hie  illud  quidem  accidit, 
quod  in  numerorum  summa,  &  multiplica- 
tione  experimur,  ut  nimirum  quocunque  ordine 


Consensus  e  j  u  s 
methodi  cum  com- 
muni  per  parallelo- 
gramma. 


FIG.  40. 


Demonstr  a  t  i  o 
generalis  methodi. 


accipiantur   numeri,    vel    singuli,  ut 

addantur  numero  jam  invento,  vel  ipsum  multiplicent,  vel  plurium  aggregata  seorsum 
addita,  vel  multiplicata  ;  semper  ad  eundem  demum  deveniatur  numerum  post  omnes, 
qui  dati  fuerant,  adhibitos  semel  singulos ;  ac  in  summa  patet  facile  deveniri  eodem,  &  in 
multiplication  potest  res  itidem  demonstrari  etiam  generaliter,  sed  ea  hue  non  pertinent. 
Pertinet  autem  hue  magis  aliud  ejusmodi  exemplum  petitum  a  compositione  virium,  in 
qua  itidem  si  multse  vires  componantur  communi  methodo  componendo  inter  se  duas  per 
diagonalem  parallelogrammi,  cujus  latera  eas  exprimant,  turn  hanc  diagonalem  cum  tertia, 
&  ita  porro ;  quocunque  ordine  res  procedat,  semper  ad  eandem  demum  post  omnes  adhibitas 
devenitur.  Hujusmodi  compositione  plurimarum  virium  generali  jam  indigebimus,  &  ad 
absolutam  demonstrationem  requiritur  generalis  expressio  compositionis  virium  quotcunque, 
qua  uti  soleo.  Compono  nimirum  generaliter  motus,  qui  sunt  virium  effectus,  &  ex  effectu 
composite  metior  vim,  ut  e  spatiolo,  quod  dato  tempusculo  vi  aliqua  percurreretur,  solet 
ipsa  vis  simplex  quselibet  sestimari.  Assumo  illud,  quod  &  rationi  est  consentaneum,  & 
experimentis  constat,  &  facile  etiam  demonstratur  consentire  cum  communi  methodo  com- 
ponendi vires,  ac  motus  per  parallelogramma,  nimirum  punctum  solicitatum  simul  initio 
cujusvis  tempusculi  actione  conjuncta  virium  quarumcunque,  quarum  directio,  &  magnitude 
toto  tempusculo  perseveret  eadem,  fore  in  fine  ejus  tempusculi  in  eo  loci  puncto,  in  quo  esset, 
si  singulae  eadem  intensitate,  &  directione  egissent  aliae  post  alias  totidem  tempusculis,  quot 
sunt  vires,  cessante  omni  nova  solicitatione,  &  omni  velocitate  jam  producta  a  vi  qualibet  post 
suum  tempusculum  :  turn  rectam,  quae  conjungit  primum  illud  punctum  cum  hoc  postremo, 
assume  pro  mensura  vis  ex  omnibus  compositse,  quae  cum  eadem  perseveret  per  totum 
tempusculum  ;  punctum  mobile  utique  per  unicam  illam  eandem  rectam  abiret.  Quod 
si  &  velocitatem  aliquam  habuerit  initio  illius  tempusculi  jam  acquisitam  ante  ;  assume 
itidem,  fore  in  eo  puncto  loci,  in  quo  esset,  si  altero  tempusculo  percurreret  spatiolum,  ad 
quod  determinatur  ab  ilia  velocitate,  altero  spatiolum,  ad  quod  determinatur  a  vi,  sive  aliis 
totidem  tempusculis  percurreret  spatiola,  ad  quorum  singula  determinatur  a  viribus  singulis. 

258.  Hue  recidere  methodum  compon- 
endi per  parallelogramma  facile  constat ;  si 
enim  in  fig.  41  componendi  sint  plures  motus, 
vel  vires  expressae  a  rectis  PA,  PB,  PC,  &c,  & 
incipiendo  a  binis  quibusque  PA,  PB,  eae  com- 
ponantur per  parallelogrammum  PAMB,  turn 
vis  composita  PM  cum  tertia  PC  per  parallelo- 
grammum PMNC,  &  ita  porro  ;  [121]  patet, 
ad  idem  loci  punctum  N  per  haec  parallelo- 
gramma definitum  debere  devenire  punctum 
mobile,  quod  prius  percurrat  PA,  turn  AM  par- 
allelam,  &  aequalem  PB  ;  turn  MN  parallelam, 
&  aequalem  PC,  atque  ita  porro  additis  quot- 
cunque aliis  motibus,  vel  viribus,  quae  per 


FIG.  41. 


N 


nova  parallela,  &  aequalia  parallelogrammorum  latera  debeant  componi. 

259.  Deveniretur  quidem  ad  idem  punctum  N,  si  alio  etiam  ordine  componerentur 
ii  motus,  vel  vires,  ut  compositis  viribus  PA,  PC  per  parallelogrammum  PAOC,  turn  vi 
PO  cum  vi  PB  per  novum  parallelogrammum,  quod  itidem  haberet  cuspidem  in  N  ;  sed 
eo  deveniretur  alia  via  PAON.  Hoc  autem  ipsum,  quod  tarn  multis  viis,  quam  multas  diversae 
plurium'compositiones  motuum,  ac  virium  exhibere  possunt,  eodem  semper  deveniri  debeat,  sic 
generaliter  demonstro.  Si  assumantur  ultra  omnia  puncta,  ad  quse  per  ejusmodi  compositiones 
deveniri  potest,  planum  quodcunque  ;  ubi  punctum  mobile  percurrit  lineolam  pertinentem 
ad  quencunque  determinatum  motum,  habet  eundem  perpendicularem  accessum  ad  id 
planum,  vel  recessum  ab  eo,  quocunque  tempusculo  id  fiat,  sive  aliquo  e  prioribus,  sive 


A  THEORY  OF  NATURAL  PHILOSOPHY  199 

F,  would  again  be  obtained  as  the  centre  of  gravity.  Now,  unless  it  had  been  already  proved 
in  general  that  there  always  was  one  centre  of  gravity,  &  only  one,  it  would  be  necessary 
here  to  demonstrate  afresh  that  the  new  point  of  section  was  the  same  as  the  first  one. 
But  to  do  this  for  every  single  instance  would  be  an  endless  task  ;  for  diverse  ways  of  joining 
the  masses  come  to  the  same  thing  as  diverse  orders  of  joining  up  letters  to  form  words  ; 
&  I  have  already,  in  Art.  114,  remarked  upon  the  immense  number  of  these  even  with  a 
small  number  of  letters. 

257.  Indeed  the  same  thing  happens  in  the  case  of  addition  &  multiplication;    for  A  similar  difficulty 
we  find,  for  instance,  no  matter  what  the  order  is  in  which  the  numbers  are  taken,  whether  a^ulnOT^roduct 
they  are  taken  singly,  &  added  to  the  number  already  obtained,  or  multiplied,  or  whether  of    several    num- 
the  addition  or  multiplication  is  made  with  a  group  of  several  of  them  ;   the  same  number  ^y  ^)i^Iso  in,  * 
is  arrived  at  finally  after  all  those  that  have  been  given  have  been  used  each  once.     Now  from  several  forces ; 
in  addition  it  is  easily  seen  that  the  result  obtained  is  the  same  ;   &  for  multiplication  also  the     m(*hod     of 

.  '  ..       ,  ,       .  .  t  compounding  them 

the  matter  can  be  easily  demonstrated ;   but  we  are  not  concerned  with  these  proofs  here,  all  at  one  time. 

Moreover,  there  is  another  example  of  this  sort  that  is  far  more  suitable  for  the  present 

occasion,  derived  from  the  composition  of  forces.     In  this,  if  several  forces  are  compounded 

in  the  ordinary  manner,  by  compounding  two  of  them  together  by  means  of  the  diagonal 

of  the  parallelogram  whose  sides  represent  the  forces,  &  then  this  diagonal  with  a  third 

force,  &  so  on.     In  whatever  order  the  operations  are  performed  we  always  arrive  at  the 

same  force  finally,  after  all  the  given  forces  have  been  used.     We  shall  now  need  a  general 

composition  of  very  many  forces,  &  for  rigorous  proof  we  must  have  a  general  representation 

for  the  composition  of  any  number  of  forces,  such  as  the  one  I  usually  employ.      Thus,  I 

in  general  compound  the  motions,  which  are  the  effects  of  the  forces,  &  measure  the  force 

from  the  resultant  of  the  effects ;  so  that  any  simple  force  is  usually  estimated  by  the  small 

interval  of  space  through  which  the  force  moves  its  point  of  application  in  a  given  short 

interval  of  time.     I  make  an  assumption,  which  is  not  only  a  reasonable  one,  but  is  also 

verified  by  experiment,  &  further  one  which  can  be  easily  shown  to  agree  with  the  usual 

method  for  the  composition  of  forces  &  motions  by  means  of  the  parallelogram.     Thus, 

I  assume  that  a  point,  which  is  influenced  simultaneously,  at  the  beginning  of  any  short 

interval  of  time  by  the  joint  action  of  any  forces  whatever,  whose  directions  &  magnitudes 

continue  unchanged  during  the  whole  of  the  interval,  will  be  at  the  end  of  the  interval 

in  the  same  position  in  space,  as  if  each  of  the  forces  had  acted  independently,  one  after 

another,  with  the  same  intensity  &  in  the  same  direction,  during  as  many  intervals  of  time 

as  there  are  forces ;    where  each  fresh  influence  &  the  velocity  already  produced  by  any 

one  of  the  forces  ceases  at  the  end  of  the  interval  that  corresponds  to  it.     Then  I  take  the 

straight  line  which  joins  the  initial  point  to  the  final  point  as  the  measure  of  the  force  that 

is  the  resultant  of  them  all,  &  that  this  force  will  be  represented  by  this  same  straight  line 

during  the  whole  of  the  interval  of  time,  &  that  the  moving  point  will  traverse  in  every 

case  that  straight  line  &  that  one  only.     But  if,  moreover,  at  the  beginning  of  the  interval 

of  time,  the  point  should  have  a  velocity  previously  acquired,  then  I  also  assume  that  it 

would  occupy  that  position  in  space  that  it  would  have  occupied  if  during  another  interval 

of  time  it  had  passed  over  an  .interval  of  space,  determined  by  this  other  velocity,  which 

is  itself  determined  by  the  force  ;    or  if  it  had  passed  over  as  many  intervals  of  spaces  in 

as  many  intervals  of  time  as  there  are  forces  determining  the  initial  velocity. 

258.  It  is  easily  seen  that  the  method  of  composition  by  means  of  the  parallelogram  Agreement  of  this 
comes  to  the  same  thing.     For,  if,  in  Fig.  41,  the  several  motions  or  forces  to  be  compounded  J^^j^^f*11   ^ 
are  represented  by  PA,  PB,  PC,  &c. ;  &,  beginning  with  any  two  of  them,  PA  &  PB,  these  of    the6  pL™Uek>S 
are  compounded  by  means  of  the  parallelogram  PAMB,  then  the  resultant  force  PM  is  gram- 
compounded  with  a  third  PC  by  means  of  the  parallelogram  PMNC,  &  so  on  ;   it  is  clear 

that  the  moving  point  must  reach  the  same  point  of  space,  N,  determined  by  these 
parallelograms,  as  it  would  have  done  if  it  had  traversed  PA,  then  AM  parallel  &  equal  to 
PB,  &  then  MN  parallel  &  equal  to  PC  ;  &  so  on,  for  any  number  of  additional  motions 
or  forces,  which  have  to  be  compounded  by  fresh  straight  lines  equal  &  parallel  to  the  sides 
of  the  parallelograms. 

259.  Now  the  same  point  N  would  be  reached  also,  if  these  motions  or  forces  were  General   proof    of 
compounded  in  another  order,  say,  by  first  compounding  PA  &  PC  by  means  of  the 
parallelogram  PAOC,  then  the  force  PO  with  the  force  PB  by  another  parallelogram,  which 

has  its  fourth  vertex  at  N,  although  the  point  is  reached  by  another  path  PAON.  The 
fact  that  the  same  point  is  bound  to  be  reached,  by  each  of  the  many  paths  that  correspond 
to  the  many  different  orders  of  compounding  several  motions  or  forces,  I  prove  in  general 
as  follows.  Imagine  a  plane  drawn  beyond  any  point  that  could  be  reached  owing  to 
compositions  of  this  kind  ;  then,  when  a  moving  point  traverses  a  short  path  corresponding 
to  any  given  motion,  there  is  the  same  perpendicular  approach  towards  the  plane,  or  recession 
from  it,  in  whichever  of  the  short  intervals  of  time  it  takes  place,  whether  one  of  those  at 


200  PHILOSOPHIC  NATURALIS  THEORIA 

aliquo  e  postremis,  vel  mediis.  Nam  ea  lineola  ex  quocunque  puncto  discedat,  ad  quod 
deventum  jam  sit,  habet  semper  eandem  &  longitudinem,  &  directionem,  cum  eidem  e 
componentibus  parallela  esse  debeat,  &  sequalis.  Quare  summa  ejusmodi  accessuum,  ac 
summa  recessuum  erit  eadem  in  fine  omnium  tempusculorum,  quocunque  ordine  dispon- 
antur  lineolae  hae  parallels,  &  sequales  lineolis  componentibus,  adeoque  etiam  id,  quod 
prodit  demendo  recessuum  summam  a  summa  accessuum,  vel  vice  versa,  erit  idem,  &  distantia 
puncti  postremi,  ad  quod  deventum  est  ab  illo  eodem  piano,  erit  eadem.  Inde  autem 
sponte  jam  fruit  id,  quod  demonstrandum  erat,  nimirum  punctum  illud  esse  idem  semper. 
Si  enim  ad  duo  puncta  duabus  diversis  viis  deveniretur,  assumpto  piano  perpendiculari  ad 
rectam,  quae  ilia  duo  puncta  jungeret,  distantia  perpendicularis  ab  ipso  non  esset  utique 
eadem  pro  utroque,  cum  altera  distantia  deberet  alterius  esse  pars. 


-  Porro  similis   admodum    est    etiam  methodus,   qua   utor   ad   demonstrandum 
manente  etiam  ubi  praeclarissimum  Newtoni  theorema,  in  quod  coalescunt  simul  duo,  quae  superius  innui,  & 
v^esntmutusenqac  ^uc  reducuntur.     Si  quotcunque  materice  puncta  utcunque  disposita,  6?  in  quotcunque  utcunque 
ejus     demonstra-  disjunctas  massas  coalescentia  habeant  velocitates  quascunque  cum  directionibus  quibuscunque, 
y  -prceterea  urgeantur  viribus  mutuis  quibuscunque,  quce  in  binis  quibusque  punctis  cequaliter 
agant  in  plagas  oppositas  ;    centrum  commune  gravitatis  omnium  vel  quiescet,  vel  movebitur 
uniformiter  in  directum  eodem  motu,  quern  haberet,  si  nulla  adesset  mutua  punctorum  actio  in 
se  invicem.     Hoc  autem  theorema  sic  generaliter,  &  admodum  facile,  ac  luculenter  demon- 
stratur.     [122]  Concipiamus    vires    singulas    per    quodvis    determinatum    tempusculum 
servare  directiones    suas,  &  magnitudines  :    in  fine  ejus    tempusculi    punctum  materiae 
quodvis  erit  in  eo  loci  puncto,  in  quo  esset,  si  singularum  virium  effectus,  vel  effectus 
velocitatis  ipsius  illi  tempusculo  debitus,  haberentur  cum  eadem  sua  directione,  &  magnitudine 
alii  post  alios  totidem  tempusculis,  quot  vires  agunt.     Assumantur  jam  totidem  tempuscula, 
quot  sunt  punctorum  binaria  diversa  in  ea  omni  congerie,  &  praeterea  unum,  ac  primo 
tempusculo  habeant  omnia  puncta  motus  debitos  velocitatibus  illis  suis,  quas  habent  initio 
ipsius,  singula  singulos  ;  turn  assignato  quovis  e  sequentibus  tempusculis  cuivis  binario, 
habeat  binarium  quodvis  tempusculo  sibi  respondente  motum  debitum  vi  mutuae,  quae 
agit  inter  bina  ejus  puncta,  ceteris  omnibus  quiescentibus.      In  fine  postremi  tempusculi 
omnia  puncta  materiae  erunt  in  hac  hypothesi  in  iis  punctis  loci,  in  quibus  revera  esse  debent 
in  fine  unici  primi  tempusculi  ex  actione  conjuncta  virium  omnium  cum  singulis  singulorum 
velocitatibus. 


-  Concipiatur  jam  ultra  omnia  ejusmodi  puncta  planum  quodcunque.  Primo 
ex  illis  tot  assumptis  tempusculis  alia  puncta  accedent,  alia  recedent  ab  eo  piano,  &  summa 
omnium  accessuum  punctorum  omnium  demptis  omnibus  recessibus,  si  qua  superest,  vel 
vice  versa  summa  recessuum  demptis  accessibus,  divisa  per  numerum  omnium  punctorum, 
aequabitur  accessui  perpendiculari  ad  idem  planum,  vel  recessui  centri  gravitatis  communis  ; 
cum  summa  distantiarum  perpendicularium  tarn  initio  tempusculi,  quam  in  fine,  divisa 
per  eundem  numerum  exhibeat  ipsius  communis  centri  gravitatis  distantiam  juxta  num. 
246.  Sequentibus  autem  tempusculis  manebit  utique  eadem  distantia  centri  gravitatis 
communis  ab  eodem  piano  nunquam  mutata  ;  quia  ob  sequales  &  contraries  punctorum 
motus,  alterius  accessus  ab  alterius  recessu  aequali  eliditur.  Quamobrem  in  fine  omnium 
tempusculorum  ejus  distantia  erit  eadem,  &  accessus  ad  planum  erit  idem,  qui  esset,  si 
solae  adfuissent  ejusmodi  velocitates,  quae  habebantur  initio  ;  adeoque  etiam  cum  omnes 
vires  simul  agunt,  in  fine  illius  unici  tempusculi  habebitur  distantia,  quas  haberetur,  si 
vires  illae  mutuae  non  egissent,  &  accessus  aequabitur  summae  accessuum,  qui  haberentur  ex 
solis  velocitatibus,  demptis  recessibus.  Si  jam  consideretur  secundum  tempusculum  in 
quo  simul  agant  vires  mutuse,  &  velocitates ;  debebunt  considerari  tria  genera  motuum  : 
primum  eorum,  qui  proveniunt  a  velocitatibus,  quae  habebantur  initio  primi  tempusculi ; 
secundum  eorum,  qui  proveniunt  a  velocitatibus  acquisitis  actione  virium  durante  per 
primum  tempusculum  ;  tertium  eorum,  qui  proveniunt  a  novis  actionibus  virium  mutuarum, 
quae  ob  mutatas  jam  positiones  concipiantur  aliis  directionibus  agere  per  totum  secundum 
tempusculum.  Porro  quoniam  hi  posteriorum  duorum  generum  motus  [123]  sunt  in 
singulis  punctorum  binariis  contrarii,  &  aequales ;  illi  itidem  distantiam  centri  gravitatis 
ab  eodem  piano,  &  accessum,  vel  recessum  debitum  secundo  tempusculo  non  mutant ; 


A  THEORY  OF  NATURAL  PHILOSOPHY  201 

the  commencement,  or  one  of  those  at  the  end,  or  one  in  the  middle.  For  the  short  line, 
whatever  point  it  has  for  its  beginning  &  whatever  point  it  finally  reaches,  must  always 
have  the  same  length  &  direction  ;  for  it  is  bound  to  be  parallel  &  equal  to  the  same  one 
of  the  components.  Hence  the  sum  of  these  approaches,  &  the  sum  of  these  recessions, 
will  be  the  same  at  the  end  of  the  whole  set  of  intervals  of  time,  no  matter  in  what  order  . 
these  little  lines,  which  are  parallel  &  equal  to  the  component  lines,  are  disposed.  Hence 
also,  the  result  obtained  by  taking  away  the  sum  of  the  recessions  from  the  sum  of  the 
approaches,  or  conversely,  will  be  the  same  ;  &  the  distance  of  the  ultimate  point  reached 
from  the  plane  will  be  the  same.  Thus  there  follows  immediately  what  was  required  to 
be  proved,  namely,  that  the  point  is  the  same  point  in  every  case.  For,  if  two  points  could 
be  reached  by  any  two  different  paths,  &  a  plane  is  taken  perpendicular  to  the  line  joining 
those  two  points,  then  it  is  impossible  for  the  perpendicular  distance  from  this  plane  to  be 
exactly  the  same  for  both  points,  since  the  one  distance  must  be  a  part  of  the  other. 

260.  Further,  the  method,  which  I  make  use  of  to  prove  a  most  elegant   theorem  Theorem     relating 
of   Newton,  is  exactly  similar  ;  in  it  the  two  noted  above  are  combined,  &  come  to  the  *?_J;h,?,  P?rman?*t 

,.  -  .  It-  I  1-  J        •  C  J  T 

same  thing.      //  any   number  of  points   of  matter,  disposed   in   any  manner,  t5   coalescing  of     gravity    even 

to  form  any  number  of  separate  masses  in  any  manner ,  have  any  velocities  in  any  direction;  JJ^utu offeree's 

^f  *'/,  in  addition,  the  points  are  under  the  influence  of  any  mutual  forces  whatever,  these  forces  acting ;    the   first 

acting  on  each  pair  of  points  equally  in  opposite  directions  ;  then  the  common  centre  of  gravity  steps  of  thc  Proof- 

of  the  whole  is  either  at  rest,  or  moves  uniformly  in  a  straight  line  with  the  same  motion  as  it 

would  have  if  there  were  no  mutual  action  of  the  points  upon  one  another.     Now  this  theorem 

is  quite  easily  &  clearly  proved  in  all  generality  as  follows.     Suppose  that  each  force  maintains 

its  direction  &  magnitude  during  any  given  short  intervals  of  time  ;  at  the  end  of  the  interval 

any  point  of  matter  will  occupy  that  point  of  space,  which  it  would  occupy  if  the  effects 

for  each  of  the  forces  (i.e.,  the  effect  of  each  velocity  corresponding  to  that  interval  of  time) 

were  obtained,  one  after  another,  in  as  many  intervals  of  time  as  there  are  forces  acting, 

whilst  each  maintains  its  own   direction  &   magnitude   the  same   as    before.     Now  take 

as    many  small  intervals    of   time  as  there  are  different   pairs  of   points  in  the  whole 

group,  &  one  interval  in  addition  ;   &  in  the  first  interval  of  time  let  all  the  points  have 

the  motions  due  to  the  velocities  that  they  have  at  the  beginning  of  the  interval  of  time 

respectively.     Then,  any  one  of  the  subsequent  intervals  of  time  being  assigned  to  any 

chosen  pair  of  points,  let  any  pair  have,  in  the  interval  of  time  proper  to  it,  that  motion 

which  is  due  to  the  mutual  force  that  acts  between  the  two  points  of  that  pair,  whilst  all 

the  others  remain  at  rest.     Then  at  the  end  of  the  last  of  these  intervals  of  time,  each 

point  of  matter  will  be,  according  to  this  hypothesis,  at  that  point  of  space  which  it  is 

bound  to  occupy  at  the  end  of  a  single  first  interval  of  time,  under  the  conjoint  action  of 

all  the  mutual  forces,  each  having  its  corresponding  velocity. 

261.  Now  imagine  a  plane  situated  beyond  all  points  of  this  kind.     Then,  in  the  first  Continuation      of 
place,  for  these  little  intervals  of  time  of  which  we  have  assumed  the  number  stated,  some  the  demonstratlon- 
of  the  points  will  approach  towards,  &  some  recede  from  the  plane  ;  &  the  sum  of  all  these 
approaches  less  the  sum  of  all  the  recessions,  if  the  former  is  the  greater,  &  conversely,  the 

sum  of  the  recessions  less  the  sum  of  the  approaches,  divided  by  the  number  of  all  the  points, 
will  be  equal  to  the  perpendicular  approach  of  the  common  centre  of  gravity  to  the  plane, 
or  the  recession  from  it.  For,  by  Art.  246,  the  sum  of  the  perpendicular  distances, 
both  at  the  beginning  &  at  the  end  of  the  interval  of  time  will  represent  the  distance  of 
the  common  centre  of  gravity  itself.  Further,  in  subsequent  intervals,  this  distance  of 
the  common  centre  of  gravity  from  the  plane  will  remain  in  every  case  quite  unchanged  ; 
because,  on  account  of  the  equal  &  opposite  motions  of  pairs  of  points,  the  approach  of  the 
one  will  be  cancelled  by  the  equal  recession  of  the  other.  Hence,  at  the  end  of  all  the  intervals 
the  distance  of  the  centre  of  gravity  will  be  the  same,  &  its  approach  towards  the  plane 
will  be  the  same,  as  it  would  have  been  if  there  had  existed  no  velocities  except  those 
which  it  had  at  the  beginning  of  the  interval ;  thus,  too,  when  all  the  forces  act  together, 
at  the  end  of  the  single  interval  of  time  there  will  be  obtained  that  distance,  which  would 
have  been  obtained  if  the  mutual  forces  had  not  been  acting  ;  &  the  approach  will  be  equal 
to  the  sum  of  the  approaches,  less  the  recessions,  acquired  from  the  velocities  alone.  If 
now  we  would  consider  a  second  interval  of  time,  in  which  we  have  acting  the  mutual  forces, 
&  the  velocities ;  we  shall  have  to  consider  three  kinds  of  motions.  Firstly,  those  that  come 
from  the  velocities  which  exist  at  the  beginning  of  the  interval ;  secondly,  those  which 
arise  from  the  velocities  acquired  through  the  action  of  forces  lasting  throughout  the  first 
interval ;  &  thirdly,  those  which  arise  from  the  new  actions  of  the  mutual  forces,  which  may 
be  assumed  to  be  acting  in  fresh  directions,  due  to  the  change  in  the  positions  of  the  points 
during  the  whole  of  this  second  interval.  Further,  since  the  latter  of  the  last  two  kinds, 
of  motion  are  equal  &  opposite  for  each  pair  of  points,  these  two  kinds  also  will  not  change 
the  distance  of  the  centre  of  gravity  from  the  plane  &  the  approach  towards  it  or  recession 


202  PHILOSOPHIC  NATURALIS  THEORIA 

sed  ea  habentur,  sicuti  haberentur,  si  semper  durarent  solae  illse  velocitates,  quae  habebantur 
initio  primi  tempusculi ;  &  idem  redit  argumentum  pro  tempusculo  quocunque  :  singulis 
advenientibus  tempusculis  accedet  novum  motuum  genus  durantibus  cum  sua  directione, 
&  magnitudine  velocitatibus  omnibus  inductis  per  singula  praecedentia  tempuscula,  ex 
quibus  omnibus,  &  ex  nova  actione  vis  mutuae,  componitur  quovis  tempusculo  motus 
puncti  cujusvis  :  sed  omnia  ista  inducunt  motus  contraries,  &  sequales,  adeoque  summa 
accessuum,  vel  recessuum  ortam  ab  illis  solis  initialibus  velocitatibus  non  mutant. 

Progressus  ulterior.  262.  Quod  si  jam  tempusculorum  magnitudo  minuatur  in  infinitum,  aucto  itidem 

in  infinitum  intra  quodvis  finitum  tempus  eorundem  numero,  donee  evadat  continuum 
tempus,  &  continua  positionum,  ac  virium  mutatio  ;  adhuc  centrum  gravitatis  in  fine 
continui  temporis  cujuscunque,  adeoque  &  in  fine  partium  quarumcunque  ejusdem 
temporis,  habebit  ab  eodem  piano  distantiam  perpendicularem,  quam  haberet  ex  solis 
velocitatibus  habitis  initio  ejus  temporis,  si  nullae  deinde  egissent  mutuae  vires  ;  &  accessus 
ad  illud  planum,  vel  recessus  ab  eo,  aequabitur  summae  omnium  accessuum  pertinentium 
ad  omnia  puncta  demptis  omnibus  recessibus,  vel  vice  versa.  Is  vero  accessus,  vel  recessus 
assumptis  binis  ejus  temporis  partibus  quibuscunque,  erit  proportionalis  ipsis  temporibus. 
Nam  singulorum  punctorum  accessus,  vel  recessus  orti  ab  illis  velocitatibus  initialibus 
perseverantibus,  adeoque  ab  motu  aequabili,  sunt  in  ratione  eadem  earundem  temporis 
partium  ;  ac  proinde  &  eorum  summae  in  eadem  ratione  sunt. 

Demonstration  is  2&3'  In.de  vero  prona  jam  est  theorematis  demonstratio.     Ponamus  enim,  centrum 

finis-  gravitatis  quiescere  quodam  tempore,  turn  moveri  per  aliquod  aliud  tempus.     Debebit 

utique  aliquo  momento  temporis  esse  in  alio  loci  puncto,  diverse  ab  eo,  in  quo  erat  initio 
motus.  Sumatur  pro  prima  e  duabus  partibus  temporis  continui  pars  ejus  temporis,  quo 
punctum  quiescebat,  &  pro  secunda  tempus  ab  initio  motus  usque  ad  quodvis  momentum, 
quo  centrum  illud  gravitatis  devenit  ad  aliud  aliquod  punctum  loci.  Ducta  recta  ab 
initio  ad  finem  hujusce  motus,  turn  accepto  piano  aliquo  perpendiculari  ipsi  productae 
ultra  omnia  puncta,  centrum  gravitatis  ad  id  planum  accederet  secunda  continui  ejus 
temporis  parte  per  intervallum  aequale  illi  rectae,  &  nihil  accessisset  primo  tempore,  adeoque 
accessus  non  fuissent  proportionales  illis  partibus  continui  temporis.  Quamobrem  ipsum 
commune  gravitatis  centrum  vel  semper  quiescit,  vel  movetur  semper.  Si  autem  movetur, 
debet  moveri  in  directum.  Si  enim  omnia  puncta  loci,  per  quje  transit,  non  jacent  in 
directum,  sumantur  tria  in  dire-[i24]-ctum  non  jacentia,  &  ducatur  recta  per  prima  duo, 
quas  per  tertium  non  transibit,  adeoque  per  ipsam  duci  poterit  planum,  quod  non  transeat 
per  tertium,  turn  ultra  omnem  punctorum  congeriem  planum  ipsi  parallelum.  Ad  id 
secundum  nihil  accessisset  illo  tempore,  quo  a  primo  loci  puncto  devenisset  ad  secundum, 
&  eo  tempore,  quo  ivisset  a  secundo  ad  tertium,  accessisset  per  intervallum  sequale  distantiae 
a  priore  piano,  adeoque  accessus  iterum  proportionales  temporibus  non  fuissent.  Demum 
motus  erit  aequabilis.  Si  enim  ultra  omnia  puncta  concipiatur  planum  perpendiculare 
rectae,  per  quam  movetur  ipsum  centrum  commune  gravitatis,  jacens  ad  earn  partem,  in 
quam  id  progreditur,  accessus  ad  ipsum  planum  erit  totus  integer  motus  ejusdem  centri ; 
adeoque  cum  ii  accessus  debeant  esse  proportionales  temporibus ;  erunt  ipsis  temporibus 
proportionales  motus  integri ;  &  idcirco  non  tantum  rectilineus,  sed  &  uniformis  erit  motus ; 
unde  jam  evidentissime  patet  theorema  totum. 


Coraiiarium    de  264.  Ex  eodem  fonte,  ex  quo  profluxit  hoc  generale  theorema,  sponte  fluit  hoc  aliud 

quantitate      motus  .  .'  r.       „,        ,  ,r 

in  eandem  piagam  ut  consectanum  :    quantitas  motus  in  Mundo  conservatur  semper  eadem,  si  ea  computetur 
conservata    in  secunaum  directionem  quacunque  ita,  ut  motus  secundum  directionem  oppositam  consider etur 

Mundo.  , .  •*  .  7 .  c  •        • 

ut  negativus,  e-jusmodi  motuum  contranorum  summa  subtracta  a  summa  directorum.  01  enim 
consideretur  eidem  direction!  perpendiculare  planum  ultra  omnia  materiae  puncta, 
quantitas  motus  in  ea  directione  est  summa  omnium  accessuum,  demptis  omnibus  recessibus, 
quae  summa  tempusculis  aequalibus  manet  eadem,  cum  mutuae  vires  inducant  accessus, 
&  recessus  se  mutuo  destruentes ;  nee  ejusmodi  conservation!  obsunt  liberi  motus  ab  anima 
nostra  producti,  cum  nee  ipsa  vires  ullas  possit  exerere,  nisi  quae  agant  in  partes  oppositas 
aequaliter  juxta  num.  74. 

^Equaiitas  actionis  265.  Porro  ex  illo  Newtoniano  theoremate  statim  jam  profluit  lex  actionis,  &  reactionis 

&     reactionis    in    ggqualium  pro  massis  omnibus.     Nimirum  si  duae  massae  quaecunque  in  se  invicem  agant 

massis  inde  orta.  .\.  X  .       „     .  .         ,  i_«        •*  vi_  i  •          -11 

viribus  quibuscunque  mutuis,  &  inter  smgula  punctorum  binana  aequalibus ;    bmas  illae 


A  THEORY  OF  NATURAL  PHILOSOPHY  203 

from  it  corresponding  to  the  second  interval.  Hence,  these  will  be  the  same  as  they  would 
have  been,  if  those  velocities  that  existed  at  the  beginning  of  the  first  interval  had  persisted 
throughout ;  &  the  same  argument  applies  to  any  interval  whatever.  Each  interval  as  it 
occurs  will  yield  a  fresh  kind  of  motions,  all  the  velocities  induced  during  each  of  the  preceding 
intervals  remaining  the  same  in  direction  &  magnitude  ;  &  from  all  of  these,  &  the  fresh 
action  of  the  mutual  force,  there  is  compounded  for  any  interval  the  motion  of  any  point. 
But  all  the  latter  induce  equal  &  opposite  motions  in  pairs  of  points ;  &  thus  the  sum  of 
the  approaches  or  recessions  arising  from  the  velocities  alone  are  unchanged  by  the  mutual 
forces. 

262.  Now  if  the  length  of  the  interval  of  time  is  indefinitely  diminished,  the  number  Further   steps    in 
of  intervals  in  any  given  finite  time  being  thus  indefinitely  increased,  until  we  acquire  the 
continuous  time,  &  continuous  change  of  position  &  forces ;   still  the  centre  of  gravity  at 

the  end  of  any  continuous  time,  &  thus  also  at  the  end  of  any  parts  of  that  time,  will  have 
that  perpendicular  distance  from  the  plane,  which  it  would  have  had,  due  to  the  velocities 
that  existed  at  the  beginning  of  the  time,  if  no  mutual  forces  had  been  acting.  The  approach 
towards  the  plane,  or  the  recession  from  it,  will  be  equal  to  the  sum  of  all  the  approaches 
corresponding  to  all  the  points  less  the  sum  of  all  the  recessions,  or  vice  versa.  Indeed, 
any  two  parts  of  the  time  being  taken,  this  approach  or  recession  will  be  proportional  to  these 
parts  of  the  time.  For  the  approach  or  recession,  for  each  of  the  points,  arising  from  the 
velocities  that  persist  throughout  &  thus  also  from  uniform  motion,  is  proportional  for  all 
parts  of  the  time  ;  &  hence  also,  their  sums  are  proportional. 

263.  The  complete  proof  now  follows  immediately  from  what  has  been  said  above.  Conclusion  of  the 
For,  let  us  suppose  that  the  centre  of  gravity  is  at  rest  for  a  certain  time,  &  then  moves  demonstration, 
for  some  other  time.     Then  at  some  instant  of  time  it  is  bound  to  be  at  some  other 

point  of  space  different  from  that  in  which  it  was  at  the  beginning  of  the  motion.  Of 
two  parts  of  continuous  time,  let  us  take  as  the  first  part  of  the  time,  that  in  which  the 
point  is  at  rest ;  &  for  the  second  part,  the  time  between  the  beginning  of  the  motion  & 
the  instant  when  the  centre  of  gravity  reaches  some  other  point  of  space.  Draw  a  straight 
line  from  the  beginning  to  the  end  of  this  motion,  &  take  any  plane  perpendicular  to  this 
line  produced  beyond  all  the  points  ;  then  the  centre  of  gravity  would  approach  towards 
the  plane,  in  the  second  part  of  the  continuous  time,  through  an  interval  equal  to  the 
straight  line,  but  in  the  first  part  of  the  time  there  would  have  been  no  approach  at  all ; 
hence  the  approaches  would  not  have  been  proportional  to  those  parts  of  the  continuous 
time.  Hence  the  centre  of  gravity  is  always  at  rest,  or  is  always  in  motion.  Further, 
if  it  is  in  motion,  it  must  move  in  a  straight  line.  For,  if  all  points  of  space,  through 
which  it  passes,  do  not  lie  in  a  straight  line,  take  three  of  them  which  are  not  collinear  ; 
&  draw  a  straight  line  through  the  first  two,  which  does  not  pass  through  the  third  ;  then 
it  will  be  possible  to  draw  through  this  straight  line  a  plane  which  will  not  pass  through 
the  third  point ;  &  consequently,  a  plane  parallel  to  it  beyond  the  whole  group  of  points. 
To  this  second  plane  there  will  be  no  approach  at  all  for  the  time,  during  which  the 
centre  of  gravity  would  travel  from  the  first  point  of  space  to  the  second  ;  &  for  that 
time,  during  which  it  would  go  from  the  second  point  to  the  third,  there  would  be  an 
approach  through  an  interval  equal  to  its  distance  from  the  first  plane  ;  &  thus,  once 
again,  the  approaches  would  not  be  proportional  to  the  times.  Lastly,  the  motion  will 
be  uniform.  For,  if  we  imagine  a  plane  drawn  beyond  all  the  points,  perpendicular  to 
the  straight  line  along  which  the  centre  of  gravity  moves,  &  on  that  side  to  which  there 
is  approach,  then  the  approach  to  that  plane  will  be  the  whole  of  the  entire  motion  of 
the  centre  ;  hence,  since  these  approaches  must  be  proportional  to  the  times,  the  whole 
motions  must  be  proportional  to  the  times ;  &  therefore  the  motion  must  not  only  be 
rectilinear,  but  also  uniform.  Thus,  the  whole  theorem  is  now  perfectly  clear. 

264.  From  the  same  source  as  that  from  which  we  have  drawn  the  above  general  theorem,  Corollary    with 
there  is  obtained  immediately  the  following  also,  as  a  corollary.     The  quantity  of  motion  conservation*  of *«ie 
in  the  Universe  is  maintained  always  the  same,  so  long  as  it  is  computed  in  some  given  direction  quantity  of  motion 
in  such  a  way  that  motion  in  the  opposite  direction  is  considered  negative,  £5"  the  sum  of  the  ^given^rection  "* 
contrary  motions  is  subtracted  from  the  sum  of  the  direct  motions.     For,  if  we  consider  a  plane 
perpendicular  to  this  direction  lying  beyond  all  points  of  matter,  the  quantity  of  motion 

in  this  direction  is  the  sum  of  all  the  approaches  with  the  sum  of  the  recessions  subtracted  ; 
this  sum  remains  the  same  for  equal  times,  since  the  mutual  forces  induce  approaches  & 
recessions  that  cancel  one  another.  Nor  is  such  conservation  affected  by  free  motions 
that  are  the  result  of  our  will ;  since  it  cannot  exert  any  forces  either,  except  such  as  act 
equally  in  opposite  directions,  as  was  proved  in  Art.  74. 

265.  Further,  from  the  Newtonian  theorem,  we  have  immediately  the  law  of  equal  Equality  of  action 
action  &  reaction  for  all  masses.     Thus,  if  any  two  masses  act  upon  one  another  with  any  massesa  the°  result 
mutual  forces,  which  are  also  equal  for  each  pair  of  points,  the  two  masses  will  acquire,  of  this  theorem. 


204 


PHILOSOPHIC  NATURALIS  THEORIA 


massae  acquirent  ab  actionibus  mutuis  summas  motuum  aequales  in  partes  contrarias,  & 
celeritates  acquisitae  ab  earum  centris  gravitatis  in  partes  oppositas,  componendae  cum 
antecedentibus  ipsarum  celeritatibus,  erunt  in  ratione  reciproca  massarum.  Nam  centrum 
commune  gravitatis  omnium  a  mutuis  actionibus  nihil  turbabitur  per  hoc  theorema,  & 
sive  ejusmodi  vires  agant,  sive  non  agant,  sed  solius  inertiae  effectus  habeantur  ;  semper 
ab  eodem  communi  gravitatis  centro  distabunt  ea  bina  gravitatis  centra  hinc,  &  inde  in 
directum  ad  distantias  reciproce  proportionales  massis  ipsis  per  num.  253.  Quare  si  praeter 
priores  motus  ex  vi  inertiae  uniformes,  ob  actionem  mutuam  adhuc  magis  ad  hoc  commune 
centrum  accedet  alterum  ex  iis,  vel  ab  eo  recedet  ;  accedet  &  alterum,  [125]  vel  recedet, 
accessibus,  vel  recessibus  reciproce  proportionalibus  ipsis  massis.  Nam  accessus  ipsi,  vel 
recessus,  sunt  differentiae  distantiarum  habitarum  cum  actione  mutuarum  virium  a  distantiis 
habendis  sine  iis,  adeoque  erunt  &  ipsi  in  ratione  reciproca  massarum,  in  qua  sunt  totae 
distantiae.  Quod  si  per  centrum  commune  gravitatis  concipiatur  planum  quodcumque, 
cui  quaepiam  data  directio  non  sit  parallela  ;  summa  accessuum,  vel  recessuum  punctorum 
omnium  massae  utriuslibet  ad  ipsum  secundum  earn  directionem  demptis  oppositis,  quae 
est  summa  motuum  secundum  directionem  eandem,  aequabitur  accessui,  vel  recessui  centri 
gravitatis  ejus  massae  ducto  in  punctorum  numerum  ;  accessus  vero,  vel  recessus  alterius 
centri  ad  accessum,  vel  recessum  alterius  in  directione  eadem,  erit  ut  secundus  numerus 
ad  primum  ;  nam  accessus,  &  recessus  in  quavis  directione  data  sunt  inter  se,  ut  accessus, 
vel  recessus  in  quavis  alia  itidem  data  ;  &  accessus,  ac  recessus  in  directione,  quae  jungit 
centra  massarum,  sunt  in  ratione  reciproca  ipsarum  massarum.  Quare  productum  accessus, 
vel  recessus  centri  primae  massae  per  numerum  punctorum,  quae  habentur  in  ipsa,  aequatur 
producto  accessus,  vel  recessus  secundae  per  numerum  punctorum,  quae  in  ipsa  continentur  ; 
nimirum  ipsae  motuum  summae  in  ilia  directione  computatorum  aequales  sunt  inter  se, 
in  quo  ipsa  actionis,  &  reactionis  aequalitas  est  sita. 


molhbus. 


inde   leges  coliisi-  266.  Ex  hac  actionum,  &  reactionum  aequalitate  sponte  profluunt  leges  collisionis 

^L  :  i/^^P  corporum,  quas  ex  hoc  ipso  principio  Wrennus  olim,  Hugenius,  &  Wallisius  invenerunt 

m    corpon-  •  *   *  •  i  XT  i       XT  *  T*  •       •    • 

bus  eiasticis,  &  simul,  ut  in  hac  ipsa  lege  Naturae  exponenda  Newtonus  etiam  memorat  Pnncipiorum 
jjj^  T  _  Ostendam  autem,  quo  pacto  generales  formulae  inde  deducantur  tarn  pro  directis 
collisionibus  corporum  mollium,  quam  pro  perfecte,  vel  pro  imperfecte  elasticorum. 
Corpora  mollia  dicuntur  ea,  quae  resistunt  mutationi  figurae,  seu  compressioni,  sed  compressa 
nullam  exercent  vim  ad  figuram  recuperandam,  ut  est  cera,  vel  sebum  :  corpora  elastica, 
quae  figuram  amissam  recuperare  nituntur  ;  &  si  vis  ad  recuperandam  sit  aequalis  vi  ad 
non  amittendam  ;  dicuntur  perfecte  elastica,  quae  quidem,  ut  &  perfecte  mollia,  nulla, 
ut  arbitror,  sunt  in  Natura  ;  si  autem  imperfecte  elastica  sunt,  vis,  quae  in  amittenda, 
ad  vim,  quae  in  recuperanda  figura  exercetur,  datam  aliquam  rationem  habet.  Addi  solet 
&  tertium  corporum  genus,  quae  dura  dicunt,  quae  nimirum  figuram  prorsus  non  mutent  ; 
sed  ea  itidem  in  Natura  nusquam  sunt  juxta  communem  sententiam,  &  multo  magis  nulla 
usquam  in  hac  mea  Theoria.  Adhuc  qui  ipsa  velit  agnoscere,  is  mollia  consideret,  quae 
minus,  ac  minus  comprimantur,  donee  compressio  evadat  nulla  ;  &  ita  quae  de  mollibus 
dicentur,  aptari  poterunt  duris  multo  meliore  jure,  quam  alii  elasticorum  leges  ad  ipsa 
transferant,  considerando  elasticitatem  infinitam  ita,  ut  figura  nee  mutetur,  nee  se  restituat  ; 
[126]  nam  si  figura  non  mutetur,  adhuc  concipi  poterit,  impenetrabilitatis  vi  amissus 
motus,  ut  amitteretur  in  compressione  ;  sed  ad  supplendam  vim,  quae  exeritur  ab  eiasticis 
in  recuperanda  figura,  non  est,  quod  concipi  possit,  ubi  figura  recuperari  non  debet.  Porro 
unde  corpora  mollia  sint,  vel  elastica  hie  non  quaero  ;  id  pertinet  ad  tertiam  partem, 
quanquam  id  ipsum  innui  superius  num.  199  ;  sed  leges  quae  in  eorum  collisionibus  observari 
debent,  &  ex  superiore  theoremate  fluunt,  expono.  Ut  autem  simplicior  evadat  res, 
considerabo  globes,  atque  hos  ipsos  circumquaque  circa  centrum,  in  eadem  saltern  ab 
ipso  centro  distantia,  homogeneos,  qui  primo  quidem  concurrant  directe  ;  nam  deinde 
ad  obliquas  etiam  collisiones  faciemus  gradum. 


Praeparatio  pro  col-          267.  Porro  ubi  globus  in  globum  agit,  &  ambo  paribus  a  centro  distantiis  homogenei 
pianorum,8l0^Trc™-'  sunt,  facile  constat,  vim  mutuam,  quse  est  summa  omnium  virium,  qua  singula  alterius 
puncta  agunt  in  singula  puncta  alterius,  habituram  semper  directionem,  quae  jungit  centra  ; 


lorum- 


A  THEORY  OF  NATURAL  PHILOSOPHY  205 

as  a  result  of  the  mutual  actions,  sums  of  motions  that  are  equal  in  opposite  directions ; 
&  the  velocities  acquired  by  their  centres  of  gravity  in  opposite  directions,  being  compounded 
of  the  foregoing  velocities,  will  be  in  the  inverse  ratio  of  the  masses.  For,  by  the  theorem, 
the  common  centre  of  gravity  of  the  whole  will  not  be  disturbed  in  the  slightest  degree 
by  the  mutual  actions,  whether  such  forces  act  or  whether  they  do  not,  but  only  the  effects 
of  inertia  will  be  obtained ;  hence  the  two  centres  of  gravity  will  always  be  distant  from 
this  common  centre  of  gravity,  one  on  each  side  of  it,  in  a  straight  line  with  it,  at  distances 
that  are  reciprocally  proportional  to  the  masses,  as  was  proved  in  Art.  253.  Hence,  if  in 
addition  to  the  former  uniform  motions  due  to  the  force  of  inertia,  one  of  the  two  masses, 
on  account  of  the  mutual  action,  should  approach  still  nearer  to  the  common  centre,  or 
recede  still  further  from  it ;  then  the  other  will  either  approach  towards  it  or  recede  from 
it,  the  approaches  or  recessions  being  reciprocally  proportional  to  the  masses.  For  these 
approaches  or  recessions  are  the  differences  between  the  distances  that  are  obtained  when 
there  is  action  of  mutual  forces  &  the  distances  when  there  is  not ;  &  thus,  they  too  will 
be  in  the  inverse  ratio  of  the  masses,  such  as  the  whole  distances  are.  But  if  we  imagine 
a  plane  drawn  through  the  common  centre  of  gravity,  &  that  some  given  direction  is  not 
parallel  to  it,  then  the  sum  of  the  approaches  or  recessions  of  all  the  points  of  either  of  the 
masses  with  respect  to  this  plane,  the  opposites  being  subtracted  (which  is  the  same  thing 
as  the  sum  of  the  motions  in  this  direction)  will  be  equal  to  the  approach  or  the  recession 
of  the  centre  of  gravity  of  that  mass  multiplied  by  the  number  of  points  in  it.  But  the 
approach  or  recessions  of  the  centre  of  the  one  is  to  the  approach  or  recession  of  the  centre 
of  the  other,  in  the  same  direction,  as  the  second  number  is  to  the  first ;  for  the  approaches 
or  recessions  in  any  given  direction  are  to  one  another  as  the  approaches  or  recessions  in  any 
other  given  direction  ;  &  the  approaches  or  recessions  along  the  line  joining  the  two  masses 
are  inversely  proportional  to  the  masses.  Therefore  the  product  of  the  approach  or  recession 
of  the  centre  of  the  first  mass,  multiplied  by  the  number  of  points  in  it,  is  equal  to  the 
approach  or  recession  of  the  centre  of  the  second  mass,  multiplied  by  the  number  of  points 
that  are  contained  in  it.  Thus  the  sums  of  the  motions  in  the  direction  under  consideration 
are  equal  to  one  another ;  &  in  this  is  involved  the  equality  of  action  &  reaction. 

266.  From  this  equality  of  action  &  reaction  there  immediately  follow  the  laws  for  Hence  the  laws  for 
collision  of  bodies,  which  some  time  ago  Wren,  Huygens  &  Wallis  derived  from  this  very  Jetton  '  be tween 
principle  at  about  the  same  time,  as  Newton  also  mentioned  in  the  first  book  of  the  Principia,  ^e  forc?s  for  elas- 
when  expounding  this  law  of   Nature.     Now  I  will  show  how  general  formulas  may  be  bodies"*16 
derived  from  it,  both  for  the  direct  collision  of  soft  bodies,  &  also  for  perfectly  or  imperfectly 

elastic  bodies.  By  soft  bodies  are  to  be  understood  those,  which  resist  deformation  of 
their  shapes,  or  compression  ;  but  which,  when  compressed,  exert  no  force  tending  to 
restore  shape  ;  such  as  wax  or  tallow.  Elastic  bodies  are  those  that  endeavour  to  recover 
the  shape  they  have  lost ;  &  if  the  force  tending  to  restore  shape  is  equal  to  that  tending 
to  prevent  loss  of  shape,  the  bodies  are  termed  perfectly  elastic  ;  &,  just  as  there  are  no 
perfectly  soft  bodies,  there  are  none  that  are  perfectly  elastic,  according  to  my  thinking, 
in  Nature.  Lastly,  they  are  imperfectly  elastic,  if  the  force  exerted  against  losing  shape 
bears  to  the  force  exerted  to  restore  it  some  given  ratio.  It  is  usual  to  add  a  third  class  of 
bodies,  namely,  such  as  are  called  hard  ;  &  these  never  alter  their  shape  at  all ;  but  these 
also,  even  according  to  general  opinion,  never  occur  in  Nature ;  still  less  can  they 
exist  in  my  Theory.  Yet,  if  anyone  wishes  to  take  account  of  such  bodies,  they  could 
consider  them  as  soft  bodies  which  are  compressed  less  &  less,  until  the  compression  finally 
becomes  evanescent ;  in  this  way,  whatever  is  said  about  soft  bodies  could  be  adapted  to 
hard  bodies  with  far  more  justification  than  there  is  for  applying  some  of  the  laws  of 
elastic  bodies  to  them,  by  considering  that  there  is  infinite  elasticity  of  such  a  nature  that 
the  figure  neither  suffers  change  nor  seeks  to  restore  itself.  For,  if  the  figure  remains 
unchanged,  it  is  yet  possible  to  consider  the  motion  lost  due  to  the  force  of  impenetrability, 
&  that  thus  it  would  be  lost  in  compression  ;  but  to  supply  the  force  which  in  elastic  bodies 
is  exerted  for  the  recovery  of  shape,  there  is  nothing  that  can  be  imagined,  when  there 
is  necessarily  no  recovery  of  shape.  Further,  what  are  the  causes  of  soft  or  elastic  bodies, 
I  do  not  investigate  at  present ;  this  relates  to  the  third  part,  although  I  have  indeed  mentioned 
it  above,  in  Art.  199.  But  I  set  forth  the  laws  which  have  to  be  observed  in  collisions 
between  them,  these  laws  coming  out  immediately  from  the  theorem  given  above.  Moreover 
to  make  the  matter  easier,  I  consider  spheres,  &  these  too  homogeneous  round  about  the 
centre,  at  any  rate  for  the  same  distance  from  that  centre  ;  &  these  indeed  will  in  the  first 
place  collide  directly ;  for  from  direct  collision  we  can  proceed  to  oblique  impact  also. 

267.  Now,  where  one  sphere  acts  upon  another,  &  both  of  them  are  homogeneous  ^^era'tlo^  "'f 
at  equal  distances  from  their  centres,  it  is  readily  shown  that  the  mutual  force,  which  is  collisions  of  spheres, 
the  sum  of  all  the  forces  with  which  each  of  the  points  of  the  one  acts  on-  each  of  the  points  Planes  &  cirdes- 
of  the  other,  must  always  be  in  the  direction  of  the  line  joining  the  two  centres.     For, 


206  PHILOSOPHISE  NATURALIS  THEORIA 

nam  in  ea  recta  jacent  centra  ipsorum  globorum,  quse  in  eo  homogeneitatis  casu  facile 
constat,  esse  centra  itidem  gravitatis  globorum  ipsorum  ;  &  in  eadem  jacet  centrum  com- 
mune gravitatis  utriusque,  ad  quod  viribus  illis  mutuis,  quas  alter  globis  exercet  in  alterum, 
debent  ad  se  invicem  accedere,  vel  a  se  invicem  recedere  ;  unde  fit,  ut  motus,  quos 
acquirunt  globorum  centra  ex  actione  mutua  alterius  in  alterum,  debeant  esse  in  directione, 
quae  jungit  centra.  Id  autem  generaliter  extendi  potest  etiam  ad  casum,  in  quo  concipiatur, 
massam  immensam  terminatam  superficie  plana,  sive  quoddam  immensum  planum  agere 
in  globum  finitum,  vel  in  punctum  unicum,  ac  vice  versa  :  nam  alterius  globi  radio  in 
infinitum  aucto  superficies  in  planum  desinit  ;  &  radio  alterius  in  infinitum  imminuto, 
globus  abit  in  punctum.  Quin  etiam  si  massa  quaevis  teres,  sive  circa  axem  quendam 
rotunda,  &  in  quovis  piano  perpendicular!  axi  homogenea,  vel  etiam  circulus  simplex, 
agat,  vel  concipiatur  agens  in  globum,  vel  punctum  in  ipso  axe  constitutum  ;  res  eodem 
redit. 

Formulae  pro  cor-  268.  Praecurrat  jam  globus  mollis  cum  velocitate  minore,  quem  alius  itidem  mollis 

pore   moih    incur-  consequatur  cum  maiore  ita,  ut  centra  ferantur  in  eadem  recta,  quae  ilia  coniungit,  &  hie 

rente    in   molle     .  *     .  .1,  ,.   .  ,,..,.  T     .  .,  .          .J,      °  r 

lentius  progrediens  demum  mcurrat  in  ilium,  quae  dicitur  colhsio  directa.  Is  incursus  mini  quidem  non  net 
m  eandem  piagam.  per  immediatum  contactum,  sed  antequam  ad  contactum  deveniant,  vi  mutua  repulsiva 
comprimentur  partes  posteriores  praecedentis,  &  anteriores  sequentis,  qua;  compressio  net 
semper  major,  donee  ad  aequales  celeritates  devenerint  ;  turn  enim  accessus  ulterior  desinet, 
adeoque  &  ulterior  compressio  ;  &  quoniam  corpora  sunt  mollia,  nullam  aliam  exercent 
vim  mutuam  post  ejusmodi  compressionem,  sed  cum  aequali  ilia  velocitate  pergunt  moveri 
porro.  Haec  aequalitas  velocitatis,  ad  quam  reducuntur  ii  duo  glo-[i27]-bi,  una  cum 
asqualiate  actionis,  &  reactionis  aequalium,  rem  totam  perficient.  Sit  enim  massa,  sive 
quantitas  materiae,  globi  prascurrentis  =  q,  insequentis  =  Q  ;  celeritas  illius  =  c,  hujus 
=  C  :  quantitas  motus  illius  ante  collisionem  erit  cq,  hujus  CQ  ;  nam  celeritas  ducta  per 
numerum  punctorum  exhibet  summam  motuum  punctorum  omnium,  sive  quantitatem 
motus  ;  unde  etiam  fit,  ut  quantitas  motus  per  massam  divisa  exhibeat  celeritatem.  Ob 
actionem,  &  reactionem  aequales,  haec  quantitas  erit  eadem  etiam  post  collisionem,  post 
quam  motus  totus  utriusque  massae,  erit  CQ  +  cq.  Quoniam  autem  progrediuntur  cum 
aequali  celeritate  ;  celeritas  ilia  habebitur  ;  si  quantitas  motus  dividatur  per  totam 

quantitatem  materias  ;    quae    idcirco  erit    -Q—  —  .     Nimirum  ad  habendam  velocitatem 

communem  post  collisionem,  oportebit  ducere  singulas  massas  in  suas  celeritates,  & 
productorum  summam  dividere  per  summam  massarum. 

Ejus   extensio   ad  269.  Si  alter  globus  'q  quiescat  ;    satis  erit  illius   celeritatem  c  considerare  =  o  :    &  si 

rTufs  "amissa  'vlei  moveatur  rnotu  contrario  motui  prioris  globi  ;    satis  erit  illi  valorem  negativum  tribuere  ; 

acquisita.  ut  adeo  &  hie,  &  in  sequentibus  formula  inventa  pro  illo  primo  casu  globorum  in  eandem 

progredientium  piagam,  omnes  casus  contineat.     In  eo  autem  si  libeat  invenire  celeritatem 

amissam    a    globo    Q,  &  celeritatem  acquisitam    a    globo    q,  satis    erit    reducere  singulas 

formulas 

c      CQ+cg   kCQ+c9_ 
"  " 


ad  eundem  denominatorem,  ac  habebitur 

Cg  —eg,      CQ—  cQ 

Q   +  q          '     Q   +  q  ' 

ex  quibus  deducitur  hujusmodi  theorema  :  ut  summa  massarum  ad  massam  alteram,  ita 
differentia  celeritatum  ad  celeritatem  ab  altera  acquisitam,  quae  in  eo  casu  accelerabit  motum 
praecurrentis  &  retardabit  motum  consequentis. 

Transitus  ad  eias-  270.  Ex  hisce,  quae  pertinent  ad  corpora  mollia,  facile  est  progredi  ad  perfecte  elastica. 

ies'  In  iis  post  compressionem  maximam,  &  mutationem  figurse  inductam  ab  ipsa,  quae  habetur, 
ubi  ad  aequales  velocitates  est  ventum,  agent  adhuc  in  se  invicem  bini  globi,  donee  deveniant 
ad  figuram  priorem,  &  haec  actio  duplicabit  effectum  priorem.  Ubi  ad  sphaericam  figuram 
deventum  fuerit,  quod  fit  recessu  mutuo  oppositarum  superficierum,  quae  in  compressione 
ad  se  invicem  accesserant,  pergent  utique  a  se  invicem  recedere  aliquanto  magis  eaedem 
superficies,  &  figura  producetur,  sed  opposita  jam  vi  mutua  inter  partes  ejusdem  globi 
incipient  retrahi,  &  productio  perget  fieri,  sed  usque  lentius,  donee  ad  maximam  quandam 
productionem  de-[i28]-ventum  fuerit,  quae  deinde  incipiet  minui,  &  globus  ad  sphaericam 
accedet  iterum,  ac  iterum  comprimetur  quodam  oscillatorio,  ac  partium  trepidatione 
hinc,  &  inde  a  figura  sphasrica,  ut  supra  vidimus  etiam  duo  puncta  circa  distantiam  limitis 


A  THEORY  OF  NATURAL  PHILOSOPHY  207 

in  that  straight  line  lie  the  centres  of  the  two  spheres ;  &  these  in  the  case  of  homogeneity 
are  easily  shown  to  be  also  the  centres  of  gravity  of  the  spheres.  Also  in  this  straight  line 
lies  the  common  centre  of  gravity  of  both  spheres ;  &  to,  or  from,  it  the  spheres  must  approach 
or  recede  mutually,  owing  to  the  action  of  the  mutual  forces  with  which  one  sphere  acts 
upon  the  other.  Hence  it  follows  that  the  motion,  which  the  centres  of  the  spheres  acquire 
through  the  mutual  action  of  one  upon  the  other,  is  bound  to  be  along  the  line  which  joins  the 
centres.  The  argument  can  also  be  extended  generally,  even  to  include  the  case  in  which 
it  is  supposed  that  an  immense  mass  bounded  by  a  plane  surface,  or  an  immense  plane 
acts  upon  a  finite  sphere,  or  on  a  single  point,  or  vice  versa  ;  for,  if  the  radius  of  either  of 
the  spheres  is  increased  indefinitely,  the  surface  ultimately  becomes  a  plane,  &  if  the  radius  of 
either  becomes  indefinitely  diminished,  the  sphere  degenerates  into  a  point.  Moreover,  if  any 
round  mass,  or  one  contained  by  a  surface  of  rotation  round  an  axis  and  homogeneous  in  any 
plane  perpendicular  to  that  axis,  or  even  a  simple  circle,  act,  or  is  supposed  to  act  upon  a 
sphere  or  point  situated  in  the  axis ;  it  comes  to  the  same  thing. 

268.  Now  suppose  that  a  soft  body  proceeds  with  a  less  velocity  than  another  soft  F°™uiae  for  a  soft 

i  •   i     •     f   1 1        •         •         •  i      •          •  i  i  i     •  body      impinging 

body  which  is  following  it  with  a  greater  velocity,  in  such  a  manner  that  their  centres  are  upon  another  soft 

travelling  in  the  same  straight  line,  namely  that  which  joins  them  ;   &  finally  let  the  latter  more^iow^in^thf 

impinge  upon  the  former  ;    this  is  termed  direct  impact.     This  impact,  in  my  opinion  same  direction. 

indeed,  does  not  come  about  by  immediate  contact,  but,  before  they  attain  actual  contact, 

the  hinder  parts  of  the  first  body  &  the  foremost  parts  of  the  second  body  are  compressed 

by  a  mutually  repulsive  force  ;   &  this  compression  becomes  greater  &  greater  until  finally 

the  velocities  become  equal.     Then  further  approach  ceases,  &  therefore  also  further 

compression  ;    &,  since  the  bodies  are  soft,  they  exercise  no  further  mutual  force  after 

such  compression,  but  continue  to  move  forward  with  that  equal  velocity.     This  equality 

in  the  velocity,  to  which  the  two  spheres  are  reduced,  together  with  the  equality  of  action 

&  reaction,  finishes  off  the  whole  matter.     For,  supposing  that  the  mass  or  quantity  of 

matter  of  the  foremost  sphere  is  equal  to  q,  that  of  the  latter  to  Q  ;   the  velocity  of  the 

former  equal  to  c,  &  that  of  the  latter  to  C.     Then  the  quantity  of  motion  of  the  former 

before  impact  is  cq,  &  that  of  the  latter  is  CQ  ;  for  the  velocity  multiplied  by  the  number 

of  points  represents  the  sum  of  the  motions  of  all  the  points,  i.e.,  the  quantity  of  motion, 

&  in  the  same  way  the  quantity  of  motion  divided  by  the  mass  gives  the  velocity.     Now, 

since  the  action  &  reaction  are  equal  to  one  another,  this  quantity  will  be  the  same  even 

after  impact ;    hence  after  impact  the  whole  motion  of  both  the  masses  together  will  be 

equal  to  CQ  +  cq.     Further,  since  they  are  travelling  with  a  common  velocity,  this  velocity 

will  be  the  result  obtained  on  dividing  the  quantity  of  motion  by  the  whole  quantity  of 

matter  ;    &  it  will  therefore  be  equal  to  (CQ  +  ^?)/(Q  +  ?)•     That  is  to  say,  to  obtain 

the  common  velocity  after  impact,  we  must  multiply  each  mass  by  its  velocity,  &  divide 

the  sum  of  these  products  by  the  sum  of  the  masses. 

269.  If  one  of  the  two  spheres  is  at  rest,  all  that  need  be  done  is  to  put  its  velocity  c  Extension    to   ail 
equal  to  zero  ;   also,  if  it  is  moving  in  a  direction  opposite  to  that  of  the  first  sphere,  we  ordained!0"*7  k 
need  only  take  the  value  of  c  as  negative.     Thus,  both  here  &  subsequently,  the  formula 

found  for  the  first  case,  in  which  the  spheres  are  moving  forward  in  the  same  direction, 
includes  all  cases.  Again,  if  in  this  case,  we  wish  to  find  the  velocity  lost  by  the  sphere 
Q,  &  the  velocity  gained  by  the  sphere  q,  we  need  only  reduce  the  two  formulae 
C  —  (CQ  +  f?)/(Q  +  ?)  &  (CQ  +  cq)/(Q  +  q)  —  f  to  a  common  denominator,  when  we 
shall  obtain  the  formulae  (Cq—  cq)/(Q  +  q)  &  (CQ  —  cQ)/(Q  +  q).  From  these  there 
can  be  derived  the  theorem  : — The  sum  of  the  masses  is  to  either  of  the  masses  as  the  difference 
between  the  velocities  is  to  the  velocity  acquired  by  the  other  mass  ;  in  the  present  case  there 
will  be  an  increase  of  velocity  for  the  foremost  body  &  a  decrease  for  the  hindmost. 

270.  From  these  theorems  relating  to  soft  bodies  we  can  easily  proceed  to  those  that  Transition  to  im- 
are  perfectly  elastic.     For  such  bodies,  after  the  maximum  compression  has  taken  place,  tic-0  bodies^6 

&  the  alteration  in  shape  consequent  on  this  compression,  which  is  attained  when  equality 
of  the  velocities  is  reached,  the  two  spheres  still  continue  to  act  upon  one  another,  until 
the  original  shape  is  attained  ;  &  this  action  will  duplicate  the  effect  of  the  first  action.  When 
the  spherical  shape  is  once  more  attained,  as  this  takes  place  through  a  mutual  recession 
of  the  opposite  surfaces  of  the  spheres,  which  during  compression  had  approached  one 
another,  these  same  surfaces  in  each  sphere  will  continue  to  recede  from  one  another  still 
somewhat  further,  &  the  shape  will  be  elongated  ;  but  the  mutual  force  between  the  parts  of 
each  sphere  is  now  changed  in  direction  &  the  surfaces  begin  to  be  drawn  together  again. 
Hence  elongation  will  continue,  but  more  slowly,  until  a  certain  maximum  elongation  is 
attained  ;  this  then  begins  to  be  diminished  &  the  sphere  once  more  returns  to  a  spherical 
shape,  once  more  is  compressed  with  a  sort  of  oscillatory  motion  &  forward  &  backward 
vibration  of  its  parts  about  the  spherical  shape  ;  exactly  as  was  seen  above  in  the  case  of 
two  points  oscillating  to  &  fro  about  a  distance  equal  to  that  corresponding  to  a  limit-point 


208  PHILOSOPHIC  NATURALIS  THEORIA 

cohassionis  oscillare  hinc,  &  inde  ;  sed  id  ad  collisionem,  &  motus  centrorum  gravitatis 
nihil  pertinebit,  quorum  status  a  viribus  mutuis  nihil  turbatur  ;  actio  autem  unius  globi 
in  alterum  statim  cessabit  post  regressum  ad  figuram  sphaericam,  post  quern  superficies 
alterius  postica  &  alterius  antica  in  centra  jam  retracts  ulteriore  centrorum  discessu  a 
se  invicem  incipient  ita  distare,  ut  vires  in  se  invicem  non  exerant,  quarum  effectus  sentiri 
possit  ;  &  hypothesis  perfecte  elasticorum  est,  ut  tantus  sit  mutuae  actionis  effectus  in 
recuperanda,  quantus  fuit  in  amittenda  figura. 

-    Duplicate   igitur    effectu,    globus    ammittet    celeritatem  *Cq~  2~J  ,  &   globus  q 


acquiret  celeritatem  *  Q~2fQ-     Quare  illius  celeritas  post  collisionem  erit  C  -  -  2C? 
slve    CQi«f.    hujus?vero 


eandem  plagam,  vel  globus  alter  quiescet,  vel  fient  in  plagas  oppositas  ;    prout  determinatis 
valoribus  Q,q,  C,c,  formulae  valor  evaserit  positivus,  nullus,  vel  negativus. 
Formula?    pro  im-  272.  Quod  si  elasticitas  fuerit  imperfecta,  &  vis  in  amittenda  ad  vim  in  recuperanda 

perfecte  elasticis.       c  t        •     •        i-  •  j  •  rr  •      •        i      rr  •     •  •  •> 

ngura  fuerit  in  aliqua  ratione  data,  ent  &  effectus  pnons  ad  effectum  postenons  itidem  in 
ratione  data,  nimirum  in  ratione  subduplicata  prioris.  Nam  ubi  per  idem  spatium  agunt 
vires,  &  velocitas  oritur,  vel  extinguitur  tota,  ut  hie  respectiva  velocitas  extinguitur  in 
compressione,  oritur  in  restitutione  figurae,  quadrata  velocitatum  sunt  ut  areae,  quas 
describunt  ordinatae  viribus  proportionales  juxta  num.  176,  &  hinc  areae  erunt  in  ratione 
virium,  si,  viribus  constantibus,  sint  constantes  &  ordinatae,  cum  inde  fiat,  ut  scalae  celeri- 
tatum  ab  iis  descriptae  sint  rectangula.  Sit  igitur  rationis  constantis  illarum  virium  ratio 
subduplicata  m  ad  «,  &  erit  effectus  in  amittenda  figura  ad  summam  effectuum  in  tota 

collisione,  ut  m  ad  m  -\-  n,  quae  ratio  si  ponantur  esse  i  ad  r,  ut  sit  r  =  -     -  satis  erit, 

effectus  illos  inventos  pro  globis  mollibus,  sive  celeritatem  ab  altero  amissam,  ab  altero 
acquisitam,  non  duplicare,  ut  in  perfecte  elasticis,  sed  multiplicare  per  r,  ut  habeantur 
velocitates  acquisitae  in  partes  contrarias,  &  componendse  cum  velocitatibus  [129]  prioribus. 

Erit  nimirum  ilia  quae  pertinet  ad  globum  Q  =  —  ^—  ^,  &  quae  pertinet  ad  globum 
q,  erit  =  —  j^  —i  adeoque  velocitas  illius  post  congressum  erit  C  —  '—ft-  —•>  & 

hujus    c  -\-  —  ;    quae    formulae   itidem  reducuntur    ad    eosdem    denominatores  ; 

ac  turn  ex  hisce  formulis,  turn  e  superioribus  quam  plurima  elegantissima  theoremata 
deducuntur,  quae  quidem  passim  inveniuntur  in  elementaribus  libris,  &  ego  ipse  aliquanto 
uberius  persecutus  sum  in  Supplements  Stayanis  ad  lib.  2,  §  2  ;  sed  hie  satis  est,  fundamenta 
ipsa,  &  primarias  formulas  derivasse  ex  eadem  Theoria,  &  ex  proprietatibus  centri  gravitatis, 
ac  motuum  oppositorum  sequalium,  deductis  ex  Theoria  eadem  ;  nee  nisi  binos,  vel  ternos 
evolvam  casus  usui  futures  infra,  antequam  ad  obliquam  collisionem,  ac  reflexionem  motuum 
gradum  faciam. 

Casus,   in  quo  27'?.  Si  elobus  perfecte  elasticus  incurrat  in  globum  itidem  quiescentem,  erit,  c  =  o, 

globus     perfecte  2r    _  ' 

elasticus  mcurrit  in  adeoque    velocitas    contrana   priori    pertmens  ad    incurrentem,    quae    erat  —£•  -  ",  erit 

alium.  Q  +  q 

zCq  ,     .  .  .  .  2CQ  —  2<rQ  .          2CQ 

^r—  -  ;      velocitas     acquisita     a  quiescente,     quae    erat  —  ,     erit     ~—  -  ;     unde 

U  +  y  Q  +  q  U  +  y 

habebitur  hoc  theorema  :  ut  summa  massarum  ad  duflam  massam  quiescentis,  vel  incurrentis, 
ita  celeritas  incurrentis,  ad  celeritatem  amissam  a  secundo,  vel  acquisitam  a  primo  ;  &  si 
massae  aequales  fuerint,  fit  ea  ratio  aequalitatis  ;  ac  proinde  globus  incurrens  totam  suam 
velocitatem  amittit,  acquirendo  nimirum  aequalem  contrariam,  a  qua  ea  elidatur,  &  globus 
quiescens  acquirit  velocitatem,  quam  ante  habuerat  globus  incurrens. 

Casus  triplex  globi  274.  Si  globus  imperfecte  elasticus  incurrat  in  globum  quiescentem  immensum,  & 

numI^nimob\ie.pla"  °lui  habeatur  pro  absolute  infinite,  cujus  idcirco  superficies  habetur  pro  plana,  in  formula 

velocitatis  acquisitae  a  globo  quiescente  ~?r-  —  ,  cum  evanescat  Q  respectu  q  absolute 

infiniti,  &  proinde  ^  -  evadat  =  o,  tota  formula  evanescit,  adeoque  ipse  haberi  potest 
pro  piano  immobili.  In  formula  vero  velocitatis,  quam  in  partem  oppositam  acquiret 
globus  incurrens,  -—-^  —,  evadit  f=  o,  [130]  &  Q  evanescit  itidem  respectu  q. 

Hinc  habetur  ^',  sive  rC,  nimirum  ob  r  =  ^?  fit  C^±-"\X  C,  cujus  prima  pars  —  x  C, 


A  THEORY  OF  NATURAL  PHILOSOPHY  209 

of  cohesion.  However,  this  has  nothing  to  do  with  the  impact  or  the  motion  of  the  centres 
of  gravity,  nor  are  their  states  affected  in  the  slightest  by  the  mutual  forces.  Again,  the 
action  of  one  sphere  on  the  other  will  cease  directly  after  return  to  the  spherical  shape  ; 
for  after  that  the  hindmost  surface  of  the  one  &  the  foremost  surface  of  the  other,  being 
already  withdrawn  in  the  direction  of  their  centres,  will  through  a  further  recession  of 
the  centres  from  one  another  begin  to  be  so  far  distant  from  one  another  that  they  will 
not  exert  upon  one  another  any  forces  of  which  the  effects  are  appreciable.  We  are  left 
with  the  hypothesis,  for  perfectly  elastic  solids,  that  the  effect  of  their  action  on  one  another 
is  exactly  the  same  in  amount  during  alteration  of  shape  &  recovery  of  it. 

271.  Hence,  the  effect  being  duplicated,  the  sphere  Q  will  lose  a  velocity  equal  to  Formula  for    per- 
(iCq  -  2cq)/(Q  +  ?),  &  the  sphere  q  will  gain  a  velocity  equal  to  (zCQ  —  2Cy)/(Q  +  q).  fect'yelastic **"«•• 
Hence,   the  velocity  of  the  former  after  impact  will  be  C  —  (zCq  —  2tq)/(Q  +  <?)  or 

(CQ  —  Cq  +  2cq)/(Q  +  ?),  &  the  velocity  of  the  latter  will  be  c  +  (zCQ  —  2«rQ)/(Q  +  q) 
or  (cq  —  cQ  +  2CQ)/(Q  -j-  q).  The  motions  will  be  in  the  original  direction,  or  one  of  the 
spheres  may  come  to  rest,  or  the  motions  maybe  in  opposite  directions,  according  as  formula, 
given  by  the  values  of  Q,  q,  C,  &  c,  turns  out  to  be  positive,  zero,  or  negative. 

272.  But  if  the  elasticity  were  imperfect,  &  the  force  during  loss  of  shape  were  in  Formula;  for  impei  - 
some  given  ratio  to  the  force  during  recovery  of  shape,  then  the  effect  corresponding  to  fectly  elastic  bodies. 
the  former  would  also  be  in  a  given  ratio  to  the  effect  due  to  the  latter,  namely,  in  the 
subduplicate  ratio  of  the  first  ratio.     For,  when  forces  act  through  the  same  interval  of 

space,  &  velocity  is  generated,  or  is  entirely  destroyed,  as  here  the  relative  velocity  is 
destroyed  during  compression  &  generated  during  recovery  of  shape,  the  squares  of  the 
velocities  are  proportional  to  the  areas  described  by  the  ordinates  representing  the  forces, 
as  was  proved  in  Art.  176.  Hence  these  areas  are  proportional  to  the  forces,  if,  the  forces 
being  constant,  the  ordinates  also  are  constant ;  for  from  that  it  is  easily  seen  that  the 
measures  of  the  velocities  described  by  them  are  rectangles.  Suppose  then  that  the 
subduplicate  ratio  of  the  constant  ratio  of  the  forces  be  m  :  n ;  then  the  ratio  of  the  effect 
during  loss  of  shape  to  the  sum  of  the  effects  during  the  whole  of  the  impact  will  be  m  :  m  +  n. 
If  we  call  this  ratio  I  :  r,  so  that  r  =  (m  -f-  ri)/m,  we  need  only,  instead  of  doubling  the  effects 
found  for  soft  bodies,  or  the  velocity  lost  by  one  sphere  or  gained  by  the  other,  multiply 
these  effects  by  r,  in  order  to  obtain  the  velocities  acquired  in  opposite  directions,  which 
are  to  be  compounded  with  the  original  velocities.  Thus,  that  for  the  sphere  Q  will  be 
equal  to  (rCq  —  rcg)/(Q  -f-  q),  &  that  for  the  sphere  q  will  be  (rCQ  —  r<rQ)/(Q  +  q). 
Hence,  the  velocity  of  the  former  after  impact  will  be  C  —  (rCq  —  rcq)/(Q  -f-  q)  &  the 
velocity  of  the  latter  will  be  c  +  (VCQ  —  rcQ)/(Q  +  q)  ;  &  these  formulje  also  can  be 
reduced  to  common  denominators.  From  these  formula,  as  well  as  from  those  proved 
above,  a  large  number  of  very  elegant  theorems  can  be  derived,  such  as  are  to  be  found 
indeed  everywhere  in  elementary  books.  I  myself  have  followed  the  matter  up  somewhat 
more  profusely  in  the  Supplements  to  Stay's  Philosophy,  in  Book  II,  §  2.  But  here  it  is 
sufficient  that  I  should  have  derived  the  fundamentals  themselves,  together  with  the  primary 
formulae,  from  one  &  the  same  Theory,  &  from  the  properties  of  the  centre  of  gravity  & 
of  equal  &  opposite  motions,  which  are  also  derived  from  the  same  theory.  Except  that 
I  will  consider  below  two  or  three  cases  that  will  come  in  useful  in  later  work,  before  I  pass 
on  to  oblique  impact  &  reflected  motions. 

273.  If  a  perfectly  elastic  sphere  strikes  another,  &  the  second  sphere  is  at  rest,  then  Case  of  a  perfectly 
c  =  o,  &  the  velocity,  in  the  direction  opposite  to  the  original  velocity,  for  the  striking  f^mrfhe™  St"k" 
body,  which  was  (zCq  —  2cq)/(Q  +  q),  will  in  this  case  be  2Cq/(Q  -f-  q)  ;  whilst  the  velocity 

gained  by  the  body  that  was  at  rest,  which  was  shown  to  be  (2CQ  —  2cQ)/(Q  +  q),  will 
be  2CQ/(Q  -|-  q).  Hence  we  have  the  following  theorem.  As  the  sum  of  the  masses  is 
to  twice  the  mass  of  the  body  at  rest,  or  to  the  body  that  impinges  upon  it,  so  is  the  velocity  of 
the  impinging  body  to  the  velocity  lost  by  the  second  body,  or  to  that  gained  by  the  first.  If 
the  masses  were  equal  to  one  another,  this  ratio  would  be  one  of  equality  ;  hence  in  this 
case  the  impinging  body  loses  the  whole  of  its  velocity,  that  is  to  say  it  acquires  an  equal 
opposite  velocity  which  cancels  the  original  velocity  ;  &  the  sphere  at  rest  acquires  a  velocity 
equal  to  that  which  the  impinging  sphere  had  at  first. 

274.  If  an  imperfectly  elastic  sphere  impinges  on  an  immense  sphere  at  rest,  which  Threefold   case  of 
may  be  considered  as  absolutely  infinite,  &  therefore  its  surface  may  be  taken  to  be  a  plane  ;  onPanre immovable 
then,  in  the  formula  for  the  velocity  gained  by  a  sphere  at  rest,  (rCQ  —  rcQ)/(Q  +  q),  plane. 

since  Q  vanishes  in  comparison  with  q  which  is  absolutely  infinite,  &  thus  Q/(Q  +  q)  =  o, 
the  whole  formula  vanishes,  &  therefore  the  immense  sphere  can  be  taken  to  be  an  immovable 
plane.  Now,  in  the  formula  for  the  velocity  which  the  impinging  sphere  acquires  in  the 
opposite  direction  to  its  original  motion,  namely,  (rCq  —  rcq)/(Q  -j-  q),  we  have  c  =  o, 
&  Q  also  vanishes  in  comparison  with  q.  Hence  we  obtain  rCq/q,  or  rC ;  that  is  to  say, 
since  r  =  (m  -f  ri)/m,  we  have  C  X  (m  +  n)/m,  of  which  the  first  part,  C  X  m/m,  or  C, 


210 


PHILOSOPHIC  NATURALIS  THEORIA 


Summa  quadra- 
torum  velocitatis 
ductorum  in  massas 
manens  in  perfecte 
elasticis. 


sive  C,  est  ilia,  quse  amittitur,  sive  acquiritur  in  partem  oppositam  in  comprimenda  figura, 
&  —  X  C  est  ilia,  quse  acquiritur  in  recuperanda,  ubi  si  fit  n  =  o,  quod  accidit  nimirum 
in  perfecte  mollibus ;  habetur  sola  pars  prima  ;  si  m  =  n,  quod  accidit  in  perfecte  elasticis, 
est  -  -  x  C  =  C,  secunda  pars  aequalis  primae  ;  &  in  reliquis  casibus  est,  ut  m  ad  n,  ita 

ilia  pars  prima  C,  sive  praecedens  velocitas,  quae  per  primam  partem  acquisitam  eliditur, 
ad  partem  secundam,  quae  remanet  in  plagam  oppositam.  Quamobrem  habetur  ejusmodi 
theorema.  Si  incurrat  ad  perpendiculum  in  planum  immobile  globus  perfecte  mollis,  acquirit 
velocitatem  contrariam  cequalem  suce  priori,  &  quiescat ;  si  perfecte  elasticus,  acquirit  duplam 
suce,  nimirum  cequalem  in  compressione,  qua  motus  omnis  sistitur,  &  cequalem  in  recuperanda 
figura,  cum  qua  resilit  ;  si  fuerit  imperfecte  elasticus  in  ratione  m  ad  n,  in  ilia  eadem  ratione 
erit  velocitas  priori  suce  contraria  acquisita,  dum  figura  mutatur,  quce  priorem  ipsam  velocitatem 
extinguit,  ad  velocitatem,  quam  acquirit,  dum  figura  restituitur,  W  cum  qua  resilit. 

275.  Est  &  aliud  theorema  aliquanto  operosius,  sed  generale,  &  elegans,  ab  Hugenio 
inventum  pro  perfecte  elasticis,  quod  nimirum  summa  quadratorum  velocitatis  ductorum 
in  massas  post  congressum  remaneat  eadem,  quae  fuerat  ante  ipsum.  Nam  velocitates 

C  -  ^-  X   (C  -  c),     &     c  +  ,,**-  X  (C-f) ;     quadrata    ducta 


post    congressum    sunt 


in  massas  continent  singula  ternos  terminos  :    primi  erunt  QCC+  qcc  ;    secundi  erunt 


(-CC+  CO  X 


cc)    X 


4Q? 


Q+  q 

postremi 
Q  +  q 

2Cc  +  cc),  sive  simul 


&(cC-cc)    X 
4Q?? 


erunt 


X 


quorum  summa  evadit  (—  CC 

(CC    -    2CC    +    CC), 


(Q  +  ?) 

[131]  X  (CC  -  2Cc  +  cc),  vel  _ 


X 


2Cc— 

(CC- 


(Q+f) 

(Q  +  g)«      *----  "  *~  '  ""  '  -  QT~,  ^C 

quod    destruit   summam   secundi   terminorum   binarii,  remanente   sola   ilia 


QCC  +  qcc, 
Sed  haec  aequalitas  nee 


Collisionis  obliquae 
communis  metho- 
du  s  per  virium 
resolutionem. 


Compositio  virium 
resolu  t  i  o  n  i  s  u  b- 
stituta. 


summa  quadratorum  velocitatum  prsecedentium  ducta  in  massas. 
habetur  in  mollibus,  nee  in  imperfecte  elasticis. 

276.  Veniendo  jam  ad  congressus  obliques,  deveniant  dato  tempore  bini  globi  A,  C 
in  fig.  42  per  rectas  quascunque  AB,  CD,  quae  illorum  velocitates  metiantur,  m  B,  &  D  ad 
physicum  contactum,  in  quo  jam  sensibi- 

lem  effectum  edunt  vires  mutuae.      Com- 

muni  methodo  collisionis  effectus  sic  de- 

finitur.     Junctis  eorum  centris  per  rectam 

BD,  ducantur,  ad   earn  productam,  qua 

opus   est,  perpendicula  AF,  CH,  &  com- 

pletis  rectangulis  AFBE,  CHDG  resolvan- 

tur  singuli  motus  AB,  CD  in  binos ;   ille 

quidem  in  AF,  AE,  sive  EB,  FB,  hie  vero 

in  CH,  CG,  sive  GD,  HD.     Primus  utro- 

bique  manet  illaesus ;  secundus  FB,  &  HD 

collisionem   facit  directam.      Inveniantur 

per  legem  collisionis  directas  velocitates  BI, 

DK,  quse  juxta  ejusmodi  leges  superius 

expositas  haberentur  post  collisionem  di- 

versae  pro    diversis    corporum    speciebus, 

&  componantur  cum  velocitatibus  expositis 

per  rectas  BL,  DQ  jacentes  in  directum  cum  EB,  GD,  &  illis   aequales.      His  peractis 

expriment  BM,  DP  celeritates,  ac  directiones  motuum  post  collisionem. 

277.  Hoc  pacto  consideratur  resolutio  motuum,  ut  vera  quaedam  resolutio  in  duos, 
quorum  alter  illaesus  perseveret,  alter  mutationem  patiatur,  ac  in  casu,  quern  figura  exprimit, 
extinguatur  penitus,  turn  iterum  alius  producatur.     At  sine    ulla  vera    resolutione    res 
vere  accidit  hoc  pacto.     Mutua  vis,  quse  agit  in  globos  B,  D,  dat  illis  toto  collisionis  tempore 
velocitates  contrarias  BN,  DS  aequales  in  casu,  quern  figura  exprimit,  binis  illis,  quarum 
altera  vulgo  concipitur  ut  elisa,  altera  ut  renascens.     Ese  compositae  cum  BO,  DR  jacentibus 
in  directum  cum  AB,  CD,  &  aequalibus  iis  ipsis,  adeoque  exprimentibus  effectus  integros 
praecedentium  velocitatum,  exhibent  illas  ipsas  velocitates  BM,  DP.     Facile  enim  patet, 
fore  LO  aequalem  AE,  sive  FB,  adeoque  MO  aequalem  NB,  &  BNMO  fore  parallelogram- 
mum  ;     ac   eadem   demonstratione   est   itidem   parallelogrammum   DRPS.     Quamobrem 
nulla  ibi  est  vera  resolutio,  sed  sola  compositio  motuum,  perseverante  nimirum  velocitate 
priore  per  vim  inertiae,  &  ea  composita  cum   nova   velocitate,  quam  generant  vires,  quse 
agunt  in  collisione. 


FIG.  42. 


A  THEORY  OF  NATURAL  PHILOSOPHY  211 

is  the  part  that  is  lost,  or  acquired  in  the  opposite  direction  to  the  original  velocity,  during 
the  compression,  &  C  X  n/m  is  the  part  that  is  acquired  during  the  recovery  of  shape. 
In  this,  if  n  =  o,  which  is  the  case  for  perfectly  soft  bodies,  there  is  only  the  first  part  ; 
if  m  —  n,  which  is  the  case  for  perfectly  elastic  bodies,  then  C  X  n/m  will  be  equal  to  C,  and 
the  second  part  is  equal  to  the  first  part  ;  &  in  all  other  cases  as  m  is  to  n,  so  is  the  first  part 
C,  or  the  original  velocity,  which  is  cancelled  by  the  first  part  of  the  acquired  velocity, 
to  the  second  part,  which  is  the  final  velocity  in  the  opposite  direction.  Hence  we  have 
the  following  theorem.  //  a  perfectly  soft  sphere  impinges  perpendicularly  upon  an  immovable 
plane,  it  will  acquire  a  velocity  equal  &  opposite  to  its  original  velocity,  &  will  be  brought 
to  rest.  If  the  body  is  perfectly  elastic,  it  will  acquire  a  velocity  double  of  its  original  velocity 
but  in  the  opposite  direction,  that  is  to  say,  an  equal  velocity  during  compression,  by  which  the 
whole  of  the  motion  ceases,  &  an  equal  velocity  during  recovery  of  shape,  with  which  it  rebounds. 
If  it  were  imperfectly  elastic,  the  ratio  being  equal  to  that  of  m  to  n,  the  velocity  acquired  in 
the  opposite  direction  to  its  original  velocity  whilst  the  shape  is  being  changed,  by  which  the 
original  velocity  is  cancelled,  will  bear  this  same  ratio  to  the  velocity  acquired  whilst  the  shape 
is  being  restored,  that  is,  the  velocity  with  which  it  rebounds. 

275.  There  is  also  another  theorem,  which  is  rather  more  laborious,  but  it  is  a  general  The  sum  of  the 
&  elegant  theorem,  discovered  by  Huygens  for  perfectly  elastic  solids.  Namely,  that  the  Velocities.  °  each 
sum  of  the  squares  of  the  velocities,  each  multiplied  by  the  corresponding  mass,  remains  multiplied  by  the 
the  same  after  the  impact  as  it  was  before  it.  Now,  the  velocities  after  impact  are  remains'11  unaltered 

C  —      2^      X  (C  —  c),  &  c  +  £~     X  (C  —  f)  ;    the  squares  of  these,  multiplied  by  the  perfectly  elastic 

^i  +  q  {J.  ~T-  q  bodies. 


masses    contain    three    terms    each  ;     the    first    are    QCC    &    qcc  :     the    second    are 


(  —  CC  +  Cf)    X  &    (fC  -  cc)  X          ->     &    the    sum    of    these    reduces    to 


(-  CC  +  aCr  -  «)  X  :    the  last  are  (CC  -  2Q  +  «),  &  -  X 

(CC-2Cc  +  w)»      or      added      together    4(Q  +  $  *  Q?    X    (CC-2Cc+cc),      or 
X  (CC  —  2Cc  4-  cc),  which  will  cancel  the  sum  of  the  second  terms  ;    hence  all 


that  remains  is  QCC  +  qcc,  the  sum  of  the  squares  of  the  original  velocities,  each  multiplied 
by  the  corresponding  mass.  This  equality  does  not  hold  good  for  soft  bodies,  nor  yet  for 
imperfectly  elastic  bodies. 

276.  Coming  now  to  oblique  impacts,  suppose  that,  in  Fig.  42,  the  two  spheres  A  &  The  usual  method 
C  at  some  given  time,  moving  along  any  straight  lines  AB,  CD,  which  measure  their  velocities, 

come  into  physical  contact  in  the  positions  B  &  D,  where  the  mutual  forces  now  produce  lution  of  forces. 

a  sensible  effect.     In  the  usual  method  the  effect  of  the  impact  is  usually  determined  as 

follows.     Join  their  centres  by  the  line  BD,  &  to  this  line,  produced  if  necessary,  draw 

the  perpendiculars  AF,  CH,  &  complete  the  rectangles  AFBE,  CHDG  ;    resolve  each  of 

the  motions  AB,  CD  in  two,  the  former  into  AF,  AE,  or  EB,  FB,  &  the  latter  into  CH, 

CG,  or  GD,  HD.     In  either  pair,  the  first  remains  unaltered  ;    the  second,  FB,  &  HD, 

give  the  effect  of  direct  impact.     The  direct  velocities  BI,  DK  are  found  by  the  law  of 

impact  ;  &  these,  according  to  laws  of  the  kind  set  forth  above,  will  after  impact  be  different 

for  different  kinds  of  bodies.     They  are  compounded  with  velocities  represented  by  the 

straight  lines  BL,  DQ,  which  are  in  the  same  straight  lines  as  EB,  GD  respectively,  &  equal 

to  them.     This  being  done,  BM,  DP  will  represent  the  velocities  &  the  directions  of  motion 

after  collision. 

277.  In  this  method,  there  is  considered  to  be  a  resolution  of  motions,  as  if  there  were  Composition  of 
a  certain  real  resolution  into  two  parts,  of  which  the  one  part  persisted  unchanged,  &  the  *orces    substituted 

•T        j      i  „  r.  r  1  •  i    •  A  ?  ,       for  resolution. 

other  part  suffered  alteration  ;  &  m  the  case,  for  which  the  figure  has  been  drawn,  the 
latter  is  altogether  destroyed  &  a  fresh  motion  is  again  produced.  But  the  matter  really 
proceeds  without  any  real  resolution  in  the  following  manner.  The  mutual  force  acting 
upon  the  spheres  B,  D,  gives  to  them  during  the  complete  time  of  impact  opposite  velocities 
BN,  DS,  which  are  also  equal,  in  the  case  for  which  the  figure  is  drawn,  to  those  two,  of 
which  the  one  is  considered  to  be  destroyed  &  the  other  to  be  produced.  These  motions, 
compounded  with  BO,  DR,  drawn  in  the  directions  of  AB,  CD  &  equal  to  them,  &  thus 
representing  the  whole  effects  of  the  original  velocities,  will  represent  the  velocities  BM, 
DP.  For  it  is  easily  seen  that  LO  is  equal  to  AE,  or  FB  ;  &  thus  MO  is  equal  to  NB,  & 
BNMO  will  be  a  parallelogram  ;  in  the  same  manner  it  can  be  shown  that  DRPS  is  a 
parallelogram.  Therefore,  there  is  in  reality  no  true  resolution,  but  only  a  composition 
of  motions,  the  original  velocity  persisting  throughout  on  account  of  the  force  of  inertia  ;  & 
this  is  compounded  with  the  new  velocity  generated  by  the  forces  which  act  during  the  impact. 


212 


PHILOSOPHIC  NATURALIS  THEORIA 


M 


tiTiU°Sltsubstft°uta  278-  Jdem  etiam  mihi^accidit,  ubi  oblique  globus  incurrit  in  planum,  sive  consideretur 

etiam   ubi  globus  motus,  qui  haberi  debet  deinde,  sive  percussionis  obliquae  energia  respectu  perpendicularis 
immobile.11 1         '  Deveniat  in  fig.  43  globus  A  cum  directione  obliqua  AB  ad  planum  [132]  CD  consideratum 

ut  immobile,  quod  contingat  physice  in  N,&  concipiatur  planum  GI  parallelum  priori  ductum 

per  centrum  B,  ad  quod  appellet  ipsum  centrum,  &  a  quo  resiliet,  si  resilit.     Ducta   AF 

perpendiculari  ad  GI,  &  completo  par- 

allelogrammo     AFBE,     in     communi 

methodo  resolvitur  velocitas  AB  in  duas 

AF,  AE  ;  sive  FB,  EB,  primam  dicunt 

manere    illaesam,   secundam   destrui  a 

resistentia  plani :  turn  perseverare  illam 

solam  per  BI  aequalem  ipsi  FB  ;  si  corpus 

incurrens  sit  perfecte  molle,  vel  componi 

cum  alia  in  perfecte  elasticis  BE  aequali 

priori  EB,   in  imperfecte  elasticis  Be, 

quae  ad  priorem  EB  habeat  rationem 

datam,  &  percurrere  in  primo  casu  BI, 

in  secundo  BM,  in  tertio  EOT.     At  in 

mea  Theoria   globus  a  viribus  in  ilia 

minima  distantia  agentibus,  quae  ibi  sunt 

repulsivae,    acquirit    secundum    direc- 

tionem  NE  perpendicularem  piano  re- 

pellenti  CD  in  primo  casu  velocitatem 

BE,  aequalem  illi,  quam  acquireret,  si 

cum  velocitate  EB  perpendiculariter  advenisset  per  EB,  in  secundo  BL  ejus  duplam,  in 

tertio  BP,  quae  ad  ipsam  habeat  illam  rationem  datam  r  ad  I,  sive  m  -f-  n  ad  m,  &  habet 

deinde  velocitatem  compositam  ex  velocitate  priore  manente,  ac  expressa  per  BO  aequalem 

AB,  &  positam  ipsi  in  directum,  ac  ex  altera  BE,  BL,  BP,  ex  quibus  constat,  componi  illas 

ipsas  BI,  BM,  Bwz,  quas  prius ;  cum  ob  IO  aequalem  AF,  sive  EB,  &  IM,  Im  aequales  BE, 

BI?,  sive  EL,  EP,  totae  etiam  BE,  BP,  BL  totis  OI,  OM,  Om  sint  aequales,  &  parallelae. 
ubique  in  hac          279.  Res  mihi  per  compositionem  virium  ubique  eodem  redit,  quo  in  communi 
tionerrf  resolution!  methodo  per  earum  resolutionem.     Resolutionem  solent   vulgo   admittere  in   motibus, 
substitui,     easque  quos  vocant  impeditos,  ubi  vel  planum  subiectum,  vel  ripa  ad  latus  procursum  impediens, 

sibi  mvicem    aequi-       ,.   •      n      •  i      •  1    ci  •  j    i  -n     •       -i_ 

vaiere.  ut  m  nuviorum  alveis,  vel  filum,  aut  virga  sustentans,  ut  in  pendulorum  oscillatiombus, 

impedit  motum  secundum  earn  directionem,  qua  agunt  velocitates  jam  conceptas,  vel 
vires ;  ut  &  virium  resolutionem  agnoscunt,  ubi  binae,  vel  plures  etiam  vires  unius  cujusdam 
vis  alia  directione  agentis  effectum  impediunt,  ut  ubi  grave  a  binis  obliquis  planis  sustinetur, 
quorum  utrumque  premit  directione  ipsi  piano  perpendiculari,  vel  ubi  a  pluribus  filis 
elasticis  oblique  sitis  sustinetur.  In  omnibus  istis  casibus  illi  velocitatem,  vel  vim  agnoscunt 
vere  resolutam  in  duas,  quarum  utrique  simul  ilia  unica  velocitas,  vel  vis  aequivaleat,  ex 
illis  veluti  partibus  constituta,  quarum  si  altera  impediatur,  debeat  altera  perseverare,  vel 
si  impediatur  utraque,  suum  utraque  effectum  edat  seorsum.  At  quoniam  id  impedi- 
mentum  in  mea  Theoria  nunquam  habebitur  ab  immediato  contactu  plani  rigidi  subjecti, 
nee  a  virga  vere  rigida,  &  inflexili  sustentante,  sed  semper  a  viribus  mutuis  repulsivis  in 
primo  casu,  attractivis  in  secundo  ;  semper  habebitur  nova  velocitas,  vel  vis  aequalis,  & 
contraria  illi,  quam  communis  methodus  elisam  dicit,  quae  cum  [133]  tota  velocitate,  vel 
vi  obliqua  composita  eundem  motum,  vel  idem  aequilibrium  restituet,  ac  idem  omnino 
erit,  in  effectuum  computatione  considerare  partes  illas  binas,  &  alteram,  vel  utramque 
impeditam,  ac  considerare  priorem  totam,  aut  velocitatem,  aut  vim,  compositam  cum  iis 
novis  contrariis,  &  aequalibus  illi  parti,  vel  illis  partibus,  quae  dicebantur  elidi.  In  id  autem, 
quod  vel  inferne,  vel  superne  motum  massae  cujuspiam  impedit,  vel  vim,  non  aget  pars 
ilia  prioris  velocitatis,  vel  illius  vis,  quae  concipitur  resoluta,  sed  velocitas  orta  a  vi  mutua, 
&  contraria  velocitati  illi  novae  genitae  in  eadem  massa,  a  vi  mutua,  vel  ipsa  vis  mutua,  quae 
semper  debet  agere  in  partes  contrarias,  &  cui  occasionem  praebet  ilia  determinata  distantia 
major,  vel  minor,  quam  sit,  quae  limites,  &  aequilibrium  constitueret. 


"mom  2%°-  ^  quidem  abunde  apparet  in  ipso  superiore  exemplo.     Ibi  in  fig.  43  globus 

incurrente  in  pia-  (quem  concipamus  mollem)  advenit  oblique  per  AB,  &  oblique  impeditur  a  piano  ejus 
progressus.  Non  est  velocitas  perpendicularis  AF,  vel  EB,  quae  extinguitur,  durante  AE, 
vel  FB,  uti  diximus ;  nee  ilia  ursit  planum  CD.  Velocitas  AB  occasionem  dedit  globo 
accedendi  ad  planum  CD  usque  ad  earn  exiguam  distantiam,  in  qua  vires  variae  agerent ; 


A  THEORY  OF  NATURAL  PHILOSOPHY 


213 


278.  The  same  thing  comes  about  in  my  theory,  when  a  sphere  impinges  obliquely 
on  a  plane,  whether  the  motion  which  it  must  have  after  impact  is  under  consideration, 
or  whether   we   are    considering   the   energy  of   oblique  percussion  with  regard  to  the 
perpendicular  to  the  plane.     Thus,  in  Fig.  43,  suppose  a  sphere  A  to  move  along  the  oblique 
direction  AB  &  to  arrive  at  the  plane  CD,  which  is  considered  to  be  immovable,  &  with 
which  the  sphere  makes  physical  contact  at  the  point  N.     Now  imagine  a  plane  GI,  parallel 
to  the  former,  to  be  drawn  through  the  centre  B  ;   to  this  plane  the  centre  of  the  sphere 
will  attain,  &  rebound  from  it,  if  there  is  any  rebound.     After  drawing  AF  perpendicular 
to  GI  &  completing  the  parallelogram  AFBE,  the  usual  method  continues  by  resolving 
the  velocity  AB  into  the  two  velocities  AF,  AE,  or  FB,  EB  ;    of  these,  the  first  is  stated 
to  remain  constant,  whilst  the  second  is  destroyed  by  the  resistance  of  the  plane  ;    &  all 
that  remains  after  impact  is  represented  by  BI,  which  is  equal  to  FB,  if  the  body  is  soft ; 
or  that  this  is  compounded  with  another  represented  by  BE,  equal  to  the  original  velocity 
EB,  in  the  case  of  perfectly  elastic  bodies ;  and  in  the  case  of  imperfectly  elastic  bodies, 
it  is  compounded  with  Bi?,  which  bears  a  given  ratio  to  the  original  EB.     Then  the  sphere 
will  move  off,  in  the  first  case  along  BI,  in  the  second  case  along  BM,  &  in  the  third  case 
along  Em.     But,  according  to  my  Theory  the  sphere,  on  account  of  the  action  of  forces 
at  those  very  small  distances,  which  are  in  that  case  repulsive,  acquires  in   the  direction 
NE  perpendicular  to  the  repelling  plane  CD,  in  the  first  case  a  velocity  BE  equal  to  that 
which  it  would  have  acquired  if  it  had  travelled  along  EB  with  a  velocity  EB  at  right 
angles  to  the  plane  ;  in  the  second  case,  it  acquires  a  velocity  double  of  this,  namely  BL, 
&  in  the  third  a  velocity  BP,  which  bears  to  BE  the  given  ratio  r  to  I,  i.e.,  m  -f-  n  :  m. 
After  impact  it  has  a  velocity  compounded  of  the  original  velocity  which  persists,  expressed 
by  BO  equal  to  AB,  &  drawn  in  the  same  direction  as  AB,  with  another  velocity,  either 
BE,  BL,  or  BP  ;  from  which  it  is  easily  shown  that  there  results  either  BI,  BM,  or  EOT, 
just  as  in  the  usual  method.     For,  since  IO,  AF,  or  EB,  &  IM,  Im  are  respectively  equal 
to  BE,  Be,  or  EL,  EP ;   hence  the  wholes  BE,  BP,  BL  are  also  respectively  equal  to  the 
wholes  OI,  OM,  Om,  &  are  parallel  to  them. 

279.  The  matter,  in  my  hands,  comes  to  the  same  thing  in  every  case  with  composition 
of  forces,  as  in  the  usual  method  is  obtained  by  resolution.  In  the  usual  method  it  is  customary 
to  admit  resolution  for  motions  which  are  termed  impeded,  for  instance,  when  a  bordering 
plane,  or  a  bank,  impedes  progress  to  one  side,  as  in  the  channels  of  rivers ;   a  string,  or  a 
sustaining  rod,  as  in  the  oscillations  of  pendulums  hinders  motion  in  the  direction  in  which 
the  velocities  or  forces  are  in  that  case  supposed  to  be  acting.     In  a  similar  manner,  they 
recognize  resolution  of  forces,  when  two,  or  even  more  forces  impede  the  effect  of  some 
one  force  acting  in  another  direction  ;  for  instance,  when  a  heavy  body  is  sustained  by  two 
inclined  planes,  each  of  which  exerts  a  pressure  on  the  body  in  a  direction  perpendicular 
to  itself,  or  when  such  a  body  is  suspended  by  several  elastic  strings  in  inclined  positions. 
In  all  these  cases,  the  velocity  of  force  is  taken  to  be  really  resolved  into  two  ;   to  both  of 
these  taken  together  the  single  velocity  or  force  will  be  equivalent,  being  as  it  were  compounded 
of  these  parts,  of  which  if  one  is  impeded,  the  other  will  still  persist,  or  if  both  are  impeded, 
they  will  each  produce  their  own  effect  separately.     Now,  since  in  my  Theory  there  never 
is  such  impediment,  caused  by  an  immediate  contact  with  the  bordering  plane,  nor  by 
a  truly  rigid  or  inflexible  sustaining  rod,  but  always  considered  to  be  due  to  mutual  forces, 
that  are  repulsive  in  the  first  case  &  attractive  in  the  second  case,  a  new  velocity  or  force, 
equal  &  opposite  to  that  which  is  in  the  usual  theory  supposed  to  be  destroyed,  is  obtained. 
This  velocity,  or  force,  combined  with  the  whole  oblique  velocity  or  force,  will  give  the 
same  motion  or  the  same  equilibrium  ;    &  it  will  come  to  exactly  the  same  thing,  when 
computing  the  effects,  if  we  consider  the  two  velocities,  or  forces,  either  one  or  the  other, 
or  both,  to  be  impeded,  as  it  would  to  consider  the  original  velocity,  or  force,  to  be  com- 
pounded with   the   new  velocities,  or   forces,  which  are   opposite   in   direction   &   equal 
to  that  part  or  parts  which  are  said  to  be  destroyed.     Moreover,  upon  the  object  which 
hinders  the  motion,  or  force,  of  any  mass  upwards  or  downwards,  it  is  not  the  part  of  the 
original  velocity,  or  force,  which  is  said  to  be  resolved,  that  will  act ;  but  it  is  the  velocity 
arising  from  the  mutual  force,  opposite  in  direction  to  that  velocity  which  is  newly  generated 
in  the  mass  by  the  mutual  force,  or  the  mutual  force  itself.     This  must  always  act  in  opposite 
directions ;  &  is  governed  by  the  given  distance,  greater  or  less  than  that  which  gives  the 
limit-points  &  equilibrium. 

280.  This  fact  indeed  is  seen  clearly  enough  in  the  example  given  above.     There,  in  Fig. 
43,  the  sphere,  which  we  will  suppose  to  be  soft,  travels  obliquely  along  AB,  &  its  progress  is 
impeded,  also  obliquely,  by  the  plane.     It  is  not  true  that  the  perpendicular  velocity  AF,  or 
EB  is  destroyed,  whilst  AE,  or  FB  persists,  as  we  have  already  proved  ;  nor  was  there  any 
direct  pressure  from  it  on  the  plane  CD.     The  velocity  AB  made  the  sphere  approach  the 
plane  CD  to  within  a  very  small  distance  from  it,  at  which  various  forces  come  into  action  ; 


Composition  sub- 
stituted also  for 
resolution  i  n  the 
case  of  a  sphere 
impinging  on  an 
immovable  plane. 


In  every  case,  in 
my  Theory,  com- 
position  is  used 
instead  of  resolu- 
tion ;  &  these  are 
equivalent  to  one 
another. 


A  case  in  point 
where  a  soft  sphere 
impinges  on  an 
immovable  plane. 


2I4 


PHILOSOPHISE  NATURALIS  THEORIA 


Aliud  globi  descen- 
dentis  per  planum 
inclinatum. 


Aliud    in    pendulo. 


Alia  ratio  com- 
ponondi  vires  in 
eodem  casu. 


Aliud  exemplum  in 
globo  sustentato  a 
binis  planis.  Diffi- 
cultas  communis 
methodi  in  eodem. 


FIG.  44. 


donee  ex  omnibus  actionibus  conjunctis  impediretur  accessus  ad  ipsum  planum,  sive 
perpendicularis  distantiae  ulterior  diminutio.  Illae  vires  agent  simul  in  directione  perpen- 
dicular! ad  ipsum  planum  juxta  num.  266  :  debebunt  autem,  ut  impediant  ejusmodi 
ulteriorem  accessum,  producere  in  ipso  globo  velocitatem,  quae  composita  cum  tota  BO 
perseverante  in  eadem  directione  AB,  exhibeat  velocitatem  per  BI  parallelam  CD.  Quoniam 
vero  triangula  rectangula  AFB,  BIO  aequalia  erunt  necessario  ob  AB,  BO  aequales ;  erit 
BEIO  parallelogrammum,  adeoque  velocitas  perpendicularis,  quae  cum  priore  velocitate 
BO  debeat  componere  velocitatem  per  rectam  parallelam  piano,  debebit  necessario  esse 
contraria,  &  aequalis  illi  ipsi  EB  perpendiculari  eidem  piano,  in  quam  resolvunt  vulgo 
velocitatem  AB.  Interea  vero  vis,  quae  semper  agit  in  partes  contrarias  aequaliter,  urserit 
planum  tantundem,  &  omnes  in  eo  produxerit  effectus  illos,  qui  vulgo  tribuuntur  globo 
advenienti  cum  velocitate  ejusmodi,  ut  perpendicularis  ejus  pars  sit  EB. 

281.  Idem  accidet  etiam  in  reliquis  omnibus  casibus  superius  memoratis.     Descendat 
globus  gravis  per  planum  inclinatum  CD  (fig.  44)  oblique,  quod  in  communi  sententia 
continget  hunc  in  modum.      Resolvunt  gravita- 

tem  BO  in  duas,  alteram  BR  perpendicularem 
piano  CD,  qua  urgetur  ipsum  planum,  quod  eum 
sustinet ;  alteram  BI,  parallelam  eidem  piano, 
quse  obliquum  descensum  accelerat.  In  mea 
Theoria  gravitas  cogit  globum  semper  magis  ac- 
cedere  ad  planum  CD  ;  donee  distantia  ab  eodem 
evadat  ejusmodi ;  ut  vires  mutuae  [134]  repul- 
sivae  agant,  &  ilia  quidem,  quae  agit  in  B,  sit 
ejusmodi  ut  composita  cum  BO  exhibeat  BI 
parallelam  piano  ipsi,  adeoque  non  inducentem 
ulteriorem  accessum,  sit  autem  perpendicularis 
piano  ipsi.  Porro  ejusmodi  est  BE,  jacens  in 
directum  cum  RB,  &  ipsi  aequalis,  cum  nimirum 
debeat  esse  parallela,  &  aequalis  OI.  Vis  autem 
aequalis  ipsi,  &  contraria,  adeoque  expressa  per  RB,  urgebit  planum. 

282.  Quod  si  grave  suspensum  in  fig.  45  filo,  vel  virga  BC  debeat  oblique  descendere 
per  arcum  circuli  BD  ;    turn  vero  in  communi  methodo  gravitatem  BO  itidem  resolvunt 
in  duas  BR,  BI,  quarum  prima  filum,  vel  virgam  tendat,  &  elidatur,  secunda  acceleret 
descensum  obliquum,  qui  fieret  ex  velocitate  concepta  per  rectam  BA  perpendicularem 
BC,  ac  praeterea  etiam  tensionem  fili  agnoscunt  ortum 

a  vi  centrifuga,  quae  exprimitur  per  DA  perpendicu- 
larem tangenti.  At  in  mea  Theoria  res  hoc  pacto 
procedit.  Globus  ex  B  abit  ad  D  per  vires  tres  com- 
positas  simul  cum  velocitate  praecedente  ;  prima  e 
viribus  est  vis  gravitatis  BO  ;  secunda  attractio  versus 
C  orta  a  tensione  fili,  vel  virgae,  expressa  per  BE  paral- 
lelam, &  aequalem  OI,  adeoque  RB,  quae  solae  compo- 
nerent  vim  BI ;  tertia  est  attractio  in  C  expressa  per 
BH  aequalem  AD  orta  itidem  a  tensione  fili  respond- 
ente  vi  centrifugae,  &  incurvante  motum.  Adest  prae- 
terea  velocitas  praecedens,  quam  exprimit  BK  aequalis 
IA,  ut  sit  BI  aequalis  KA.  His  viribus  cum  ea  veloci- 
tate simul  agentibus  erit  globus  in  D  in  fine  ejus  tem- 
pusculi,cui  ejusmodi  effectus  illarum  virium  respond- 
ent. Nam  ibi  debet  esse,  ubi  esset,  si  aliae  ex  illis  causis  agerent  post  alias  :  gravitate  agente 
veniret  per  BO,  vi  BE  abiret  per  OI,  velocitate  BK  abiret  per  IA  ipsi  aequalem,  vi  BH 
abiret  per  AD.  Quamobrem  res  tota  itidem  peragitur  sola  compositione  virium,  &  motuum. 

283.  Porro  si  sumatur  EG  aequalis  BH  ;    turn  tota  attractio  orta  a  tensione  fili  erit 
BG,  quae  prius  considerata  est  tanquam  e  binis  partibus  in  directum  agentibus  composita, 
ac  res  eodem  redit ;    nam  si  prius  componantur  BH,  &  BE  in  BG  (quo  casu  tota  BG  ut 
unica  vis  haberetur),  turn  BO,  ac  demum  BK,  ad  idem  punctum  D  rediretur  juxta  generalem 
demonstrationem,  quam  dedi  num.  259.     Jam  vero  vi  expressa  per  totam  BG  attraheretur 
ad  centrum  suspensionis  C  ab  integra  tensione  fili,  ubi  pars  EG,  vel  BH  ad  partem  BE 
habet  proportionem  pendentem  a  celeritate  BK,  ab  angulo  RBO,  ac  a  radio  CB  ;   sed  ista 
meae  Theoriae  cum  omnium  usitatis  Mechanicae  elementis  communia  sunt,  posteaquam 
compositionis  hujus  cum  ilia  resolutione  aequivalentia  est  demonstrata. 

284.  Quae  de  motu  diximus  facto  vi  oblique,  sed  non  penitus  impedita,  eadem  in 
aequilibrio  habent  locum,  ubi  omnis  impeditur  motus.     Innitatur  globus  gravis  B  in  fig. 
46  binis  planis  AC,  CD,  quae  accurate,  vel  in  mea  Theoria  [135]  physice  solum,  contingat 


B 


FIG.  45. 


A  THEORY  OF  NATURAL  PHILOSOPHY  215 

then,  under  the  combined  actions  of  all  the  forces  further  approach  toward  the  plane, 
or  further  diminution  of  the  perpendicular  distance  from  it,  is  impeded.  The  forces  act 
together  in  the  direction  perpendicular  to  the  plane,  as  was  shown  in  Art.  266  ;  &  they 
must,  in  order  to  impede  further  approach  of  this  kind,  produce  in  the  sphere  itself  a 
velocity  which,  compounded  with  the  whole  velocity  that  persists  throughout,  namely  BP, 
in  the  direction  of  AB,  will  give  a  velocity  represented  by  BI  parallel  to  CD.  But,  since 
the  right-angled  triangles  AFB,  BIO  are  necessarily  congruent  on  account  of  the  equality 
of  AB  &  BO,  it  follows  that  BEIO  is  a  parallelogram.  Hence,  the  perpendicular  velocity, 
which  has,  when  combined  with  the  original  velocity  BO,  to  give  a  resultant  represented 
by  a  straight  line  parallel  to  the  plane,  must  of  necessity  be  equal  &  opposite  to  that  represented 
by  EB,  also  perpendicular  to  the  plane,  into  which  commonly  the  velocity  AB  is  resolved. 
Meanwhile,  the  force,  which  always  acts  equally  in  opposite  directions,  would  act  on  the 
plane  to  precisely  the  same  extent,  &  all  those  effects  would  be  produced  on  it,  which  are 
commonly  attributed  to  the  sphere  striking  it  with  a  velocity  of  such  sort  that  its  perpendicular 
part  is  EB. 

281.  The  same  thing  happens  also  in  the  rest  of  the  cases  mentioned  above.     Let  a  Another    case  in 
heavy  sphere  descend  along  the  inclined  plane  CD,  in  Fig.  44 ;  the  descent  takes  place,  ^ere   descending 
according  to  the  customary  idea,  in  the  following  manner.     Gravity,  represented  by  BO,  along  an  inclined 
is  resolved  into  two  parts,  the  one,  BR,  perpendicular  to  the  plane  CD,  acts  upon  the  plane  plane' 

&  is  resisted  by  it ;  the  other,  BI,  parallel  to  the  plane,  accelerates  the  oblique  descent. 
According  to  my  Theory,  gravity  forces  the  sphere  to  approach  the  plane  CD  ever  nearer 
&  nearer,  until  the  distance  from  it  becomes  such  as  that  for  which  the  repulsive  forces 
come  into  action  ;  that  which  acts  on  B  is  such  that,  when  combined  with  BO,  will  give 
a  velocity  represented  by  BI  parallel  to  the  plane,  &  thus  does  not  induce  further  approach  ; 
moreover  it  is  perpendicular  to  the  plane.  Further,  it  is  such  as  BE,  lying  in  the  same 
straight  line  as  RB,  &  equal  to  it,  because  indeed  it  must  be  parallel  &  equal  to  OI.  Lastly, 
a  force  that  is  equal  &  opposite,  &  so  represented  by  BR,  will  act  upon  the  plane. 

282.  But  if,  in  Fig.  45,  a  heavy  body  is  suspended  by  a  string  or  rod  BC,  it  is  bound  to  The    pendulum  is 
descend  obliquely  along  the  circular  arc  BD.     Now,  in  the  usual  method,  gravity,  represented  ^°^CT     C*SG    m 
by  BO,  is  again  resolved  into  two  parts,  BR  &  BI  ;   the  first  of  these  exerts  a  pull  on  the 

string  or  rod  &  is  destroyed  ;  the  second  accelerates  the  oblique  descent,  which  would 
come  about  through  a  velocity  supposed  to  act  along  BA  perpendicular  to  BC  ;  in  addition 
to  these,  account  is  taken  of  the  tension  of  the  string  arising  from  a  centrifugal  force,  which 
is  represented  by  DA  perpendicular  to  the  tangent.  But,  according  to  my  Theory,  the 
matter  goes  in  this  way.  The  sphere  passes  from  B  to  D,  under  the  action  of  three  forces 
compounded  with  the  original  velocity.  The  first  of  these  forces  is  gravity,  BO  ;  the  second 
is  the  attraction  towards  C  arising  from  the  tension  of  the  string  or  rod,  &  represented 
by  BE,  parallel  &  equal  to  OI,  &  thus  also  to  RB,  these  two  alone,  taken  together,  give  a 
force  BI.  The  third  is  an  attraction  towards  C,  represented  by  BH,  equal  to  AD,  arising 
also  from  the  tension  of  the  string  corresponding  to  the  centrifugal  force  &  incurving  the 
motion.  In  addition  to  these,  we  have  the  original  velocity,  represented  by  BK,  equal 
to  IA,  so  that  BI  is  equal  to  KA.  If  such  forces  as  these  act  together  with  this  velocity, 
the  sphere  will  arrive  at  D  at  the  end  of  the  interval  of  time  to  which  such  effects  of  the 
forces  correspond.  For  it  must  reach  that  point  at  which  it  would  be,  if  all  these  causes 
acted  one  after  the  other  ;  &,  with  gravity  acting,  it  would  travel  along  BO  ;  with 
the  force  BE  acting  it  would  pass  along  OI ;  with  the  velocity  BK,  it  would  traverse  IA, 
which  is  equal  to  BK  ;  &  with  the  force  BH  acting,  it  would  go  from  A  to  D.  Hence, 
in  this  case  also,  the  whole  matter  is  accomplished  with  composition  alone,  for  forces 
&  motions. 

283.  Further,  if  EG  is  taken  equal  to  BH ;    then  the  whole  attraction  arising  from  Another  manner  of 
the  tension  of  the  string  will  be  BG,  which  previously  was  considered  only  as  being  com-  f  °™epsouin'dS1| 
pounded  of  two  parts  acting  in  the  same  straight  line  ;   &  it  comes  to  the  samejthing  just  considered. 

as  before.  For,  if  BH  &  BE  are  first  of  all  compounded  into  BG  (in  which  case  BG  is  reckoned 
as  a  single  force),  then  BO  is  taken  into  account,  &  finally  BK  ;  we  shall  be  led  to  the  same 
point  D,  according  to  the  general  demonstration  I  gave  in  Art.  259.  Now  we  have  an 
attraction  to  the  centre  of  suspension  C  due  to  a  force  expressed  by  the  whole  BG,  where 
the  part  of  it,  EG,  or  BH,  bears  to  the  part  BH  a  ratio  that  depends  upon  the  velocity 
BK,  the  angle  RBO,  &  the  radius  CB.  The  results  of  my  Theory  are  in  agreement  with 
the  elementary  principles  of  Mechanics  accepted  by  everyone  else,  as  soon  as  the  equivalence 
of  my  composition  with  their  resolution  has  been  demonstrated. 

284.  The  same  things  hold  good  in  the  case  of  equilibrium,  where  all  motion  is  impeded,  ^^  su^ 
as  those  we  have  already  spoken  of  with  respect  to  motion  derived  from  a  force  acting  by    two    planes 
obliquely,  but  not  altogether  impeded.     In  Fig.  46,  a  heavy  sphere  is  supported  by  two  i^ai^ 
planes  AC,  CD,  which  actually,  or  in  my  Theory  physically  only,  it  touches  at  H  &  F ;  this  case. 


216  PHILOSOPHISE  NATURALIS  THEORIA 

in  H,  &  F,  &  gravitatem  referat  recta  verticalis  BO,  ac  ex  puncto  O  ad  rectas  BH,  BF 
ducantur  rectse  OR,  OI  parallels  ipsis  BF,  BH,  &  producta  sursum  BK  tantundem,  ducantur 
ex  K  ipsis  BF,  BH  parallels  KE,  KL  usque  ad  easdem  BH,  BF  ;  ac  patet,  fore  rectas  BE, 
BL  aequales,  &  contrarias  BR,  BI.  In  communi  methodo  resolutionis  virium  concipitur 
gravitas  BO  resoluta  in  binas  BR,  BI,  quarum  prima  urgeat  planum  AC,  secunda  DC  ; 
&  quoniam  si  angulus  HCF  fuerit  satis  acutus  ;  erit  itidem  satis  acutus  angulus  R,  qui 
ipsi  aequalis  esse  debet,  cum  uterque  sit  complementum  HBF  ad  duos  rectos,  alter  ob 
parallelogrammum,  alter  ob  angulos  BHC,  BFC  rectos  ;  fieri  potest,  ut  singula  latera 
BR,  RO,  sive  BI,  sint,  quantum  libuerit,  longiora  quam  BO  ;  vires  singulas,  quas  urgent 
ilia  plana,  possunt  esse,  quantum  libuerit,  majores,  quam  sola  gravitas  :  mirantur  multi, 
fieri  posse,  ut  gravitas  per  solam  ejusmodi  applicationem  tantum  quodammodo  supra  se 
assurgat,  &  effectum  tanto  majorem  edat. 

Soiutio    in    ipsa  285.    DifEcultas    ejusmodi  in  communi  etiam  sententia  evitari  facile  potest  exemplo 

methodo  communi  *  ,.  •        i  •    *        •  i  *....  «  .  i*  •  11 

in  hac  Theoria  vectls>  de  quo  agemus  infra,  in  quo  sola  applicatio  vis  in  multo  majore  distantia  collocatae 
nuiium  ipsi  difficul-  multo  majorem  effectum  edit.  Verum  in  mea  Theoria  ne  ullus  quidem  difficultati  est 
locus.  Non  resolvitur  revera  gravitas  in  duas  vires  BR,  BI,  quarum  singulae  plana  urgeant, 
sed  gravitas  inducit  ejusmodi  accessum  ad  ea  plana,  in  quo  vires  repulsivse  perpendiculares 
ipsis  planis  agentes  in  globum  componant  vim  BK  aequalem,  &  contrariam  gravitati  BO, 
quam  sustineat,  &  ulteriorem  accessum  impediat.  Ad  id  praestandum  requiruntur  illas 
vires  BE,  BL  aequales,  &  contrarias  hisce  BR,  BI,  quae  rem  conficiunt.  Sed  quoniam  vires 
sunt  mutuae,  habebuntur  repulsiones  agentes  in  ipsa  plana  contrarias,  &  aequales  illis  ipsis 
BE,  BL,  adeoque  agent  vires  expressae  per  illas  ipsas  BR,  BI,  in  quas  communis  methodus 
gravitatem  resolvit. 

Aliud  in  giobo  sus-  286.  Quod  si  globus  gravis  P  in  fig.  47  e  filo  BP  pendeat,  ac  sustineatur  ab  obliquis 

>iqms.  £jjg  ^g^  j-j-g^  exprimat  autem  BH  gravitatem,  &  sit  BK  ipsi  contraria,  &  aequalis,  ac  sint 
HI,  KL  parallelae  DB,  &  HR,  KE  parallelae  filo  AB  ;  communis  methodus  resolvit  gravi- 
tatem BH  in  duas  BR,  BI,  quas  a  filis  sustineantur,  &  ilia  tendant  ;  sed  ego  compono  vim 
BK  gravitati  contrariam,  &  aequalem  e  viribus  BE,  BL,  quas  exerunt  attractivas  puncta 
fili,  quae  ob  pondus  P  delatum  deorsum  sua  gravitate  ita  distrahuntur  a  se  invicem,  donee 
habeantur  vires  attractivae  componentes  ejusmodi  vim  contrariam,  &  aequalem  gravitati. 

Conciusio    gene-          287.  Quamobrem  per  omnia  casuum  diversorum  genera  pervagati  jam  vidimus,  nullam 
rails   pro    hac  esse  USpiam  ;n  mea  Theoria  veram  aut  virium,  aut  motuum  resolutionem,  sed  omnia 

theona,  quae  omnia  r      ,  ,  ,  .   .  ,        r       ,-. 

per  solam  prorsus    phasnomena    pendere  a  sola  compositione  virium,  &    motuum,  adeo-[i3oj-que 


compositionem.  naturam  eodem  ubique  modo  simplicissimo  agere,  componendo  tantummodo  vires,  & 
motus  plures,  sive  edendo  simul  eum  effectum,  quern  ederent  illae  omnes  causae  ;  si  aliae 
post  alias  effectus  ederent  suos  aequales,  &  eandem  habentes  directionem  cum  iis,  quos 
singulae,  si  solae  essent,  producerent.  Et  quidem  id  generale  esse  Theorise  meae,  patet 
vel  ex  eo,  quod  nulli  possunt  esse  motus  ex  parte  impediti,  ubi  nullus  est  immediatus 
contactus,  sed  in  libero  vacuo  spatio  punctum  quodvis  liberrime  movetur  parendo  simul 
velocitati,  quam  habet  jam  acquisitam,  &  viribus  omnibus,  quae  ab  aliis  omnibus  pendent 
materiae  punctis. 

Resolutio    tantum  288.  Quanquam  autem  habeatur  revera  sola  compositio  virium  ;    licebit  adhuc  vires 

mente  concept  a  imaginatione   nostra   resolvere  in   plures,   quod  saepe   demonstrationes   theorematum,   & 

ssepe  utilis  ad  con-        -°.  ..  ix.  .  i  T  •  jjoi 

trahendas  solu-  solutionem  problematum  contrahet  mirum  in  modum,  ac  expeditiores  reddet,  &  elegant- 
tiones.  iores  ;   nam  licebit  pro  unica  vi  assumere  vires  illas,  ex  quibus  ea  componeretur.     Quoniam 

enim  idem  omnino  effectus  oriri  debet,  sive  adsit  unica  vis  componens,  sive  reapse  habeantur 
simul  plures  illae  vires  componentes  ;  manifestum  est,  substitutione  harum  pro  ilia  nihil 
turbari  conclusiones,  quae  inde  deducuntur  :  &  si  post  resolutionem  ejusmodi  inveniatur 
vis  contraria,  &  aequalis  alicui  e  viribus,  in  quas  vis  ilia  data  resolvitur  ;  ilia  haberi  potest 
pro  nulla  consideratis  solis  reliquis,  si  in  plures  resoluta  fuit,  vel  sola  altera  reliqua,  si 
resoluta  fuit  in  duas.  Nam  componendo  vim,  quae  resolvitur,  cum  ilia  contraria  uni  ex  iis, 
in  quas  resolvitur,  eadem  vis  provenire  debet  omnino,  quae  oritur  componendo  simul 
reliquas,  quae  fuerant  in  resolutione  sociae  illius  elisae,  vel  retinendo  unicam  illam  alteram 
reliquam,  si  resolutio  facta  est  in  duas  tantummodo  ;  atque  id  ipsum  constat  pro  resolu- 
tione in  duas  ipsis  superioribus  exemplis,  &  pro  quacunque  resolutione  in  vires  quotcunque 
facile  demonstratur. 


A  THEORY  OF  NATURAL  PHILOSOPHY 


217 


FIG.  47. 


218 


PHILOSOPHIC  NATURALIS  THEORIA 


FIG.  47. 


A  THEORY  OF  NATURAL  PHILOSOPHY 


219 


let  the  vertical  line  BO  represent  gravity,  &  draw  from  the  point  O,  to  meet  the  straight 
lines  BH,  BF,  the  straight  lines  OR,  OI  parallel  to  BF,  BH  ;  also  producing  BK  upwards 
to  the  same  extent,  draw  through  K  the  straight  line  KE,  KL,  parallel  to  BF,  BH  to  meet 
BH  &  BF.  Then  it  is  clear  that  BE,  BL  will  be  equal  &  opposite  to  BR,  BI.  Now, 
according  to  the  usual  method  by  means  of  resolution  of  forces,  the  gravity  BO  is  supposed 
to  be  resolved  into  the  two  parts  BR,  BI,  of  which  the  first  acts  upon  the  plane  AC  &  the 
second  upon  DC.  Also  if  the  angle  HCF  is  sufficiently  acute,  then  the  angle  at  R  is  also 
sufficiently  acute  ;  for  these  angles  must  be  equal  to  one  another.  For,  each  is  the  supple- 
ment of  the  angle  HBF,  the  one  in  the  parallelogram,  the  other  on  account  of  BHC 
&  BFC  being  right  angles.  This  being  so,  it  may  happen  that  each  of  the  sides  BR,  RO, 
or  BI,  will  be  greater  than  BO,  to  any  desired  extent.  Thus  each  of  the  forces,  which 
act  upon  the  planes,  may  be  greater  than  gravity  alone,  to  any  desired  extent.  Many 
will  wonder  that  it  is  possible  that  gravity,  by  a  mere  application  of  this  kind,  surpasses 
itself  to  so  great  an  extent,  &  gives  an  effect  that  is  so  much  greater. 

285.  A  difficulty  of  this  kind  even  according  to  the  ordinary  opinion  is  easily  avoided  Answer   to  the 
by  comparing  the  case  of  the  lever,  with  which  we  will  deal  later  ;  in  it  the  mere  application  ortoar'y'  mltho'd! 
of  a  force  situated  at  a  much  greater  distance  gives  a  far  greater  effect.     But  with  my  in     my     Theory 
Theory  there  is  no  occasion  for  any  difficulty  of  the  sort.     For  there  is  no  actual  resolution  for  ^ny  difficulty?11 
of  gravity  into  the  two  parts  BR,  BI,  each  acting  on  one  of  the  planes ;  but  gravity  induces 

an  approach  to  the  planes,  to  within  the  distance  at  which  repulsive  forces  acting  perpen- 
dicular to  the  planes  upon  the  sphere  compound  into  a  force  BK,  equal  &  opposite  to  the  grav- 
ity BO  ;  this  force  sustains  the  sphere  &  impedes  further  approach  to  the  planes.  To  represent 
this,  the  forces  BE,  BL  are  required  ;  these  are  equal  &  opposite  to  BR,  BI ;  &  that  is  all  there 
need  be  said  about  the  matter.  Now,  since  the  forces  are  mutual,  there  are  repulsions  acting 
upon  the  planes,  &  these  repulsions  are  equal  &  opposite  to  BE,  BL  ;  &  thus  the  forces  acting 
are  represented  by  BR,  BI,  which  are  those  into  which  the  ordinary  method  resolves  gravity. 

286.  But  if,  in  Fig.  47,  a  heavy  sphere  P  is  suspended  by  a  string  BP,  &  this  is  held  Explanation  in^he 
up  by  inclined  strings  AB,  DB,  &  gravity  is  represented  by  BH  ;  let  BK  be  equal  &  opposite  £uspendedabyP  i™ 
to  it,  &  let  HI,  KL  be  parallel  to  the  string  DB,  &  HR,  KE  parallel  to  the  string  AB.     The  clined  strings, 
ordinary  method  resolves  the  gravity  BH  into  the  two  parts  BR,  BI,  which  are  sustained 

by  the  strings  &  tend  to  elongate  them.  On  the  other  hand,  I  compound  the  force  BK, 
equal  &  opposite  to  gravity  from  the  two  forces  BE,  BL  ;  these  attractive  forces  are  put 
forth  by  the  points  of  the  string,  which,  owing  to  the  heavy  body  P  suspended  beneath 
are  drawn  apart  by  its  gravity  to  such  a  distance  that  attractive  component  forces  are 
obtained  such  as  will  give  a  force  that  is  equal  &  opposite  to  the  gravity  of  P. 

287.  Having  thus  considered  all  sorts  of  different  cases,  we  now  see  that  there  is  nowhere  General    summing 
in  my  Theory  any  real  resolution  either  of  forces  or  of  motions ;   but  that  all  phenomena  "his  Theory?*  which 
depend  on  composition  of  forces  &  motions  alone.     Thus,  nature  in  all  cases  acts  in  the  gives     everything 
same  most  simple  manner,  by  compounding  many  forces  &  motions  only;   that  is  to  say,  akrae°mp0 

by  producing  at  one  time  that  effect,  which  all  the  causes  would  produce,  if  they  acted 
one  after  the  other,  &  each  produced  that  effect  which  was  equal  &  in  the  same  direction 
as  that  which  it  would  produce  if  it  alone  acted.  That  this  is  a  general  principle  of  my 
Theory  is  otherwise  evident  from  the  fact  that  no  motions  can  be  in  part  impeded,  where 
there  is  no  immediate  contact ;  on  the  contrary,  any  point  can  move  in  a  free  empty  space 
in  the  freest  manner,  subject  to  the  combined  action  of  the  velocity  it  has  already  acquired, 
&  to  all  the  forces  which  come  from  all  other  points  of  matter. 

288.  Now,  although  as  a  matter  of  fact  we  can  only  have  compositions  of  forces,  yet  Resolution,  ai- 
we  may  mentally  resolve  our  forces  into  several ;   &  this  will  often  shorten  the  proofs  of  m°ntii    fiction,  is 
theorems  &  the  solution  of  problems  in  a  wonderful  manner,  &  render  them  more  elegant  yet  often  useful  in 
&  less  cumbrous ;   for  we  may  assume  instead  of  a  single  force  the  forces  from  which  it  is  8 
compounded.     For,  since  the  same  effect  must  always   be  produced,  whether  a  single 
component  force  is  present,  or  whether  in  fact  we  have  the  several  component  forces  taken 

all  together,  it  is  plain  that  the  conclusions  that  are  derived  will  in  no  way  be  disturbed 
by  the  substitution  of  the  latter  for  the  former.  If  after  such  resolution  a  force  is  found, 
equal  &  opposite  to  any  one  of  the  forces  into  which  the  given  force  is  resolved,  then  these 
two  can  be  taken  together  as  giving  no  effect ;  &  only  the  rest  need  be  considered  if  the 
given  force  was  resolved  into  several  parts,  or  only  the  other  force  if  the  given  force  was 
resolved  into  two  parts.  For,  by  compounding  the  force  which  was  resolved  with  that 
force  which  is  equal  &  opposite  to  the  one  of  the  forces  into  which  it  was  resolved,  the 
same  force  must  be  obtained  as  would  arise  from  compounding  all  the  other  forces  which 
were  partners  of  the  cancelled  force  in  the  resolution,  or  from  retaining  the  single  remaining 
force  when  the  resolution  was  into  two  parts  only.  This  has  been  shown  to  be  the  case 
for  resolution  in  the  two  examples  given  above,  &  can  be  easily  proved  for  any  sort  of 
resolution  into  forces  of  any  number  whatever. 


220  PHILOSOPHIC  NATURALIS  THEORIA 

Methodus  generalis  289.  Porro  quod  pertinct  ad  resolutionem  in  plures  vires,  vel  motus,  facile  est  ex 

SiM^qwtCTiaqne.  "s>  <luse  dicta  sunt  num-  257  definire  legem,  quae  ipsam  resolutionem  rite  dirigat,  ut 
habeantur  vires,  quae  datam  aliquam  componant.  Sit  in  fig.  48,  vis  quaecunque,  vel  motus 
AP,  &  incipiendo  ab  A  ducantur  quotcunque,  &  cujuscunque  longitudinis  rectae  AB,  BC, 
CD,  DE,  EF,  FG,  GP,  continue  inter  se  connexse  ita,  ut  incipiant  ex  A,  ac  desinant  in  P  ; 
&  si  ipsis  BC,  CD,  &c.  ducantur  parallels,  &  aequales  Ac,  Ad,  &c.  ;  vires  omnes  AB,  Ac, 
Ad,  Ae,  A/,  Ag,  Ap  component  vim  AP ;  unde  patet  illud  :  ad  componendam  vim 
quamcunque  posse  assumi  vires  quotcunque,  &  quascunque,  quibus  assumptis  determinari 
poterit  una  alia  praeterea,  quae  compositionem  perficiat ;  nam  poterunt  duci  rectae  AB, 
BC,  CD,  &c.  parallelae,  &  aequales  datis  quibuscunque,  &  ubi  postremo  deventum  fuerit 
ad  aliquod  punctum  G,  satis  erit  addere  vim  expressam  per  GP. 

Eyoiutio      resolu-  [137]  200.  Eo   autem  generali   casu   continetur   particularis   casus   resolutionis   in   vires 

tionis   in  duas    L    J/J  ,        ,  ~      ,  r.  .... 

tantum.  tantummodo  duas,  quae  potest  hen  per  duo  quaevis   latera  tnanguli  cujuscunque,  ut   in 

fig.  49,  si  datur  vis  AP,  &  fiat  quodcunque  triangulum  ABP ;  vis  resolvi  potest  in  duas 
AB,  BP,  &  data  illarum  altera,  datur  &  altera,  quod  quidem  constat  etiam  ex  ipsa  com- 
positione,  seu  resolutione  per  parallelogrammum  ABPC,  quod  semper  compleri  potest, 
&  in  quo  AC  est  parallela,  &  aequalis  BP,  ac  bins  vires  AB,  AC  componunt  vim  AP  :  atque 
idem  dicendum  de  motibus. 

Cur  vis  composita  291.  Ejusmodi  resolutio  illud  etiam  palam  faciet ;    cur  vis  composita  a  viribus  non 

sit     minor     com-  m  directum  iacentibus,  sit  minor  ipsis  componentibus,  quae  nimirum  sunt  ex  parte  sibi 

ponentibus      simul    .  •«_!••  ••       *  1-1  •         •  • 

sumptis.  invicem  contranae,  &  elisis  mutuo  contrarns,  &  aequalibus,  remanet  in  vi  composita  summa 

virium  conspirantium,  vel  differentia  oppositarum  pertinentium  ad  componentes.  Si 
enim  in  fig.  50,  51,  52  vis  AP  componatur  ex  viribus  AB,  AC,  quae  sint  latera  parallelogrammi 
ABPC,  &  ducantur  in  AP  perpendicula  BE,  CF,  cadentibus  E,  &  F  inter  A,  &  P  in  fig.  50, 
in  A,  &  P  in  fig.  51,  extra  in  fig.  52  ;  satis  patet,  fore  in  prima,  &  postrema  acqualia  triangula 
AEB,  PFC,  adeoque  vires  EB,  FC  contrarias,  &  aequales  elidi ;  vim  vero  AP  in  primo 
casu  esse  summam  binarum  virium  conspirantium  AE,  AF,  aequari  unicae  AF  in  secundo, 
&  fore  differentiam  in  tertio  oppositarum  AE,  AF. 


292'  ^n  res°luti°ne  quidem  vis  crescit  quodammodo  ;  quia  mente  adjungimus  alias 
lutione :  nihii  inde  oppositas,  &  aequales,  quae  adjunctae  cum  se  invicem  elidant,  rem  non  turbant.  Sic  in 
vdivlUCi  pr°  fi§-  52  resolvendo  Ap  in  bmas  AB,  AC,  adjicimus  ipsi  AP  binas  AE,  PF  contrarias,  & 
praeterea  in  directione  perpendiculari  binas  EB,  FC  itidem  contrarias,  &  aequales.  Cum 
resolutio  non  sit  realis,  sed  imaginaria  tantummodo  ad  faciliorem  problematum  solutionem  ; 
nihil  inde  difHcultatis  afferri  potest  contra  communem  methodum  concipiendi  vires,  quas 
hue  usque  consideravimus,  &  quae  momento  temporis  exercent  solum  nisum,  sive  pressionem  ; 
unde  etiam  fit,  ut  dicantur  vires  mortuae,  &  idcirco  solum  continue  durantes  tempore 
sine  contraria  aliqua  vi,  quae  illas  elidat,  velocitatem  inducunt,  ut  causae  velocitatis  ipsius 
inductae  :  nee  inde  argumentum  ullum  desumi  poterit  pro  admittendis  illis,  quas  Leibnitius 
invexit  primus,  &  vires  vivas  appellavit,  quas  hinc  potissimum  necessario  saltern  concipiendas 
esse  arbitrantur  nonnulli,  ne  nimirum  in  resolutione  virium  habeatur  effectus  non  aequalis 
suae  causae.  Effectus  quidem  non  aequalis,  sed  proportionalis  esse  debet,  non  causae,  sed 
actioni  causae,  ubi  ejusmodi  actio  contraria  aliqua  actione  non  impeditur  vel  tota,  vel  ex 
parte,  quod  accidit,  uti  vidimus,  in  obliqua  compositione  :  ac  utcunque  &  aliae  responsiones 
sint  in  communi  etiam  sententia  pro  casu  resolutionis ;  [138]  in  mea  Theoria,  cum  ipsa 
resolutio  realis  nulla  sit,  nulla  itidem  est,  uti  monui,  difficultas. 


us. 


Satis  patere  ex  hac  2g,    £t    quidem    tarn    ex    iis,    quae    hue    usque    demonstrata    sunt,    quam    ex 

Theoria,    Vires  .  i  •  fl  T     •  •  •  •     j-   • 

Vivas  in    Natwa  quae  consequentur,  satis  apparebit,  nullum  usquam  esse  ejusmodi  virium  vivarum  indicium, 
nuttas  esse.  nullam  necessitatem  ;   cum  omnia  Naturae  phaenomena  pendeant  a  motibus,  &  sequilibrio, 

adeoque  a  viribus  mortuis,  &  velocitatibus  inductis  per  earum  actiones,  quam  ipsam  ob 
causam  in  ilia  dissertatione  De  Viribus  Fivis,  quae  hujus  ipsius  Theoriae  occasionem  mihi 
praebuit  ante  annos  13,  affirmavi,  Fires  Vivas  in  Natura  nullas  esse,  &  multa,  quae  ad  eas 
probandas  proferri  solebant,  satis  luculenter  exposui  per  solas  velocitates  a  viribus  non 
vivis  inductas. 


A  THEORY  OF  NATURAL  PHILOSOPHY 


221 


FIG.  48. 


FIG.  49. 


FIG.  50. 


FIG.  51. 


EL. 


FIG.  52. 


222 


PHILOSOPHIC  NATURALIS  THEORIA 


P 


B 


FIG.  48. 


FIG.  49. 


FIG.  50. 


FIG.  51. 


FIG.  52. 


A  THEORY  OF  NATURAL  PHILOSOPHY  223 

289.  Further,  as  regards  resolution  into  several  forces  or  motions,  it  is  easy,  from  A  general  method 
what  has  been  said  in  Art.  257,  to  determine  a  law  which  will  rightly  govern  such  resolution,  ^to^yTu^ber^f 
so  that  the  forces  which  compose  any  given  force  may  be  obtained.     In  Fig.  48,  let  AP  other  forces. 

be  any  force,  or  motion  ;  starting  from  A,  draw  any  number  of  straight  lines  of  any  length, 
AB,  BC,  CD,  DE,  EF,  FG,  GP,  continuously  joining  one  another  so  that  they  start  from 
A  &  end  up  at  P.  Then,  if  to  these  lines  AB,  BC,  &c.,  straight  lines  Ac,  Ad,  &c.,  are  drawn, 
equal  &  parallel,  all  the  forces  represented  by  AB,  Ac,  Ad,  Ae,  A/,  Ag,  Ap,  will  compound 
into  a  force  AP.  From  this  it  is  clear  that,  to  make  up  any  force,  it  is  possible  to  assume 
any  forces,  &  any  number  of  them,  &  these  being  taken,  it  is  possible  to  find  one  other  force 
which  will  complete  the  composition.  For,  the  straight  lines  AB,  BC,  CD,  &c.,  can  be 
drawn  parallel  &  equal  to  any  given  lines  whatever,  &  when  finally  they  end  up  at  some 
point  G,  it  will  be  sufficient  to  add  the  force  represented  by  GP. 

290.  Moreover  the  particular  case  of  resolution  into  two  forces  only  is  contained  in  Derivation  of  the 
the  general  case.     This  can  be  accomplished  by  means  of  any  two  sides  of  any  triangle,  t^in'6 two  ^co- 
in Fig.  49,  if  AP  is  the  given  force,  &  any  triangle  ABP  is  constructed,  then  the  force  AP  tions  only. 

can  be  resolved  into  the  two  parts  AB,  BP ;  &  if  one  of  these  is  given,  the  other  also  is 
given.  This  indeed  is  manifest  even  from  the  composition  itself,  or  from  resolution  by 
means  of  the  parallelogram  ABPC,  which  can  be  completed  in  every  case  ;  in  this  AC 
is  parallel  &  equal  to  BP,  &  the  two  forces  AB,  AC  will  compound  into  the  force  AP.  The 
same  may  be  said  with  regard  to  motions. 

291.  Such  a  resolution  also  brines  out  clearly  the  reason  why  a  force  compounded  Why  the  resultant 

..  f  ,..,  •if-i-L-Ltr.  force    is    less  than 

from  forces  that  do  not  lie  in  the  same  straight  line  is  less  than  the  sum  of  these  components.  the  two  component 
These  are  indeed  partly  opposite  to  one  another;    &,  when  the  equals  &  opposites  have  forces  taken  to- 
cancelled  one  another,  there  remains  in  the  force  compounded  of  them  the  sum  of  the  g 
forces  that  agree  in  direction  or  the  difference  of  the  opposites  which  relate  to  the  components. 
For,  in  Figs.  50,  61,  62,  if  the  force  AP  is  compounded  from  the  forces  AB,  AC,  which  are 
sides  of  the  parallelogram  ABPC,  &  BE,  CF  are  drawn  perpendicular  to  AP,  E  &  F  falling 
between  A  &  P  in  Fig.  50,  at  A  &  P  in  Fig.  51,  &  beyond  them  in  Fig.  52  ;    then  it  is 
plain  that,  in  the  first  &  last  cases  the  triangles  AEB,  PFC  are  equal,  &  thus  the  forces  EB 
&  FC,  which  are  equal  &  opposite,  cancel  one  another.     But  the  force  AP  in  the  first  case 
is  the  sum  of  the  two  forces  AE,  AF  acting  in  the  same  direction  ;  it  is  equal  to  the  single 
force  AF  in  the  second  case  ;  &  in  the  third  case  it  is  equal  to  the  difference  of  the  opposite 
forces  AE,  AF. 

292.  In  resolution  there  is  indeed  some  sort  of  increase  of  force.     The  reason  for  The  reason  why  the 
this  is  that  mentally  we  add  on  other  equal  &  opposite  forces,  which  taken  together  cancel  ij^rJase^n6  resoiu° 
one  another,  &  thus  do  not  have  any  disturbing  effect.     Thus,  in  Fig.  52,  by  resolving  tion:  no  argument 
the  force  AP  into  the  two  forces  AB,  AC,  we  really  add  to  AP  the  two  equal  &  opposite  ^  "derfved^f rom 
forces  AE,  PF,  &,  in  addition,  in  a  direction  at  right  angles  to  AP,  the  two  forces  EB,  FC,  this." 

which  are  also  equal  &  opposite.  Now,  since  resolution  is  not  real,  but  only  imaginary, 
&  merely  used  for  the  purpose  of  making  the  solution  of  problems  easier ;  no  exception 
can  be  taken  on  this  account  to  the  usual  method  of  considering  forces  such  as  we  have 
hitherto  discussed,  such  as  exert  for  an  instant  of  time  merely  a  stress  or  a  pressure  ;  for 
which  reason  they  are  termed  dead  forces,  &  because,  whilst  they  last  for  a  continuous 
time  without  any  contrary  force  to  cancel  them,  they  yet  only  produce  velocity,  they  are 
looked  upon  as  the  causes  of  the  velocity  produced.  Nor  from  this  can  any  argument  be 
derived  in  favour  of  admitting  the  existence  of  those  forces,  which  were  first  introduced 
by  Leibniz,  &  called  by  him  living  forces.  These  forces  some  people  consider  must  at 
least  be  supposed  to  exist,  in  order  that  in  the  resolution  of  forces,  for  instance,  there  should 
not  be  obtained  an  effect  unequal  to  its  cause.  Now  the  effect  must  be  proportional,  & 
not  equal ;  also  it  must  be  proportional,  not  to  the  cause,  but  to  the  action  of  the  cause, 
where  an  action  of  this  kind  is  not  impeded,  either  wholly  or  in  part,  by  some  equal  & 
opposite  action,  which  happens,  as  we  have  seen,  in  oblique  composition.  But,  whatever 
may  be  the  various  arguments,  according  to  the  usual  opinion,  to  meet  the  difficulties  in 
the  case  of  resolution,  since,  in  my  Theory,  there  is  no  real  resolution,  there  is  no  difficulty, 
as  I  have  already  said. 

293.  Indeed  it  will  be  sufficiently  evident,  both  from  what  has  already  been  proved,  it  is  sufficient  to 
as  well  as  from  what  will  follow,  that  there  is  nowhere  any  sign  of  such  living  forces,  nor  xheory  that"  there 
is  there  any  necessity.    For  all  the  phenomena  of  Nature  depend  upon  motions  &  equilibrium,  do  not    exist    in 
&  thus  from  dead  forces  &  the  velocities  induced  by  the  action  of  such  forces.     For  this  forces.e 
reason,  in  the  dissertation  De  Firibus  Vivis,  which  was  what  led  me  to  this  Theory  thirteen 

years  ago,  I  asserted  that  there  are  no  living  forces  in  Nature,  &  that  many  things  were 
usually  brought  forward  to  prove  their  existence,  I  explained  clearly  enough  by  velocities 
derived  solely  from  forces  that  were  not  living  forces. 


224  PHILOSOPHIC  NATURALIS  THEORIA 

Tiobf Ce"a stl cT to  294"  Unum  hie  proferam,  quod  pertinet  ad  collisionem  globorum  elasticorum  obliquam, 

quatuor  giobos,  quae  compositionem  resolutioni  substitutam  illustrat.  Sint  in  fig.  53  triangula  ADB. 
BH9>  GML  rectan§ula  in  D>  H>  M  «a,  ut  latera  BD,  GH,  LM,  sint  aequalia  singula 
dimidiae  basi  AB,  ac  sint  BG,  GL,  LQ  parallels  AD,  BH,  GM.  Globus  A  cum  velocitate 
AB  =  2  incurrat  in  B  in  globum  C  sibi  aequalem  jacentem  in  DB  producta  :  ex  collisione 
obliqua  dabit  illi  velocitatem  CE  =  I,  aequalem  suae  BD,  quam  amittet,  &  progredietur 
per  BG  cum  velocitate  =  AD  =  \/  3.  Ibi  eodem  pacto  si  inveniat  globum  I,  dabit  ipsi 
velocitatem  IK  =  I,  amissa  sua  GH,  &  progredietur  per  GL  cum  V  2  ;  turn  ibi  dabit, 
globo  O  velocitatem  OP  =  I,  amissa  sua  LM,  &  abibit  cum  LQ  =  I,  quam  globo  R, 
directe  in  eum  incurrens,  communicabit.  Quare,  ajunt,  ilia  vi,  quam  habebat  cum  veloci- 
tate =  2,  communicabit  quatuor  globis  sibi  aequalibus  vires,  quse  junguntur  cum 
velocitatibus  singulis  =  I  ;  ubi  si  vires  fuerint  itidem  singulas  =  i,  erit  summa  virium  = 
4,  quae  cum  fuerit  simul  cum  velocitate  =  2,  vires  sunt,  non  ut  simplices  velocitates  in 
massis  sequalibus,  sed  ut  quadrata  velocitatum. 


Ejus  explicate  in          295.  At  in  mea  Theoria  id  argumentum  nullam  sane  vim  habet.     Globus  A  non 

!dribulheviVis  t«  transfer!  in  globum  C  partem  DB  suae  velocitatis  AB  resolutse  in  duas  DB,  TB,  &  cum 

soiam     compositi-  ea  partem  suae  vis.     Agit  in  globos  vis  nova  mutua  in  partes  oppositas,  quae  alteri  imprimit 

velocitatem  CE,  alteri  BD.     Velocitas  prior  globi  A  expressa  per  BF  positam  in  directum 

cum  AB,  &  ipsi  aequalem,  componitur  cum  hac  nova  accepta  BD,  &  oritur  velocitas  BG 

minor  ipsa  BF  ob  obliquitatem  compositionis.     Eodem  pacto  nova  vis  mutua  agit  in  globos 

in  G,  &  I,  in  L,  &  O,  in  Q,  &  R,  &  velocitates  novas  primi  globi  GL,  LQ,  zero,  componuht 

velocitates   GH,  &   GN ;    LM,  &  LS  ;    LQ,  &  QL,  sine  ulla  aut  vera   resolutione,  aut 

translatione  vis  vivae,  Natura  in  omni  omnino  casu,  &  in  omni  corporum  genere  agente 

prorsus  eodem  pacto. 

Quid    notandum  296.  Sed  quod  attinet  ad  collisiones  corporum,  &  motus  [139]  reflexes,  unde  digressi 

sunTSobTcontimli1  eramus  '•>  inprimis  illud  monendum  duco  ;  cum  nulli  mihi  sint  continui  globi,  nulla  plana 
autpianacontinua',  continua  ;  pleraque  ex  illis,  quae  dicta  sunt,  habebunt  locum  tantummodo  ad  sensum, 
contactushematl°US  &  Proxime  tantummodo,  non  accurate  ;  nam  intervalla,  quae  habentur  inter  puncta, 
inducent  inaequalitates  sane  multas.  Sic  etiam  in  fig.  43.  ubi  globus  delatus  ad  B  incurrit 
in  CD,  mutatio  viae  directionis  non  fiet  in  unico  puncto  B,  sed  per  continuam  curvaturam ; 
ac  ubi  globus  reflectetur,  ipsa  reflexio  non  fiet  in  unico  puncto  B,  sed  per  curvam  quandam. 
Recta  AB,  per  quam  globus  adveniet,  non  erit  accurate  recta,  sed  proxime  ;  nam  vires  ad 
distantias  omnes  constant!  lege  se  extendunt,  sed  in  majoribus  distantiis  sunt  insensibiles ; 
nisi  massa,  in  quam  tenditur,  sit  enormis,  ut  est  totius  Terrae  massa  in  quam  sensibili  vi 
tendunt  gravia.  At  ubi  globus  advenerit  satis  prope  planum  CD  ;  incipiet  incurvari 
etiam  via  centri,  quae  quidem,  jam  attracto,  jam  repulso  globo,  serpet  etiam,  donee  alicubi 
repulsio  satis  praevaleat  ad  omnem  ejus  perpendicularem  velocitatem  extinguendam  (utar 
enim  imposterum  etiam  ego  vocabulis  communibus  a  virium  resolutione  petitis,  uti  & 
superius  aliquando  usu  fueram,  &  nunc  quidem  potiore  jure,  posteaquam  demonstravi 
aequipollentiam  verse  compositionis  virium  cum  imaginaria  resolutione),  &  retro  etiam 
motum  reflectat. 


Lex  reflexionis  297.  Et  quidem  si  vires  in  accessu  ad  planum,  ac  in  recessu  a  piano  fuerint  prorsus 
^q113!68  inter  se  ;  dimidia  curva  ab  initio  sensibilis  curvaturae  usque  ad  minimam  distantiam 
a  piano  erit  prorsus  aequalis,  &  similis  reliquae  dimidise  curvae,  quae  habebitur  inde  usque 
ad  finem  curvaturae  sensibilis,  ac  angulus  incidentiae  erit  sequalis  angulo  reflexionis.  Id 
in  casu,  quem  exprimit  fig.  43,  curva  ob  insensibilem  ejus  tractum  considerata  pro  unico 
puncto,  pro  perfecte  elasticis  patet  ex  eo,  quod  in  triangulis  rectangulis  AFB,  MIB  latera 
aequalia  circa  angulos  rectos  secum  trahant  aequalitatem  angulorum  ABF,  MBI,  quorum 
alter  dicitur  angulus  incidentiae,  &  alter  reflexionis,  ubi  in  imperfecte  elasticis  non  habetur 
ejusmodi  aequalitas,  sed  tantummodo  constans  ratio  inter  tangentem  anguli  incidentiae, 
&  tangentem  anguli  reflexionis,  quae  nimirum  ad  radios  sequales  BF,  BI  sunt  FA,  &  Im, 
&  sunt  juxta  denominationem,  quam  supra  adhibuimus  num.  272,  &  retinuimus  hue  usque, 
ut  772  ad  n. 


A  THEORY  OF  NATURAL  PHILOSOPHY 


225 


FIG.  53- 


Q 


226 


PHILOSOPHIC  NATURALIS  THEORIA 


FIG.  53. 


A  THEORY  OF  NATURAL  PHILOSOPHY 


227 


294.  I  will  bring  forward  here  one  example,  which  deals  with  the  oblique  impact  of 
elastic  spheres ;    this  will  illustrate  the  substitution  of  composition  for  resolution.     In 
Fig.  53,  let  ADB,  BHG,  GML,  be  right-angled  triangles  such  that  the  sides  BD,  GH,  LM 
are  each  equal  to  half  the  base  AB,  &  let  BG,  GL,  LQ  be  parallel  to  AD,  BH,  GM.     Suppose 
the  sphere  A,  moving  with  a  velocity  =  2,  to  impinge  at  B  upon  a  sphere  C,  equal  to  itself, 
lying  in  DB  produced.     From  the  oblique  impact,  it  will  impart  to  C  a  velocity  CE  =  i, 
which  is  equal  to  its  own  velocity  BD,  which  it  loses ;  &  it  itself  will  go  on  along  BG  with 
a  velocity  equal  to  AD  =  -\/3-     It  will  then  come  to  the  sphere  I,  will  give  to  it  a  velocity 
IK  =  I,  losing  its  own  velocity  GH,  &  will  go  on  along  GL  with  a  velocity  equal  to  \/2. 
Then  it  will  give  the  sphere  O  a  velocity  OP  =  i,  losing  its  own  velocity  LM,  &  will  go  on 
with  a  velocity  LQ  =  I.     This  it  will  give  up  to  the  sphere  R,  on  which  it  impinges  directly. 
Wherefore,  they  contest,  by  means  of  the  force  which  it  had  in  connection  with  a  velocity 

=  2,  it  will  communicate  to  four  spheres  equal  to  itself  forces,  each  of  which  is  conjoined 
with  a  velocity  =  i  ;  hence,  since,  if  each  of  the  forces  were  also  equal  to  I,  their  sum 
would  be  equal  to  4,  &  this  sum  was  at  the  same  time  connected  with  a  velocity  =  2,  it 
must  be  that  the  forces  are  not  in  the  simple  ratio  of  the  velocities  in  equal  masses  but  as 
their  squares. 

295.  But  in  my  Theory  this  argument  has  no  weight  at  all.     The  sphere  A  does  not 
transfer  to  the  sphere  C  that  part  DB  of  its  velocity  AB  resolved  into  the  two  parts  DB, 
TB  ;  &  with  it  part  of  its  force.     There  acts  on  the  spheres  a  new  mutual  force  in  opposite 
directions,  which  gives  the  velocity  CE  to  the  one  sphere,  &  the  velocity  BD  to  the  other. 
The  previous  velocity  of  the  sphere  A,  represented  by  BF  lying  in  the  same  direction  as, 
and  equal  to,  AB,  is  compounded  with  the  newly  received  velocity  BD,  and  the  velocity 
BG,  less  than  BF  on  account  of  the  obliquity  of  the  composition,  is  the  result.     In  the 
same  way,  a  new  mutual  force  acts  upon  the  spheres  at  G  &  I,  at  L  &  O,  at  Q  &  R,  &  the 
new  velocities  of  the  first  sphere,  GL,  LQ  &  zero,  are  the   resultants   of   the   velocities 
GH  &  GN,  LM  &  LS,  &  LQ  &  QL  respectively  ;  &  there  is  not  either  any  real  resolution, 
or  transference  of  living  force.    'Nature  in  every  case  without  exception,  &  for  all  classes 
of  bodies  acts  in  exactly  the  same  manner. 

296.  But  we  have  digressed  from  the  consideration  of  impact  of  bodies  &  reflected 
motions.     Returning  to  them,  I  will  first  of  all  bring  forward  a  point  to  be  noted  carefully. 
Since,  to  my  idea,  there  are  no  such  things  as  continuous  spheres  or  continuous  planes, 
many  of  the  things  that  have  been  said  are  only  true  as  far  as  we  can  observe,  &  only  very 
approximately  &  not  accurately ;   for  the  intervals,  which  exist  between  the  points,  induce 
a  large  number  of  inequalities.     So  also,  in  Fig.  43,  where  the  sphere  carried  forward  to 
B  impinges  upon  the  plane  CD,  the  change  in  the  direction  of  the  path  will  not  take  place 
at  the  single  point  B,  but  by  means  of  a  continuous  curvature.     Also  in  the  case  where 
the  sphere  is  reflected,  the  reflection  will  not  occur  at  the  single  point  B,  but  along  a  certain 
curve.     The  straight  line  AB,  along  which  the  sphere  is  approaching,  will  not  accurately 
be  a  straight  line,  but  approximately  so  ;   for  the  forces  extend  to  all  distances  according 
to  a  fixed  law,  but  at  fairly  great  distances  are  insensible,  unless  the  mass  it  is  approaching 
is  enormous,  as  in  the  case  of  the  whole  Earth,  to  which  heavy  bodies  tend  to  approach 
with  a  sensible  force.     But  as  soon  as  the  sphere  comes  sufficiently  near  to  the  plane  CD, 
the  path  to  the  centre  will  begin  to  be  curved,  &  indeed,  as  the  sphere  is  first  attracted 
&  then  repelled,  the  path  will  be  winding,  until  it  reaches  a  distance  at  which  the  repulsion 
will  be  strong  enough  to  destroy  all  its  perpendicular  velocity  (for  in  future  I  also  will  use 
the  usual  terms  derived  from  resolution  of  forces,  as  I  did  once  or  twice  in  what  has  been 
given  above  ;    &  this  indeed  I  shall  now  do  with  greater  justification  seeing  that  I  have 
proved  the  equivalence  between  true  composition  &  imaginary  resolution),  &  also  will 
reflect  the  motion. 

297.  Indeed,  if  the  forces  during  the  approach  towards  the  plane  &  those  during  the 
recession  from  it  were  exactly  equal  to  one  another,  then  the  half  of  the  curve  starting 
from  the  beginning  of  sensible  curvature  up  to  the  least  distance  from  the  plane  would 
be  exactly  equal  &  similar  to  the  other  half  of  the  curve  from  this  point  to  the  end  of  sensible 
curvature,  &  the  angle  of  incidence  would  be  equal  to  the  angle  of  reflection.     This,  in 
the  case  for  which  Fig.  43  is  drawn,  where  on  account  of  the  insensible  length  of  its  arc 
the  curve  is  considered  as  a  single  point,  is  evidently  true  for  perfectly  elastic  bodies,  from 
the  fact  that  in  the  right-angled  triangles  AFB,  MIB,  the  equal  sides  about  the  right  angles 
involve  the  equality  of  the  angles  ABF,  MBI,  of  which  the  first  is  called  the  angle  of  incidence 
&  the  second  that  of  reflection  ;    whereas,  in  imperfectly  elastic  bodies,  there  is  no  such 
equality,  but  only  a  constant  ratio  between  the  tangents  of  the  angle  of  incidence  &  the 
tangent  of  the  angle  of  reflection.     For  instance,  these  are,  measured  by  the  equal  radii 
BF,  BI,  equal  to  FA,  Im  ;  &  these  latter  are,  according  to  the  notation  used  above  in  Art. 
272,  &  retained  thus  far,  in  the  proportion  of  m  to  n. 


Oblique  impact  of 
a  sphere  on  four 
sph e r e s,  an  ex- 
ample usually 
brought  forward  in 
support  of  living 
forces. 


Its  explanation  in 
my  Theory  without 
living  forces  by 
means  of  compo- 
sition alone. 


It  is  therefore  to 
be  noted  that  there 
are  no  continuous 
spheres  or  con- 
tinuous planes,  nor 
such  a  thing  as 
mathematical  con- 
tact. 


Law  of  reflection 
for  perfectly  & 
imperfectly  elastic 
bodies. 


228  PHILOSOPHIC  NATURALIS  THEORIA 

Eadem    facta    vi  2g8.  Curvaturam  in  reflexione  exhibet  figura  154,  ubi  via  puncti  mobilis  repulsi  a  piano 

agente     in     ahqua    /~1/~.       f   \-n/-\-r^-\/r  •         T>       i  •     •         ...  JT  ..  .,  r    .       .    .        , 

distantia,  consider-  <~<~>  Ml  ArSvjJJM,  quae  circa  D,  ubi  vires  mcipiunt  esse  sensibiles,  incipit  ad  sensum  mcurvan, 
atacurvatura  &  desinit  in  eadem  distantia  circa  D.    Ea 

quidem,    si  habeatur  semper  repulsio, 

incurvatur  perpetuo  in  eandem  plagam, 

ut  figura  exhibet ;    si  vero  &  attractio 

repulsionibus   interferatur,    serpit,    uti 

monui ;  sed  si  paribus  a  piano  distantiis 

vires  aequales  sunt ;  satis  patet,  &  accu- 

ratissime  demonstrari  [140]  etiam  pos- 
set, ubi  semel  deventum  sit  alicubi,  ut 

in  Q,  ad  directionem  parallelam  piano, 

debere    deinceps    describi    arcum    QD 

prorsus  aequalem,  &  similem  arcui  QB, 

&  ita  similiter  positum   respectu  plani 

CO,   ut    ejus   inclinationes    ad   ipsum 


planum  in  distantiis  aequalibus  ab  eo, 
&  a  Q  hinc,  &  inde  sint  prorsus  aequales  ; 


R/I 
'*' 


quam  ob  causam  tangentes  BN,  DP, 
quae  sunt  quasi  continuationes  rectarum  AB,  MD,  angulos  faciunt  ANC,  MPO  aequales, 
qui  deinde  habentur  pro  angulis  incidentiae,  &  reflexionis. 

Quid,  si  planum  sit  299.  Si  planum  sit  asperum,  ut  Figura  exhibet,  &  ut  semper  contingit  in  Natura  ; 

catk^rene^Fonem  aequalitas  ilia  virium  utique  non  habetur.  At  si  scabrities  sit  satis  exigua  respectu  ejus 
lucU.  distantiae,  ad  quam  vires  sensibiles  protenduntur  ;  inaequalitas  ejusmodi  erit  perquam 

exigua,  &  anguli  incidentiae,  &  reflexionis  aequales  erunt  ad  sensum.  Si  enim  eo  intervallo 
concipiatur  sphaera  VRTS  habens  centrum  in  puncto  mobili,  cujus  segmentum  RTS  jaceat 
ultra  planum  ;  agent  omnia  puncta  constituta  intra  illud  segmentum,  adeoque  monticuli 
prominentes  satis  exigui  respectu  totius  ejus  massae,  satis  exiguam  inaequalitatem  poterunt 
inducere  ;  &  proinde  sensibilem  aequalitatem  angulorum  incidentias,  &  reflexionis  non 
turbabunt,  sicut  &  nostri  terrestres  montes  in  globo  oblique  projecto,  &  ita  ponderante, 
ut  a  resistentia  aeris  non  multum  patiatur,  sensibiliter  non  turbant  parabolicum  motum 
ipsius,  in  quo  bina  crura  ad  idem  horizontale  planum  eandem  ad  sensum  inclinationem 
habent.  Secus  accideret,  si  illi  monticuli  ingentes  essent  respectu  ejusdem  sphaerae.  Atque 
haec  quidem,  qui  diligentius  perpenderit,  videbit  sane,  &  lucem  a  vitro  satis  laevigato  resilire 
debere  cum  angulo  reflexionis  aequali  ad  sensum  angulo  incidentiae  ;  licet  &  ibi  pulvisculus 
quo  poliuntur  vitra,  relinquat  sulcos,  &  monticules,  sed  perquam  exiguos  etiam  respectu 
distantiae,  ad  quam  extenditur  sensibilis  actio  vitri  in  lucem  ;  sed  respectu  superficierum, 
quae  ad  sensum  scabrae  sunt,  debere  ipsam  lucem  irregulariter  dispergi  quaqua  versus. 
Quid  in  impactu  300.  Pariter  ubi  globus  non  elasticus  deveniat  per  AB  in  eadem  ilia  fig.  43,  &  deinde 

Hs^Tn  ^pianum1"  ^ebeat  sme  reflexione  excurrere  per  BQ,  non  describet  utique  rectam  lineam  accurate, 
veiocitas  amissa,  sed  serpet,  &  saltitabit  non  nihil  :  erit  tamen  recta  ad  sensum  :  velocitas  vero  mutabitur 
m"^u™atvira  "con*  *ta  '  ut  s^  vel°citas  P"01"  AB  ad  posteriorem  BI,  ut  radius  ad  cosinum  inclinationis  OBI 
tinua.  rectae  BO  ad  planum  CD,  ac  ipsa  velocitas  prior  ad  velocitatum  differentiam,  sive  ad  partem 

velocitatis  amissam,  quam  exprimit  IQ  determinata  ab  arcu  OQ  habente  centrum  in  B, 
erit  ut  radius  ad  sinum  versum  ipsius  inclinationis.  Quoniam  autem  imminuto  in  infinitum 
angulo,  sinus  versus  decrescit  in  infinitum  etiam  respectu  ipsius  arcus,  adeoque  summa 
omnium  sinuum  versorum  pertinentium  ad  omnes  inflexiones  infinitesimas  tempore  finito 
factas  adhuc  in  infinitum  decrescit  ;  ubi  inflexio  evadat  [141]  continua,  uti  fit  in  curvis 
continuis,  ea  summa  evanescit,  &  nulla  fit  velocitas  amissio  ex  inflexione  continua  orta,  sed 
vis  perpetua,  quae  tantummodo  ad  habendam  curvaturam  requiritur  perpendicularis  ipsi 
curvae,  nihil  turbat  velocitatem,  quam  parit  vis  tangentialis,  si  qua  est,  quae  motum  perpetuo 
acceleret,  vel  retardet  ;  ac  in  curvilineis  motibus  quibuscunque,  qui  habeantur  per  quas- 
cunque  directiones  virium,  semper  resolvi  potest  vis  ilia,  quae  agit,  in  duas,  alteram 
perpendicularem  curvas,  alteram  secundum  directionem  tangentis,  &  motus  in  curva  per 
hanc  tangentialem  vim  augebitur,  vel  retardabitur  eodem  modo,  quo  si  eaedem  vires  agerent, 
&  motus  haberetur  in  eadem  recta  linea  constanter.  Sed  hasc  jam  meae  Theoriae  communia 
sunt  cum  Theoria  vulgari. 

Theoremata  pro  301.  Communis  est  itidem  in  fig.  44,  &  45  ratio  gravitatis  absolutae  BO  ad  vim  BI,  quse 

scen^um^vef  retar-  obliquum  descensum  accelerat,  vel  ascensum  retardat,  quae  est,  ut  radius  ad  sinum  anguli 
dante  ascensum  in  BOI,  vel  OBR,  sive  cosinum  OBI.  Angulum  OBI  est  in  fig.  44,  quem  continet  directio 
&  BI>  quse  est  eadem,  ac  directio  plani  CD,  cum  linea  verticali  BO,  adeoque  angulus  OBR 
est  aequalis  inclinationi  plani  ad  horizontem,  &  angulus  idem  OBR  in  fig.  45  est  is,  quem 
continet  cum  verticali  BO  recta  CB  jungens  punctum  oscillans  cum  puncto  suspensionis. 
Quare  habentur  haec  theoremata  :  Fis  accelerans  descensum,  vel  retardans  ascensum  in  flanis 


A  THEORY  OF  NATURAL  PHILOSOPHY  229 

298.  Fig.  54  illustrates  the  curvature  in  reflection  ;  here  we  have  the  path  of  a  moving  The  case  of  a  force 
point  repelled  by  a  plane  CO  represented  by  ABQDM  ;    this,  near  B,  where  the  forces  sSbie^tance  • 
begin  to  be  sensible,  begins  to  be  appreciably  curved,  &  leaves  off  at  the  same  distance  consideration  of 
from  the  plane,  near  the  point  D.     The  path,  indeed,  if  there  is  always  repulsion,  will  be  the  path™ 
continuously  incurved  towards  the  same  parts,  as  is  shown  in  the  figure  ;   but  if  attraction 

alternates  with  repulsion,  the  path  will  be  winding,  as  I  mentioned.  However,  if  the 
forces  at  equal  distances  from  the  plane  are  equal  to  one  another,  it  is  sufficiently  clear, 
&  indeed  it  could  be  rigorously  proved,  that  as  soon  as  some  point  such  as  Q  was  reached 
where  the  direction  of  the  path  was  parallel  to  the  plane,  it  must  thereafter  describe  an 
arc  QD  exactly  equal  &  similar  to  the  arc  QB  ;  &  therefore  similarly  placed  with  .respect 
to  the  plane  CO ;  so  that  the  inclinations  of  the  parts  at  equal  distances  from  the  plane, 
&  fromQ  on  either  side,  are -exactly  equal.  Hence,  the  tangents  BN,  DP,  which  are  as  it 
were  continuations  of  the  straight  lines  AB,  MD,  will  make  the  angles  ANC,  MPO  equal  to 
one  another  ;  &  these  may  then  be  looked  upon  as  the  angles  of  incidence  &  reflection. 

299.  If  the  plane  is  rough,  as  is  shown  in  the  figure,  &  such  as  always  occurs  in  Nature,  What  if  the  plane 
there  will  in  no  case  be  this  equality  of  forces.     But  if  the  roughness  is  sufficiently  slight  tion^to 

in  comparison  with  that  distance,  over  which  sensible  forces  are  extended,  such  inequality  tion  of  light. 

will  be  very  slight,  &  the  angle  of  incidence  will  be  practically  equal  to  the  angle  of  reflection. 

For  if  with  a  radius  equal  to  that  distance  we  suppose  a  sphere  VRTS  to  be  drawn,  having 

its  centre  at  the  position  of  the  moving  point,  &  a  segment  RTS  lying  on  the  other  side  of 

the  plane  ;   then  all  the  points  contained  within  that  segment  exert  forces ;  &,  if  therefore 

the  little  prominences  are  sufficiently  small  compared  with  the  whole  mass,  they  can  only 

induce  quite  a  slight  inequality.     Hence,  they  will  not  disturb  the  sensible  equality  of  the 

angles  of  incidence  &  reflection  ;   just  as  the  mountains  on  our  Earth,  acting  on  a  sphere 

projected  in  a  direction  inclined  to  the  vertical,  &  of  such  a  weight  that  it  does  not  suffer 

much  from  the  resistance  of  the  air,  do  not  sensibly  disturb  its  parabolic  motion,  in  which 

the  two  parts  of  the  parabola  have  practically  the  same  inclination  to  the  same  horizontal 

plane.     It  would  be  quite  another  matter,  if  the  little  prominences  were  of   large  size 

compared  with  the  sphere.     Anyone  who  will  study  these  matters  with  considerable  care 

will  perceive  clearly  that  light  also  must  rebound  from  a  sufficiently  well  polished  piece  of 

glass  with  the  angle  of  reflection  to  all  intents  equal  to  the  angle  of  incidence.     Although 

it  is  true  that  the  powder  with  which  glasses  are  polished  leaves  little  furrows  &  prominences ; , 

but  these  are  always  very  slight  compared  with  the  distance  over  which  the  sensible  action 

of  glass  on  light  extends.     However,  for  surfaces  that  are  sensibly  rough,  it  will  be  perceived 

that  light  must  be  scattered  irregularly  in  all  directions. 

300.  Similarly,  when  a  non-elastic  sphere  travels  along  AB,  in  Fig.  43,  &  then  without  What  happens  in 
reflection  has  to  continue  along  BQ,  it  will  not  describe  a  perfectly  accurate  straight  line,  SlLS?9  ~«  »bh3"ft 

......  ,      ,  ,        ..  .-M    i  11    •  •    i       ""pact    in  a    soit 

but  will  wind  irregularly  to  some  extent  ;  yet  the  line  will  be  to  all  intents  a  straight  sphere ;  the  veio- 

line.      Moreover,  the  velocity  will  be  changed  in  such  a  way  that  the  previous  velocity  ma^ns'^unTm'pairwi 

AB  will  be  to  the  new  velocity  BI,  as  the  radius  is  to  the  cosine  of  OBI  the  inclination  in  continuous  cur- 

of  the  straight  line  BO  to  the  plane  CD  ;    &  the  previous   velocity  is    to   the  difference  vature- 

between  the   velocities,  i.e.,  to   the   velocity  that   is  lost,   which  is   represented  by  IQ 

determined  by  the  arc  OQ  having  its  centre  at  B,  as  the  radius  is  to- the  versine  of  the 

same  angle.     Now,  since,  when  the  angle  is  indefinitely  diminished,  the  versine  decreases 

indefinitely  with  respect  to  the  arc  itself,  &  thus  the  sum  of  all  the  versines  belonging  to 

all  the  infinitesimal  inflections  made  in  a  finite  time  still  decreases  indefinitely ;  it  follows 

that,  when  the  inflexion  becomes  continuous,  as  is  the  case  with  continuous  curves,  this 

sum  vanishes,  &  therefore  there  is  no  loss  of  velocity  arising  from  continuous  inflection. 

There  is  a  perpetual  force,  which  is  required  for  the  purpose  of  keeping  up  the  curvature, 

perpendicular  to  the  curve  itself,  &  therefore  not  disturbing  the    velocity  at  all ;  the 

velocity  arises  from  a  tangential  force,  if  there  is  any,  &  this  continuously   accelerates 

or  retards  the  motion.      In  curvilinear  motions  of  all  kinds,  due  to  forces  in  all  kinds  of 

directions,  it  is  always  possible  to  resolve  the  force  acting  into  two  parts,  one  of  them 

perpendicular  to  the  curve,  &  the  other  along  the  tangent ;   the  motion  along  the  curve 

will  be  increased  or  retarded  by  the  tangential  force,  in  precisely  the  same  manner  as  if  these 

same  forces   acted  &  the  motion  was  constantly  in  the  same  straight  line.      But  all  these 

matters  are  common  to  my  theory  and  the  usual  theory. 

301.  In  Fig.  44, 45,  there  is  a  common  ratio  between  the  absolute  gravity  BO  &  the  force  The  o'rems  with 

T>T       i  •   i  11  „      i  •  •      •  i  i  r     i        regard  to  the  force 

Bl,  which  accelerates  the  descent  or  retards  the  ascent;  &  this  ratio  is  equal  to  that  of  the  accelerating  de- 
radius  to  the  sine  of  the  angle  BOI,  or  OBR,  or  the  cosine  of  OBI.     The  angle  OBI  is,  in  scent  or  retarding 

TI.  t  ,.,.  -ill        T          •         T>T        i-i-i  IT          •  r    ascent  in  the  cases 

rig.  44,  that  which  is  contained  by  the  direction  BI,  which  is  the  same  as  the  direction  of  Of    the    inclined 
the  plane  CD,  with  the  vertical  line  BO  ;  &  thus  the  angle  OBR  is  equal  to  the  inclination  Planf    &    ° f   the 
of  the  plane  to  the  horizon  ;  &  the  same  angle  OBR,  in  Fig.  45,  is  that  which  is  contained  r 
by  the  vertical  BO  with  the  straight  line  CB,  which  joins  the  point  of  oscillation  with  the 
point  of  suspension.     Hence,  we  have  the  following  theorems.     The  force  accelerating  descent, 


230  PHILOSOPHIC  NATURALIS  THEORIA 

inclinatis,  vel  ubi  oscillatio  fit  in  arcu  circulari,  est  ad  gravitatem  absolutam,  ibi  quidem  ut 
sinus  inclinationis  ipsius  plani,  hie  vero  ut  sinus  anguli,  quern  cum  verticali  linea  continet  recta 
jungens  punctum  oscillans  cum  puncto  suspensionis,  ad  radium.  E  quorum  theorematum 
priore  fluunt  omnia,  quae  Galilaeus  tradidit  de  descensu  per  plana  inclinata  ;  ac  e  posteriore 
omnia,  quae  pertinent  ad  oscillationes  in  circulo  ;  quia  immo  etiam  ad  oscillationes  factas 
in  curvis  quibuscunque  pondere  per  filum  suspense,  &  curvis  evolutis  applicato  ;  ac  eodem 
utemur  infra  in  definiendo  centre  oscillationis. 


Appiicatio  Theoriae  302.  Hisce  perspectis,  applicanda  est  etiam  Theoria  ad  motuum  refractionem,  ubi 

tres  rasus^veioci-  continentur  elementa  mechanica  pro  refractione  luminis,  &  occurrit  elegantissimum 
tatis  normaiis  ex-  theorema  a  Newtono  inventum  hue  pertinens.  Sint  in  fig.  55  binae  superficies  AB,  CD 
'  l  t£e>  parallelae  inter  se,  &  punctum  mobile  quodpiam  extra  ilia  plana  nullam  sentiat  vim,  inter 
ipsa  vero  urgeatur  viribus  quibuscunque,  quae  tamen  &  semper  habeant  directionem 
perpendicularem  ad  ipsa  plana,  &  in  asqualibus  distantiis  ab  altero  ex  iis  asquales  sint  ubique  ; 
ac  mobile  deferatur  ad  alterum  ex  iis,  ut  AB,  directione  quacunque  GE.  Ante  appulsum 
feretur  motu  rectilineo,  &  sequabili,  cum  nulla  urgeatur  vi  :  ejus  velocitatem  exprimat  EH, 
quas  erecta  ER,  perpendiculari  ad  AB,  resolvi  poterit  in  duas,  alteram  perpendicularem 
ES,  alteram  parallelam  HS.  Post  ingressum  inter  alia  duo  [142]  plana  incurvabitur  motus 
illis  viribus,  sed  ita,  ut  velocitas  parallela  ab  iis  nihil  turbetur,  velocitas  autem  perpendicularis 
vel  minuatur,  vel  augeatur ;  prout  vires  tendent  versus  planum  citerius  AB,  vel  versus 
ulterius  CD.  Jam  vero  tres  casus  haberi  hinc  possunt ;  vel  enim  iis  viribus  tota  velocitas 
perpendicularis  ES  extinguitur,  antequam  deveniatur  ad  planum  ulterius  CD  ;  vel  perstat 
usque  ad  appulsum  ad  ipsum  CD,  sed  imminuta,  vi  contraria  praevalente  viribus  eadem 
directione  agentibus ;  vel  perstat  potius  aucta. 


Primo  reflexionem  303.  In  primo  casu,  ubi  primum  velocitas  perpendicularis  extincta  fuerit  alicubi  in 

X,  punctum  mobile  reflectet  cursum  retro  per  XI,  &  iisdem  viribus  agentibus  in  regressu, 
quae  egerant  in  progressu,  acquiret  velocitatem  perpendicularem  IL  asqualem  amissae  ES, 
quas  composita  cum  parallela  LM,  sequali  priori  HS,  exhibebit  obliquam  IM  in  recta  nova 
IK,  quam  describet  post  egressum,  &  erunt  aequales  anguli  HIL,  MES,  adeoque  &  anguli 
KIB,  GEA  ;  quod  congruit  cum  iis,  quae  in  fig.  54.  sunt  exhibita,  &  pertinent  ad 
reflexionem. 


-  3°4"  ^n  secundo  casu  prodibit  ultra  superficiem  ulteriorem  CD,  sed  ob  velocitatem 

cessu  ad  superficiem  perpendicularem  OP  minorem  priore  ES,  parallelam  vero  PN  sequalem  priori  HS,  erit 
iUd^enterefractio°  angulus  ONP  minor,  quam  EHS,  adeoque  inclinatio  VOD  ad  superficiem  in  egressu  minor 
nem,    sed    cum  inclinatione  GEA  in  ingressu.     Contra  vero  in  tertio  casu  ob  op  majorem  ES,  angulus 
udD  erit  major.     In  utroque  autem  hoc  casu  differentia  quadratorum  velocitatis  ES,  & 
OP  vel  op,  erit  constans,  per  num.  177  in  adn.  m,  quscunque   fuerit  inclinatio   GE   in 
ingressu,  a  qua  inclinatione  pendet  velocitas  perpendicularis  SE. 

sufuV°angu?i1Suici-  3°5-  Inde  autem  facile  demonstratur,  fore  sinum  anguli  incidentiae  HES,  ad  sinum 
dentiae,  ad  sinum  anguli  refracti  PON  (&  quidquid  dicitur  de  iis,  quae  designantur  litteris  PON,  erunt  com- 
munia  iis,  quae  exprimuntur  litteris  pon)  in  ratione  constanti,  quaecunque  fuerit  inclinatio 
rectae  incidentis  GE.  Sumatur  enim  HE  constans,  quae  exprimat  velocitatem  ante 
incidentiam  :  exprimet  HS  velocitatem  parallelam,  quae  erit  aequalis  rectae  PN  exprimenti 
velocitatem  parallelam  post  refractionem  ;  ac  ES,  OP  expriment  velocitates  perpendiculares 
ante,  &  post,  quarum  quadrata  habebunt  differentiam  constantem.  Sed  ob  HS,  PN  semper 
aequales,  differentia  quadratorum  HE,  ON  aequatur  differentiae  quadratorum  ES,  OP. 
Igitur  etiam  differentia  quadratorum  HE,  ON  erit  constans  ;  cumque  ob  HE  constantem 
debeat  esse  constans  ejus  quadratum  ;  erit  constans  etiam  quadratum  ON,  adeoque  constans 
etiam  ipsa  ON,  &  proinde  constans  erit  &  ratio  HE  ad  ON  ;  quas  quidem  ratio  est  eadem, 
ac  sinus  anguli  NOP  ad  sinum  HES  :  cum  enim  sit  in  quovis  triangulo  rectangulo  radius 
ad  latus  utrumvis,  ut  basis  ad  sinum  anguli  oppositi  ;  in  diversis  triangulis  rectangulis 
sunt  sinus,  ut  latera  opposita  divisa  per  [143]  bases,  sive  directe  ut  latera,  &  reciproce  ut 
bases,  &  ubi  latera  sunt  sequalia,  ut  hie  HS,  PN,  erunt  reciproce  ut  bases. 


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PHILOSOPHISE  NATURALIS  THEORIA 


R 


H     S 


L     M 


FIG.  55. 


A  THEORY  OF  NATURAL  PHILOSOPHY  233 

or  retarding  ascent,  on  inclined  planes,  or  where  there  is  oscillation  in  a  circular  arc,  is  to  the 
absolute  gravity,  in  the  first  case  as  the  sine  of  the  inclination  of  the  plane  to  the  radius,  &  in 
the  second  case  as  the  sine  of  the  angle  between  the  vertical  i3  the  line  joining  the  oscillating 
'point  to  the  -point  of  suspension,  is  to  the  radius.  From  the  first  of  these  theorems  there 
follow  immediately  all  that  Galileo  published  on  the  descent  along  inclined  planes  ;  & 
from  the  second,  all  matters  relating  to  oscillations  in  a  circle.  Moreover,  we  have  also 
all  matters  that  relate  to  oscillations  made  in  curves  of  all  sorts  by  a  weight  suspended  by  a 
string  wrapped  round  in  volute  curves  ;  &  we  shall  make  use  of  the  same  idea  later  to  define 
the  centre  of  oscillation. 

302.  These  matters  being  investigated,  we  now  have  to  apply  the  Theory  to  the  refraction  Application  of  the 
of  motions,  in  which  are  contained  the  mechanical  principles  of  the  refraction  of  light  ;  Son^the  "three 
here  also  we  find  a  most  elegant  theorem  discovered  by  Newton,  referring  to  the  subject,  cases  in  which  the 
In  Fig.  55,  let  AB,  CD  be  two  surfaces  parallel  to  one  another  ;  &  let  a  moving  point  feel 


the   action  of  no  force  when  outside  those  planes,  but  when  between   the  two  planes  diminished,  or 

suppose  it  is  subject  to  any  forces,  so  long  as  these  always  have  a  direction  perpendicular  increased- 

to  the  planes,  &  they  are  always  equal  at  equal  distances  from  either  of  them.     Suppose 

the  point  to  approach  one  of  the  planes,  AB  say,  in  any  direction  GE.     Until  it  reaches 

AB  it  will  travel  with  rectilinear  &  uniform  motion,  since  it  is  acted  upon  by  no  force  ; 

let  EH  represent  its  velocity.     Then,  if  ER  is  erected  perpendicular  to  the  plane  AB,  the 

velocity  can  be  resolved  into  two  parts,  the  one,  ES,  perpendicular  to,  &  the  other,  HS, 

parallel  to,  the  plane  AB.     After  entry  into  the  space  between  the  two  planes  the  motion 

will  be  incurved  owing  to  the  action  of  the  forces  ;  but  in  such  a  manner  that  the  velocity 

parallel  to  the  plane  will  not  be  affected  by  the  forces  ;  whereas  the  perpendicular  velocity 

will  be  diminished  or  increased,  according  as  the  forces  act  towards  the  plane  AB,  or  towards 

the  plane  CD.     Now  there  are  three  cases  possible  ;   for,  the  whole  of  the  perpendicular 

velocity  may  be  destroyed  before  the  point  reaches  the  further  plane  CD,  or  it  may  persist 

right  up  to  contact  with  the  plane  CD,  but  diminished  in  magnitude,  owing  to  a  force 

existing  contrary  to  the  forces  in  that  direction,  or  it  may  continue  still  further  increased. 

303.  In  the  first  case,  where  the  perpendicular  velocity  was  first  destroyed  at  a  point  in  the   first  case 
X,  the  moving  point  will  follow  a  return  path  along  XI  ;   &  as  the  same  forces  act  in  the  d^edCti°n  **  '"" 
backward  motion  as  in  the  forward  motion,  the  point  will  acquire  a  perpendicular  velocity 

IL,  equal  to  ES,  that  which  it  lost  ;  this,  compounded  with  the  parallel  velocity  LM, 
equal  to  the  previous  parallel  velocity  HS,  will  give  a  velocity  IM,  in  an  oblique  direction 
along  the  new  straight  line  IK,  along  which  the  point  will  move  after  egress.  Now  the 
angles  HIL,  MES  will  be  equal,  &  therefore  also  the  angles  KIB,  GEA  ;  this  agrees  with 
what  is  represented  in  Fig.  54,  &  pertains  to  reflection. 

304.  In  the  second  case,  the  point  will  proceed  beyond  the  further  surface  CD  ;   but,  In  *he  second  case 

•  1-11-       /^vV>  •  11  i  •  T-«I        i  •»        i  it  i    we  have  refraction 

since  the  perpendicular  velocity  Or  is  now  less  than  the  previous  one  ES,  whilst  the  parallel  &  nearer  approach 
velocity  is  the  same  as  the  previous  one  HS,  the  angle  ONP  will  be  less  than  the  angle  EHS,  *°rfthe.  re.fracti?g 
&  therefore  the  inclination  to  the  surface,  VOD,  on  egress,  will  be  less  than  the  inclination,  third,  refraction  & 
GEA,  on  ingress.     On  the  other  hand,  in  the  third  case,  since  op  is  greater  than  ES,  the  recession  from  the 
angle  uoD  will  be  greater  than  the  angle  GEA.     But  in  either  case,  we  here  have  the  difference 
between  the  squares  of  the  velocity  ES,  &  that  of  OP,  or  op,  constant,  as  was  shown  in 
Art.  177,  note  m,  whatever  may  be  the  inclination  on  ingress,  made  by  GE  with  the  plane, 
upon  which  inclination  depends  the  perpendicular  velocity  SE. 

305.  Further,  from  this  it  is  easily  shown  that  the  sine  of  the  angle  of  incidence  HES  The  constant  ratio 
is  to  the  sine  of  the  angle  of  refraction  HON  (&  whatever  is  said  with  regard  to  these  angles,  alguT  o^taddence 
denoted  by  the  letters  PON,  will  hold  good  for  the  angles  denoted  by  the  letters  pan},  in  to  the  sine  of  the 
a  constant  ratio,  whatever  the  inclination  of  the  line  of  incidence,  GE,  may  be.     For,  "**" 
suppose  HE,  which  represents  the  velocity  before  incidence,  to  be  constant  ;    then  HS, 
representing  the  parallel  velocity,  will  be  equal  to  PN,  which  represents  the  parallel  velocity 

after  refraction.  Now,  if  ES,  OP  represent  the  perpendicular  velocities  before  &  after 
refraction,  they  will  have  the  difference  between  their  squares  constant.  But,  since  HS, 
PN  are  equal,  the  difference  between  the  squares  of  HE,  ON  will  be  equal  to  the  difference 
between  the  squares  of  ES,  OP.  Hence  the  difference  of  the  squares  of  HE,  ON  will  be 
constant.  But,  since  HE  is  constant,  its  square  must  also  be  constant  ;  therefore  the 
square  of  ON,  &  thus  also  ON  itself,  must  be  constant.  Therefore  also  the  ratio  of  HE 
to  ON  is  constant  ;  &  this  ratio  is  the  same  as  that  of  the  sine  of  the  angle  NOP  to  the 
sine  of  the  angle  HES.  For,  since  in  any  right-angled  triangle  the  ratio  of  the  radius  to 
either  side  is  that  of  the  base  to  the  angle  opposite,  in  different  right-angled  triangles, 
the  sines  vary  as  the  sides  opposite  them  divided  by  the  bases,  or  directly  as  the  sides  & 
inversely  as  the  bases  ;  &  where  the  sides  are  equal,  as  HS,  PN  are  in  this  case,  the  sines 
vary  as  the  bases. 


234 


PHILOSOPHIC  NATURALIS  THEORIA 


&nrati?ve°io- 
citatum  reciproca 
ratbnis  sinuum. 


3°6'  °-uamobrem  in  refractionibus,  quae  hoc  modo  fiant  motu  libero  per  intervallum 
inter  duo  plana  parallela,  in  quo  vires  paribus  distantiis  ab  altero  eorum  pares  sint,  ratio 
sinus  anguli  incidentiae,  sive  anguli,  quern  facit  via  ante  incursum  cum  recta  perpendiculari 
piano,  ad  sinum  anguli  refracti,  quern  facit  via  post  egressum  itidem  cum  vertical!,  est 
constans,  quaecunque  fuerit  inclinatio  in  ingressu.  Praeterea  vero  habetur  &  illud,  fore 
celeritates  absolutas  ante,  &  post  in  ratione  reciproca  eorum  sinuum.  Sunt 
ejusmodi  velocitates  ut  HE,  ON,  quae  sunt  reciproce  ut  illi  sinus. 


enim 


Haec  q11^6111  ad  luminis  refractiones  explicandas  viam  sternunt,  ac  in  Tertia 
open'  occasionem  Parte  videbimus,  quo  pacto  hypothesis  hujusce  theorematis  applicetur  particulis  luminis. 
Sed  interea  considerabo  vires  mutuas,  quibus  in  se  invicem  agant  tres  massas,  ubi  habebuntur 
generalius  ea,  quae  pertinent  etiam  ad  actiones  trium  punctorum,  &  quae  a  num.  225,  & 
228  hue  reservavimus.  Porro  si  integrae  vires  alterius  in  alteram  diriguntur  ad  ipsa  centra 
gravitatis,  referam  hie  ad  se  invicem  vires  ex  integris  compositas  ;  sed  etiam  ubi  vires  aliam 
directionem  habeant  quancunque  ;  si  singulae  resolvantur  in  duas,  alteram,  quae  se  dirigat 
a  centre  ad  centrum  ;  alteram,  quae  sit  ipsi  perpendicularis,  vel  in  quocunque  dato  angulo 
obliqua  ;  omnia  in  prioribus  habebunt  itidem  locum. 


Consideratio  direc- 


se  mutuo  agunt. 


308.  Agant  in  se  invicem  in  fig.  56  tres  massae,  quarum  centra  gravitatis  sint  A,  B,  C, 
yi"bus  mutuis  ad  ipsa  centra  directis,  &  considerentur  inprimis  directiones  virium.  Vis 
puncti  C  ex  utraque  CV,  Cd  attractiva  erit  Ce  ;  ex  utraque  repulsiva  CY,  Ca,  erit  CZ, 
&  utriusque  directio  saltern  ad  partes  oppositas  producta  ingreditur  triangulum,  &  secat 
ilia  angulum  internum  ACB,  haec  ipsi  ad  verticem  oppositum  aCY.  Vi  CV  attractiva  in 
B,  ac  CY  repulsiva  ab  A,  habetur  CX  ;  &  vi  Cd  attractiva  in  A,  ac  Ca  repulsiva  a  B,  habetur 
Cb,  quarum  utraque  abit  extra  triangulum,  &  secat  angulos  ipsius  externos.  Primae  Ce, 
cum  debeant  respondere  attractiones  BP,  AG,  respondent  cum  attractionibus  mutuis 
BN,  AE,  vires  BO,  AF,  vel  cum  repulsionibus  BR,  AI,  vires  BQ,  AH,  ac  tarn  priores  binae, 
quam  posteriores,  jacent  ad  eandem  partem  lateris  AB,  &  vel  ambae  ingrediuntur  triangulum 
tendentes  versus  ipsum,  vel  ambae  extra  ipsum  etiam  productae  abeunt,  &  tendunt  ad 
partes  oppositas  directionis  Ce  respectu  AB.  Secundae  CZ  debent  respondere  repulsiones 
BT,  AL,  quae  cum  repulsionibus  BR,  AI,  constituunt  BS,  AK,  cum  attractionibus  BN, 
AE  constituunt  BM,  AD,  ac  tarn  priores  binae,  quam  posteriores  jacent  ad  eandem  plagam 
respectu  AB,  &  ambarum  [144]  directiones  vel  productae  ex  parte  posteriore  ingrediuntur 
triangulum,  sed  tendunt  ad  partes  ipsi  contrarias,  ut  CZ,  vel  extra  triangulum  utrinque 
abeunt  ad  partes  oppositas  direction!  CZ  respectu  AB.  Quod  si  habeatur  CX,  quam 
exponunt  CV,  CY,  turn  illi  respondent  BP,  &  AL,  ac  si  prima  conjungitur  cum  BN,  jam 
habetur  BO  ingrediens  triangulum  ;  si  BR,  turn  habetur  quidem  BQ,  cadens  etiam  ipsa 
extra  triangulum,  ut  cadit  ipsa  CX  ;  sed  secunda  AL  jungetur  cum  AI,  &  habebitur  AK, 
quae  producta  ad  partes  A  ingredietur  triangulum.  Eodem  autem  argumento  cum  vi  Cb 
vel  conjungitur  AF  ingrediens  triangulum,  vel  BS,  quae  producta  ad  B  triangulum  itidem 
ingreditur.  Quamobrem  semper  aliqua  ingreditur,  &  turn  de  reliquis  binis  redeunt,  quae 
dicta  sunt  in  casu  virium  Ce,  CZ. 


um. 


Theorema  pertinens          309.  Habetur  igitur  hoc  thcorema.     Quando  tres  masses  in  se  invicem  agunt  viribus 
ir  directis  ad,  centra  gravitatis,  vis  composita  saltern  unius  babet  directionem,  quez  saltern  producta 

ad  •partes  oppositas  secat  angulum  internum  trianguli,  i3  ipsum  ingreditur  :  reliquce  autem 
duce  vel  simul  ingrediuntur,  vel  simul  evitant,  W  semper  diriguntur  ad  eandem  plagam  respectu 
lateris  jungentis  earum  duarum  massarum  centra  ;  ac  in  primo  casu  vel  omnes  tres  tendunt  ad 
interiora  trianguli  jacendo  in  angulis  internis,  vel  omnes  tres  ad  exteriora  in  partes  triangulo 
oppositas  jacendo  in  angulis  ad  verticem  oppositis  ;  in  secundo  vero  casu  respectu  lateris 
jungentis  eas  binas  massas  tendunt  in  plagas  oppositas  ei,  in  quam  tendit  vis  ilia  prioris  masses. 


Theorema  elegan-          310.  Sed  est  adhuc  elegantius  theorema,  quod  ad  directionem  pertinet,  nimirum  : 

nens  cum^eju^de-  Omnium  trium  compositarum  virium  directiones  utrinque  products  transeunt  per  idem  punctum  : 

monstratione.  y  si  id  jaceat  intra  triangulum  ;  vel  omnes  simul  tendunt  ad  ipsum,  vel  omnes  simul  ad  partes 

ipsi  contrarias  :   si  vero  jaceat  extra  triangulum  ;   bince,  quarum  directiones  non  ingrediuntur 


A  THEORY  OF  NATURAL  PHILOSOPHY 


235 


H 


FIG.  56. 


236 


PHILOSOPHIC  NATURALIS  THEORIA 


H 


FIG.  56. 


A  THEORY  OF  NATURAL  PHILOSOPHY  237 

306.  Hence,  in  refractions,  which  arise  in  this  way  from  a  free  motion  between  two  T.he   ratio  of  ihe 
parallel  planes,  where  the  forces  at  equal  distances  from  one  or  the  other  of  them  are  equal,  the  *  ratioCC"oSfta^he 
the  ratio  of  the  sine  of  the  angle  of  incidence,  or  the  angle  made  by  the  path  before  refraction,  velocities    the 
with  a  straight  line  perpendicular  to  the  plane,  to  the  sine  of  the  angle  of  refraction,  or  the^ines. 

the  angle  made  after  refraction  with  the  vertical  also,  is  constant,  whatever  may  be  the 
inclination  at  ingress.  We  also  obtain  the  theorem  that  the  absolute  velocities  before  and 
after  refraction  are  in  the  inverse  ratio  of  the  sines.  For  such  velocities  are  represented 
by  HE,  ON  ;  &  these  are  inversely  as  the  sines  in  question. 

307.  These  facts  suggest  a  method  for  explaining  refraction  of  light ;   &  in  the  Third  Passing  on  to  the 

T>     *  u   n  ,0,  •         L*  L  »L    '1.         *u     -rut,          -.I.  u  TJ    theorem       which 

Part  we  shall  see  the  manner  in  which  the  hypothesis  of  the  above  theorem  may  be  applied  gave  rise  to  this 

to  particles  of  light.     In  the  meanwhile,  I  will  consider  the  mutual  forces,  with  which  work- 

three  masses  act  upon  one  another ;   here  we  shall  obtain  more  generally  all  those  things 

that   relate    to   the   actions   of    three   points   also,  such   as    I   reserved   from   discussion 

in  Art.  225,  228  until  now.     Further,  if  the  total  forces  of  the  one  or  the  other  are  directed 

towards  their  centres  of  gravity,  I  will  here  take  account  of  the  mutual  forces  compounded 

of  these  wholes.     But,  where  the  forces  have  any  directions  whatever,  if  each  of  them  is 

resolved  into  two  parts,  of  which  one  is  directed  from  centre  to  centre  &  the  other  is 

perpendicular  to  this  line,  or  makes  some  given  inclination  with  it,  then  also  all  things 

that  are  true  for  the  former  hold  good  also  in  this  case. 

308.  In  Fig.  56,  let  three  masses,  whose  centres  of  gravity  are  at  A,  B,  C,  act  upon  investigation  of 
one  another  with  mutual  forces  directed  to  their  centres  ;   &  first  of  all  let  the  directions  ^  fo^e^with 
of  the  forces  be  considered.     The  force  on  the  point  C,  from  the  two  attractive  forces  which  three  masses 
CV,  Cd  will  be  Ce  ;   that  from  CY,  Ca,  both  repulsive,  will  be  CZ ;    &  the  direction  of  ^ther  P°n  °De  an" 
both  of  these,  produced  backwards  in  one  case,  will  fall  within  the  triangle,  the  former 

dividing  the  angle  ACB,  &  the  latter  the  vertically  opposite  angle  aCY,  into  two  parts. 
But,  from  CV,.  attractive  towards  B,  &  CY,  repulsive  from  A,  we  obtain  CX ;  &  from  Cd, 
attractive  towards  A,  &  Ca,  repulsive  from  B,  we  have  Cb  ;  &  the  direction  of  each  of  these 
will  fall  without  the  triangle,  &  divide  its  exterior  angles  into  two  parts.  To  Ce,  the 
first  of  these,  since  we  must  have  the  corresponding  attractions  BP,  AG,  there  correspond 
the  forces  BO,  AF,  from  combination  with  the  mutual  attractions  BN,  AE  ;  or  the  forces 
BQ,'AH,  from  combination  with  the  mutual  repulsions  BR,  AI.  Both  the  former  of  these 
pairs,  &  the  latter,  lie  on  the  same  side  of  AB  ;  either  both  will  fall  within  the  triangle 
&  tend  in  its  direction,  or  both  will,  even  if  produced,  fall  without  it ;  in  each  case,  they 
will  tend  in  the  opposite  direction  to  that  of  Ce  with  respect  to  AB.  To  CZ,  the  second 
of  the  forces  on  C,  there  must  correspond  the  repulsions  BT,  AL  ;  these,  combined  with 
the  repulsions  BR,  AI,  give  the  forces  BS,  AK  ;  &  with  the  attractions  BN,  AE,  the  forces 
BM,  AD.  Both  the  former  of  these,  &  both  the  latter,  lie  on  the  same  side  of  AB  ;  & 
the  directions  of  the  two,  either  when  produced  backwards  will  fall  within  the  triangle 
but  tend  in  opposite  directions  to  that  of  CZ  with  respect  to  it,  or  they  will  fall  without 
the  triangle  &  tend  off  on  either  side  in  directions  opposite  to  that  of  CZ  with  respect  to 
AB.  Now  if  CX  is  obtained,  given  by  CV,  CY,  then  there  will  correspond  to  it  BP  &  AL  ; 
&,  if  the  first  of  these  is  compounded  with  BN,  we  shall  then  have  BO  falling  within  the 
triangle  ;  or  if  compounded  with  BR,  we  shall  have  BQ,  falling  also  without  the  triangle, 
just  as  CX  does ;  but,  in  that  case,  the  second  action  AL  will  be  compounded  with  AI, 
&  AK  will  be  obtained,  &  this  when  produced  in  the  direction  of  A  will  fall  within  the 
triangle.  By  the  same  argument,  with  the  force  Cb  there  will  be  associated  the  force  AF 
falling  within  the  triangle,  or  the  force  BS,  which  when  produced  in  the  direction  of  B 
will  also  fall  within  the  triangle.  Hence,  in  all  cases,  some  one  of  the  forces  falls  within 
the  triangle  ;  &  then  what  has  been  said  in  the  case  of  Ce,  CZ  will  apply  to  the  other  two 
forces. 

309.  We  therefore  have  the  following  theorem.     When  three  masses  act  upon  one  another  Theorern     relating 

•  .it  '          ]•        .    j    .  1^1-  1  •  i  i  i  •          ri  to  the  directions  of 

with  forces  directed  towards  their  centres  of  gravity,  the  resultant  force,  in  at  least  one  case,  the  forces. 
will  have  a  direction  which,  produced  backwards  if  necessary,  will  divide  an  internal  angle 
of  the  triangle  into  two  parts,  W  fall  within  the  triangle.  Also  the  remaining  two  forces  will 
either  both  fall  within,  or  both  without,  the  triangle  W  will  in  all  cases  be  directed  towards 
the  same  side  of  the  line  joining  the  centres  of  the  two  masses.  In  the  first  case,  all  three  forces 
either  tend  towards  the  interior  of  the  triangle,  falling  within  the  interior  angles,  or  outwards 
away  from  the  triangle,  falling  within  the  angles  that  are  vertically  opposite  to  the  interior 
angles.  In  the  second  case,  on  the  other  hand,  they  tend  to  opposite  sides,  of  the  line  joining 
the  two  masses,  to  that  towards  which  the  force  on  the  third  mass  tends. 

310.  But  there  is  a  still  more  elegant  theorem  with  regard  to  the  directions  of  the  A  still    more^eie- 
forces,  namely  : — The  directions  of  all  three  resultant  forces,  when  produced  each  way,  pass  ^^  regard^  the 
through  the  same  point.     If  this  point  lies  within-  the  triangle,  all  three  forces  tend  towards  directions   of    the 
it,  or  all  three  away  from  it  ;   but,  if  it  lies  without  the  triangle,  those  two  forces,  which  do  not  ^onstration. 


238  PHILOSOPHIC  NATURALIS  THEORIA 

triattgulum,  tendunt  ad  ipsum,  ac  tertia,  cujus  directio  triangulum  ingreditur,  tend.it  ad  Cartes 
ipsi  contrariias  ;    vel  illce  bints  ad  partes  ipsi  contr arias,  W  tertia  ad  ipsiim. 

Prima  pars,  quod  omnes  transeant  per  idem  punctum,  sic  demonstratur.  In  figura 
quavis  a  57  ad  62,  quae  omnes  casus  exhibent,  vis  pertinens  ad  C  sit  ea,  quas  triangulum 
ingreditur,  ac  reliquse  binas  HA,  QB  concurrant  in  D  :  oportet  demonstrare,  vim  etiam, 
quae  pertinet  ad  C,  dirigi  ad  D.  Sint  CV,  Cd  vires  componentes,  ac  ducta  CD,  ducatur 
VT  parallela  CA,  occurrens  CD  in  T ;  &  si  ostensum  fuerit,  ipsam  fore  aequalem  Cd  • 
res  erit  perfecta  :  ducta  enim  dT  remanebit  CVTW  parallelogramrnum,  per  cujus  diagonalem 
CT  dirigetur  vis  composita  ex  CV,  Cd.  Ejusmodi  autem  aequalitas  demonstrabitur 
considerando  rationem  CV  ad  Cd  compositam  ex  quinque  intermediis,  CV  ad  BP,  BP  ad 
PQ,  PQ,  sive  BR  ad  AI,  AI,  sive  HG  ad  AG,  AG  ad  [145]  Cd.  Prima  vocando  A,  B,  C 
massas,  quarum  ea  puncta  sunt  centra  gravitatum,  est  ex  actione,  &  reactione  aequalibus 
ratio  massae  B  ad  C  ;  secunda  sin  PQB,  sive  ABD,  ad  sin  PBQ,  sive  CBD  ;  tertia  A  ad 
B  :  quarta  sin  HAG,  sive  CAD,  ad  sin  GHA,  sive  BAD  :  quinta  C  ad  A.  Tres  rationes, 
in  quibus  habentur  massas,  componunt  rationem  BxAxCadCxBxA,  quas  est  i  ad 
i,  &  remanet  ratio  sin  ABD  x  sin  CAD  ad  sin  CBD  X  sin  BAD.  Pro  sin  ABD,  &  sin 
BAD,  ponantur  AD,  &  BD  ipsis  proportionales ;  ac  pro  sinu  CAD,  &  sin  CBD  ponantur 

sin  ACD  X  CD    „    sin  BCD  X  CD    .     .  ,  ™  .  .  .      .    ,    ,    ,  . 

._        — ,  &  — ,  ipsis  aequales  ex    ingonometna,  &  habebitur  ratio 

AD  r>D 

sin  ACD  X  CD  ad  sin  BCD  X  CD  sive  sin  ACD,  vel  CTV,  qui  ipsi  aequatur  ob  VT, 
CA  parallelas,  ad  sin  BCD,  sive  VCT,  nimirum  ratio  ejusdem  illius  CV  ad  VT.  Quare 
VT  aequatur  Cd,  CVTd  est  parallelogrammum,  &  vis  pertinens  ad  C,  habet  directionem 
itidem  transeuntem  per  D. 

Secunda  pars  patet  ex  iis,  quae  demonstrata  sunt  de  directione  duarum  virium,  ubi 
tertia  triangulum  ingreditur,  &  sex  casus,  qui  haberi  possunt,  exhibentur  totidem  figuris. 
In  fig.  57,  &  58  cadit  D  extra  triangulum  ultra  basim  AB,  in  59,  &  60  intra  triangulum, 
in  61,  &  62  extra  triangulum  citra  verticem  ad  partes  basi  oppositas,  ac  in  singulorum 
binariorum  priore  vis  CT  tendit  versus  basim,  in  posteriore  ad  partes  ipsi  oppositas.  In 
iis  omnibus  demonstratio  est  communis  juxta  leges  transformationis  locorum  geometri- 
corum,  quas  diligenter  exposui,  &  fusius  persecutus  sum  in  dissertatione  adjecta  meis 
Sectionum  Conicarum  Elementis,  Elementorum  tomo  3. 


311-  Quoniani  evadentibus  binis  HA,  O_B  parallelis,  punctum  D  abit    in    infinitum 
paraiieiarum.  &  tertia  CT  evadit  parallela  reliquis  binis  etiam  ipsa  juxta  easdem  leges ;   patet  illud  :    Si 

bince  ex  ejusmodi  directionibus  fuerint  parallels  inter  se  ;  erit  iisdem  parallela  y  tertia  : 
ac  ilia,  qua  jacet  inter  directiones  virium  transeuntes  per  reliquas  binas,  quce  idcirco  in  eo 
casu  appellari  potest  media,  habebit  directionem  oppositam  directionibus  reliquarum  conformibus 
inter  se. 

AHud  generate  ter-  312.  Patet  autem,  datis  binis  directionibus  virium,  dari  semper  &  tertiam.     Si  enim 

dTtis'wnU011'        6  illae  sint  parallelae  ;    erit  illis  parallela  &  tertia  :    si  autem  concurrant  in  aliquo  puncto  ; 

tertiam  determinant  recta  ad  idem  punctum  ducta  :  sed  oportet,  habeant  illam  conditionem, 

ut  tarn  binae,  quas  triangulum  non  ingrediantur,  quam  quae  ingrediantur,  vel  simul  tendant 

ad  illud  punctum,  vel  simul  ad  partes  ipsi  contrarias. 

Theorema   pracip.          313.  Haec  quidem  pertinent  ad  directiones  :    nunc  ipsas  earum  virium  magnitudines 

dine,   quod^oU  inter  se  comparabimus,  ubi    statim  occurret    elegantissimum    illud    theorema,  de    quo 

Open    occasionem  mentionem  fed  num.  225  :    Vires  acceleratrices  binarum  quarumvis  e  tribus  massis  in  se 

s  trVtio]expeditis."  mutuo  agentibus  sunt  in  rations  composita  ex  tribus,  [146]  nimirum  ex  directa  sinuum  angulorum 

sima-  quo  s  continet  rec  ta  jungens  ipsarum  centra  gravitatis  cum  rectisductis  ab  iisdem  centrisad  centrum 

tertice  mass<e  ;  reciproca  sinuum  angulorum,  quos  directiones  ipsarum  virium  continent  cum 

iisdem  rectis  illas  jungentibus  cum  tertia  ;    &  reciproca  massarum.     Nam  est  BQ  ad  AH 

assumptis  terminis  mediis  BR,  AI  in  ratione  composita  ex  rationibus  BQ,  ad  BR,  &  BR 

ad  AI,  &  AI  ad  AH.     Prima  ratio  est  sinus  QRB,  sive  CBA  ad  sinum  BQR,  sive  PBQ,  vel 

CBD  :  secunda  massae  A  ad  massam  B  :   tertia  sinus  IHA,  sive  HAG,  vel  CAD,  ad  sinum 

HIA,  sive  CAB  :    eae  rationes,  permutato  solo  ordine  antecedentium,  &  consequentium, 

sunt  rationes  sinus  CBA  ad  sinum  CAB,  quae  est  ilia  prima  e  rationibus  propositis  directa  ; 

sinus  CAD  ad  sinum  CBD,  quae  est  secunda  reciproca  :    &  massae  A  ad  massam  B,  quas 

est  tertia  itidem  reciproca.     Eadem  autem  est  prorsus  demonstratio  :    si  comparetur  BQ, 

vel  AH  cum  CT,  ac  in  hac  demonstratione,  ut  &  alibi  ubique,  ubi  de  sinubus  angulorum 


A  THEORY   OF  NATURAL  PHILOSOPHY 


239 


Q 


B  R 


FIG.  57. 


A, I 


E 

FIG.  59. 


R  B 


E 

Fio.  6c 


P  Q 


I  A 


E  B  R 

FIG,  61. 


H 


Fio.  62. 


240 


PHILOSOPHISE  NATURALIS  THEORIA 


B  R 


FIG.  57. 


Q  P 


A    I 


R  B 


FIG.  59 


FIG.  60. 


P  Q 


I  A 


E 

FIG.  61. 


B   R 


H 


FIG.  62. 


A  THEORY  OF  NATURAL   PHILOSOPHY  241 

fall  within  the  triangle,  tend  towards  it,  W  the  third,  whose  direction  does  not  fall  within  the 
triangle,  tends  away  from  it,  or  the  former  two  tend  away  from  the  point  y  the  third  towards  it. 
The  proof  of  the  first  part  of  the  theorem,  that  the  forces  all  pass  through  the  same  point, 
is  as  follows.  In  any  one  of  the  diagrams  from  Fig.  57  to  Fig.  62,  which  between  them 
give  all  possible  cases,  let  the  force  which  acts  on  C  be  that  which  falls  within  the  triangle  ; 
&  let  the  other  two,  HA  &  QB,  meet  in  the  point  D  ;  then  it  has  to  be  shown  that  the' 
force  which  acts  on  C,  also  passes  through  D.  Let  CV,  Cd  be  the  component  forces  ;  join 
CD  &  draw  VT  parallel  to  CA  to  meet  CD  in  T  ;  then,  if  it  can  be  shown  that  VT  is  equal 
to  Cd,  the  proposition  is  proved  ;  for,  if  dT  is  joined,  CVTd  will  be  a  parallelogram,  & 
the  force  compounded  of  CV  &  Cd  will  be  directed  along  its  diagonal.  Such  equality  will 
be  proved  by  considering  the  ratio  of  CV  to  Cd,  compounded  of  the  five  intermediate 
ratios  CV  to  BP  ;  BP  to  PQ  ;  PQ,  or  BR,  to  AI  ;  AI,  or  HG,  to  AG  ;  &  AG  to  Cd.  The 
first  of  these,  if  we  call  the  masses  A,  B,  C,  which  have  these  points  as  their  centres  of  gravity, 
will,  on  account  of  the  equality  of  action  &  reaction,  be  the  ratio  of  the  mass  B  to  the  mass 
C  ;  the  second,  the  ratio  of  the  sine  of  PQB,  or  ABD,  to  the  sine  of  PBQ,  or  CBD  ;  the 
third,  that  of  the  mass  A  to  the  mass  B  ;  the  fourth,  that  of  the  sine  of  HAG,  or  CAD, 
to  the  sine  of  GHA,  or  BAD  ;  the  fifth,  that  of  the  mass  C  to  the  mass  A.  The  three 
ratios,  in  which  the  masses  appear,  together  give  the  ratio  BxAxCtoCxBxA, 
which  is  that  of  i  to  i  ;  &  there  remains  the  ratio  of  sinAED  X  sinCAD  to  wzCBD  X 
sinEAD.  For  sinABD  &  sinEAD  substitute  AD  &  BD,  which  are  proportional  to  them  ; 
&  for  sinCAD  &  sinCED  substitute  sinACD  X  CD/AD  &  sinECD  X  CD/BD,  which  are 
equal  to  them  by  trigonometry.  There  will  be  obtained  the  ratio  of  sinACD  X  CD  to 
sinECD  X  CD,  or  sinACD  to  sinECD  ;  &,  since  VT  &  CA  are  parallel,  this  ratio  is  equal  to 
that  of  sinCTV  to  .rzWCT,  that  is,  to  the  ratio  of  CV  to  VT.  Therefore  VT  is  equal  to 
Cd,  CVT^  is  a  parallelogram,  &  the  force  on  C  has  also  its  direction  passing  through  D. 
The  second  part  is  evident  from  what  has  already  been  proved  with  regard  to  the  directions 
of  two  forces  when  the  third  falls  within  the  triangle  ;  &  the  six  possible  cases  are  shown 
in  the  six  figures.  In  Fig.  57,  58,  the  point  D  falls  without  the  triangle  on  the  far  side 
of  the  base  AB  ;  in  Fig.  59,  60,  it  falls  within  the  triangle  ;  in  Fig.  61,  62,  outside  the 
triangle  on  the  side  of  the  vertex  remote  from  the  base  ;  &  in  the  first  of  each  pair  of  figures, 
the  force  CT  tends  towards  the  base,  &  in  the  latter  away  from  it.  In  all  of  these  the 
proof  is  the  same,  having  regard  to  the  laws  of  transformation  of  geometrical  positions  ; 
these  I  have  set  forth  carefully,  &  I  investigated  them  more  minutely  in  a  dissertation 
added  as  a  supplement  to  my  Sectionum  Conicarum  Elementa,  the  third  volume  of  my 
Elementa  Matheseos. 

311.  Now,  since  the  point  D  will  go  off  to  infinity,  when  two  of  the  forces,  HA  &  Corollary  for  the 
QB,  happen  to  be  parallel,  &  the  third  also,  according  to  the  same  laws,  becomes  parallel  Directions.   pa: 

to  the  other  two,  we  have  this  theorem.  //  two  of  these  forces  are  parallel  to  one  another, 
the  third  also  is  parallel  to  them  ;  &  that  force,  which  lies  between  the  directions  of  the  other 
two,  y  consequently  in  that  case  can  be  called  the  middle  force,  has  its  direction  opposite  to 
the  directions  of  the  other  two,  which  are  in  agreement  with  one  another. 

312.  Further,  it  is  clear  that,  when  the  directions  of  two  of  the  forces  are  given,  the  Another  general 

^ 


direction  also  of  the  third  force  is  given  in  all  cases.     For  if  the  former  are  parallel,  the  tion^ot'the   third 
third  will  be  parallel  to  them  ;   &  if  the  former  meet  at  a  point,  the  straight  line  joining  force  is  given  when 
the  third  mass  to  this  point  will  determine  the  third  direction.     But  this  condition  holds  ;  thee  0^£  ^  a°e 
namely,  that  the  two  which  do  not  fall  within  the  triangle,  or  the  pair  which  do  fall  within  given. 
the  triangle,  either  both  tend  towards  the  point  D,  or  both  tend  away  from  it. 

313.  So  much  with  regard  to  directions  ;  now  we  will  go  on  to  compare  with  one  Fundamental  theo. 
another  the  magnitudes  of  these  forces.  We  immediately  come  to  that  most  elegant  Magnitude1  °  which 
theorem,  which  has  already  been  mentioned  in  Art.  225.  The  accelerating  effects  of  any  gave  rise  to  the 

,,  T      '  ,,  j  •  ,•  jjjjZ.          whole  of  this  work. 

two  masses  out  of  three  that  mutually  act  upon  one  another  are  in  a  ratio  compounded  of  three 
ratios  ;  namely,  the  direct  ratio  of  the  sines  of  the  angles  made  by  the  straight  line  joining  the 
centres  of  gravity  of  these  two  with  the  straight  lines  joining  each  of  these  to  the  centre  of  gravity 
of  the  third  mass  :  the  inverse  ratio  of  sines  of  the  angles  which  the  directions  of  the  forces  make 
with  the  straight  lines  joining  the  two  masses  to  the  third  ;  W  the  inverse  ratio  of  the  masses.  For, 
if  BR,  AI  are  taken  as  intermediary  terms,  the  ratio  of  BQ  to  AH  is  equal  to  the  ratios  com- 
pounded from  the  ratio  of  BQ  to  BR,  that  of  BR  to  AI,  &  that  of  AI  to  AH.  The  first  ratio  is 
equal  to  that  of  the  sine  of  QRB,  or  CBA,  to  the  sine  of  BQR,  or  PBQ,  or  CBD  ;  the  second 
is  that  of  the  mass  A  to  the  mass  B  ;  &  the  third  is  equal  to  that  of  the  sine  of  IHA,  or 
HAG,  or  CAD  to  the  sine  of  HIA,  or  CAB.  These  ratios  are,  by  a  simple  permutation 
of  the  antecedents  &  consequents,  as  sinCEA  is  to  sinCAE,  which  is  the  first  direct  ratio 
of  those  enunciated  ;  as  sinCAD  to  sinCED,  which  is  the  second  inverse  ratio  ;  &  as^  the 
mass  A  to  the  mass  B,  which  also  is  the  third  inverse  ratio.  Moreover  the  proof  is  precisely 
similar,  if  the  ratio  of  BQ,  or  AH,  to  CT  is  considered  ;  &  in  this  proof,  as  also  in  all  others, 


242  PHILOSOPHIC  NATURALIS  THEORIA 

agitur,  angulis  quibusvis  substitui  possunt,  uti  saepe  est  factum,  &  fiet  imposterum,  eorum 
complementa  ad  duos  rectos,  quae  eosdem  habent  sinus. 

CoroiiaHum     sim.  314.  Inde  consequitur,  esse  ejusmodi  vires  reciproce,  ut  massas  ductas  in,  suas  distantias 

FpsU.pr°  V"  S  a  tertia  massa,  y  reciproce,  ut  sinus,  quos  earum  directiones  continent  cum  iisdem  rectis  ; 
adeoque  ubi  e&  ad,  ejusmodi  rectas  inclinentur  in  angulis  (squalibus,  esse  tantummodo  reciproce, 
ut  producta  massarum  per  distantias  a  massa  tertia.  Nam  ratio  directa  sinuum  CBA,  CAB 
est  eadem,  ac  distantiarum  AC,  BC,  sive  reciproca  distantiarum  BC,  AC,  qua  substituta  pro 
ilia,  habentur  tres  rationes  reciprocje,  quas  exprimit  ipsum  theorema  hie  propositum. 
Porro  ubi  anguli  aequales  sunt,  sinus  itidem  sunt  aequales,  adeoque  eorum  sinuum  ratio 
fit  I  ad  i. 

Ratio  virium   mo.  315.  Vires  autem  matrices  sunt  in  ratione  composita  ex  binis    tantummodo,  nimirum 

directa  sinuum  angulorum,  quos  continent  distantiee  a  tertia  massa  cum  distantia  a  se  invicem  ; 
y  reciproca  sinuum  angulorum,  quos  continent  cum  iisdem  distantiis  directiones  virium  ;  vel 
in  ratione  composita  ex  reciproca  illarum  distantiarum,  y  reciproca  horum  posteriorum  sinuum  : 
ac  si  inclinationes  ad  distantias  sint  eequales,  in  sola  ratione  reciproca  distantiarum.  Nam 
vires  motrices  sunt  summae  omnium  virium  determinantium  celeritatem  in  punctis 
omnibus  secundum  earn  directionem,  secundum  quam  movetur  centrum  gravitatis  commune, 
quae  idcirco  sunt  praeterea  directe,  ut  massae,  sive  ut  numeri  punctorum  ;  adeoque  ratio 
directa,  &  reciproca  massarum  mutuo  eliduntur. 

Ratio  yirium  acce-  316.  Praeterea  vires  acceleratrices,  si  alicubi  earum  directiones  concurrunt,  sunt  ad  se 

dirituiatur1'adbl^if  ^nv^cem  ™  ratione  composita  ex  reciproca  massarum,  &  reciproca  sinuum  angulorum,  quibus 

quod     commune  inclinantur  ad  directionem  ter  tice  ;    y  vires  motrices  in  hac  poste-[i4j]-riore  tantum.     Nam 

punctum.  o{j  iatera  proportionalia  sinubus  angulorum  oppositorum,  erit  AC  X  sin  CAD  =  CD  X 

sin  CDA  ;    &  pariter  CB  x  sin  CBA  =  CD  X  sin  CDB.     Quare  ob  CD  communem, 

sola  ratio  sinuum  ADC,  BDC,  quibus  directiones  AD,  BD  inclinantur  ad  CD,  aequatur 

compositae  ex  rationibus  sinuum  CAD,  CBD,  &  distantiarum  CA,  CB,  quae  ingrediebantur 

rationem  virium  B,  &  A  ;   ac  eodem  pacto  AC  X  sin  ACD=  AD  x  sin  ADC,  &  AB  x 

sin  ABD  =  AD  X  sin  ADB,  adeoque  AC  X  sin  ACD  ad  AB  x  sin  ABD,  ut  sinus  ADC 

ad  sinum  ADB,  quibus  directiones  CD,  BD  inclinantur  ad  AD  ;  &  eadem  est  demonstratio 

pro  sinubus  ADB,  EDB  assumpto  communi  latere  BD. 

Alia  expressio  tam  317.  Si  ducatur  MO  parallela  DA,  occurrens  BD,  CD  in  M,  O,  y  compleatur  parallelo- 
qut'm11  SiSStri-  grammum  DMON  ;  erunt  vires  motrices  in  C,  B,  A  ad  se  invicem,  ut  recta  DO,  DM,  DN, 
cium  in  eodem  casu.  y  vires  acceleratrices  prteterea  in  ratione  massarum  reciproca.  Est  enim  ex  praecedenti  vis 
motrix  in  C  ad  vim  in  B,  ut  sin  BDA  ad  sin  CDA,  vel  ob  AD,  OM  parallelas,  ut  sin  DMO 
ad  sin  DOM,  nimirum  ut  DO  ad  DM,  &  simili  argumento  vis  in  C  ad  vim  in  A,  ut  DO 
ad  DN.  '  Vires  autem  motrices  divisae  per  massas  evadunt  acceleratrices.  Quamobrem  si, 
tres  vires  agerent  in  idem  punctum  cum  directionibus,  quas  habent  eee  vires  motrices,  y  essent 
Us  proportionates  ;  binee  componerent  vim  oppositam,  y  tequalem  tertice,  ac  essent  in  ^equilibria. 
Id  autem  etiam  directe  patet  :  nam  vires  BQ,  AH  componuntur  ex  quatuor  viribus  BR, 
BP,  AI,  AG,  quae  si  ducantur  in  massas  suas,  ut  riant  motrices  ;  evadit  prima  aequalis,  & 
contraria  tertiae,  quam  idcirco  elidit,  ubi  deinde  AH,  BQ  componantur  simul,  &  in  ejusmodi 
compositione  remanent  BP,  AG,  ex  quarum  oppositis,  &  aequalibus  CV,  Cd  componitur 
tertia  CT. 


Hie  debere  haberi  ^jg.  Hinc  in  hisce  viribus  motricibus  habebuntur  omnia,  quae  habentur  in  compositione 

fn'  compositione,"  virium  ;   dummodo  capiatur   [resolutio]  compositse  contraria.      Si    nimirum    resolvantur 

resolution  virium.  singulae  componentes  in  duas,  alteram  secundum  directionem  tertiae,  alteram  ipsi  perpen- 

dicularem,  hae  posteriores  elidentur,  illae  priores  confident  summam  sequalem  tertiae,  ubi 

ambae  eandem  directionem  habent,  uti  sunt  binae,  quas  simul  ingrediantur,  vel  simul  evitent 

triangulum  ;  nam  in  iis,  quarum  altera  ingreditur,  altera  evitat,  tertia  aequaretur  differen- 

tiae ;  &  facile  tam  hie,  quam  in  ratione  composita,  res  traducitur  ad  resolutionem  in 

aliam  quamcunque  directionem  datam,  praeter  directionem  tertiae,  binis  semper  elisis,  & 

reliquarum  accepta  summa  ;    si  rite  habeatur  ratio  positivorum,  &  negativorum. 

AT}  y       TTT~)  AT? 

Alia  expressio  -jg    ]?st  &  juu(j  utije  .   tres  vjres  motrices  in  C,  B,  A  sunt  inter  se,  ut  -.  '          _TA,  -.  _, 

rationum  earundem  AD   X  BD    AD 


virium. 


,      ,  y  acceler  atrices  prczterea  [148]  in  ratione  reciproca  massarum.     Nam  ex  Trigonometria 
BD 


*  BAD'  &  *  EAD 


AB        AE 
communis  :    erit  sin  ADB  ad  sin  ADE,  ut    =-   ad  ==,  vel,  ducendo  utrunque  terminum 


A  THEORY  OF  NATURAL  PHILOSOPHY 


Q 


B  R 


FIG.  57. 


A     I 


E  R  B 

FIG.  59. 


HG 


E 

FIG.  60. 


P  Q 


I  A 


B   R 


M 


FIG.  61. 


FIG.  62. 


244 


PHILOSOPHIC  NATURALIS  THEORIA 


B  R 


FIG.  57. 


Q  P 


A    I 


R  B 


FIG.  59. 


P  Q 


I  A 


E  B   R 

FIG.  6r. 


H 


FIG.  62. 


A  THEORY  OF  NATURAL  PHILOSOPHY  245 

where  sines  of  angles  are  considered,  we  can  substitute  for  any  of  the  angles,  as  often  has 
been  done,  &  as  will  be  done  hereafter,  their  supplements ;  for  these  have  the  same  sines. 

314.  Hence  we  have  the  following  corollary.     Such  accelerating  effects  are  inversely  as  simple      corollary 
the  -products  of  each  of  the  two  masses  into  its  distance  from  the  third  mass,  &  inversely  as  the 

sines  of  the  angles  between  their  directions  &  these  distances  ;  y  thus,  if  they  are  inclined 
at  equal  angles  to  these  distances,  the  effects  are  inversely  -proportional  to  the  -products  of  the 
masses  into  the  distances  from  the  third  mass  only.  For  the  direct  ratio  of  the  sines  of  the 
angles  CB A,  CAB  is  the  same  as  that  of  the  distances  AC,  BC,  or  inversely  as  the  distances 
BC,  AC ;  &  if  the  latter  is  substituted  for  the  former,  we  have  three  inverse  ratios,  which 
are  given  in  the  enunciation  of  this  corollary.  Further,  when  the  angles  are  equal,  their 
sines  are  also  equal,  &  their  ratio  is  that  of  I  to  i. 

315.  The  motive  forces  are  in  a  ratio  compounded  of  two  ratios  only,  namely,  the  direct  The   ratio  of  the 
ratio  of  the  sines  of  the  angles  the  line  joining  each  to  the  third  mass  &  the  line  joining  the  two  motlve  forces- 

to  one  another  ;  fcff  the  inverse  ratio  of  the  sines  of  the  angles  which  their  directions  make  with 
these  distances  ;  or  the  ratio  compounded  of  the  inverse  ratio  of  these  distances  W  the  inverse 
ratio  of  the  latter  sines.  Also,  if  the  inclinations  to  the  distances  are  equal  to  one  another,  the 
ratio  is  the  simple  inverse  ratio  of  the  distances.  For  the  motive  forces  are  the  sums  of  all 
the  forces  determining  velocity  for  all  points  in  the  direction  along  which  the  common 
centre  of  gravity  will  move  ;  &  hence  they  are,  other  things  apart,  directly  as  the  masses, 
or  as  the  number  of  points ;  &  thus  the  direct  &  the  inverse  ratio  of  the  masses  eliminate 
one  another. 

316.  Further,  the  accelerations,  if  their  directions  meet  at  a  point,  are  to  one  another  in  The  ratio  of  the 
the  ratio  compounded  from  the  inverse  ratio  of  the  masses,  &  the  inverse  ratio  of  the  sines  of  ^ey  "are^directed 
the  angles  between  their  directions  &  that  of  the  third.     The  motive  forces  are  in  the  latter  towards  some  com- 
ratio  only.     For,  since  the  sides  of  a    triangle  are  proportional  to  the  sines  of  the  opposite  mon  pomt' 
angles,  we  have  AC.  wzCAD  =  CD.  sinCDA,  &  similarly,  CB.  sinCBA.  =  CD.  wiCDB. 

Hence,  since  CD  is  common,  the  single  ratio  of  the  sines  of  ADC,  BDC,  the  inclinations 
of  AD,  BD,  to  CD,  is  equal  to  that  compounded  from  the  ratios  of  the  sines  of  CAD,  CBD, 
&  the  distances  CA,  CB,  which  formed  the  ratio  of  the  forces  on  B  &  A.  In  the  same 
way,  AC.  .rzwACD  =  AD.  jjwADC,  &  AB.  stnABD  =  AD.  sinADE,  &  therefore  AC.  j««ACD 
is  to  AB.  sinABD  as  the  sine  of  ADC  is  to  the  sine  of  ADB,  the  inclinations  of  CD,  BD 
to  AD.  The  proof  is  the  same  for  the  sines  of  the  angles  ADB,  EDB,  by  using  the  common 
side  DB. 

317.  //  MO  is  drawn  parallel  to  DA,  meeting  BD,  CD  in  M,  O  respectively,  &  if  the  Another  expression 
parallelogram  DMON  is  completed,  then  the  motive  forces  for  C,  B,  A  will  be  to  one  another  t°*e  forces  &  the 
as  the  straight  lines  DO,  DM,  DN  ;    &  for  the  accelerations,  we  have  in  addition  the  inverse  accelerations  in  the 
ratio  of  the  masses.     For,  from  the  preceding  article,  the  motive  force  for  C  is  to  the  motive  s 

force  for  B  as  ji'wBDA  is  to  sinCDA  ;  that  is  to  say,  since  AD,  OM  are  parallel,  as  j/wDMO 
is  to  JtwDOM,  or  as  DO  is  to  DM.  Similarly  the  force  for  C  is  to  the  force  for  B  as  DO 
is  to  DN.  Now,  the  motive  forces  divided  by  the  corresponding  masses  give  the  accelerations. 
Hence,  if  three  forces  act  at  a  point,  having  the  same  directions  as  the  motive  forces  &  propor- 
tional to  them,  the  resultant  compounded  from  any  two  of  these  will  give  a  force  equal  &  opposite 
to  the  third,  W  they  will  be  in  equilibrium.  This  is  immediately  evident ;  for,  the  forces  BQ, 
AH  are  compounded  from  the  four  forces  BR,  BP,  AI,  AG  ;  &  if  these  are  multiplied  by 
the  corresponding  masses,  so  as  to  give  the  motive  forces,  the  first  of  them  will  come  out 
equal  &  opposite  to  the  third  &  will  thus  cancel  it,  when  later  AH,  BQ  are  compounded 
together  ;  &  in  such  composition  we  are  left  with  BP,  AG ;  &  from  CV  &  Cd,  which  are 
equal  &  opposite  to  these,  the  third  force  CT  is  compounded. 

318.  Hence  for  these  motive  forces,  we  have  all  those  things  which  hold  good  in  the  For   these    forces 
composition  of  forces,  so  long  as  resolution  is  considered  to  be  the  inverse  of  composition.  ^ostfthings^hich 
Thus,  if  each  of  the  components  is  resolved  into  two  parts,  one  in  the  direction  of  the  third  hold  good  for  com- 
force,  &  the  other  perpendicular  to  it,  the  latter  will  cancel  one  another,  &  the  former  tton^'Tforces™55 
will  give  a  sum  equal  to  the  third,  when  both  have  the  same  direction,  as  is  the  case  when 

both  of  them  either  fall  within  the  triangle  or  both  of  them  are  directed  away  from  it ; 
for  those,  in  which  one  falls  within  the  triangle  &  the  other  away  from  it,  the  third  will 
be  equal  to  the  difference.  The  matter,  both  in  this,  &  in  the  ratio  compounded  of  these, 
is  easily  referred  to  a  resolution  in  any  chosen  direction  other  than  the  direction  of  the 
third,  the  two  at  right  angles  always  cancelling  one  another  &  the  sum  being  taken  of  those 
that  remain  ;  provided  due  regard  is  had  to  positives  &  negatives. 

319.  Here  is  another  useful  theorem.     The  three  motive  forces  on  C,  B,  &  A  are  in  Another  expression 
the  ratio   of   AB.ED/ADVBD,  AE/AD,  BE/BD,   W  the  accelerations  have,  in  addition,  ** 
the  inverse  ratio  of  the  masses.     For,  by  trigonometry,  we  have  AB/BD  =  JtflADB/tt'ffBAD, 

&  AE/ED  =  j*«ADE/tt«EAD.  Hence,  since  the  divisors  sinEAD,  &  ji'wEAD  are  equal, 
it  follows  that  sinADB  is  to  sinADE  as  AB/BD  is  to  AE/ED  ;  or,  multiplying  each  term 


246  PHILOSOPHIC  NATURALIS  THEORIA 

in  •£•= »  ut    AT-.  ;     ^  ad  -pp- .     Simili  autem  argumento  est  itidem  sin  BDA.    sin  BDE 
/VL'          /\U  X  rjU         AL) 

AB  x  ED      BE 
:  :  ADx?p  •  BD  ;   ex  quo  patent  omma. 

Expressio     simpli-  320.  Si  punctum  D  abeat  in  infinitum,  directionibus  virium  evadentibus  parallels  ; 

leiumi!  C  "  rati°  rectarum  ED,  AD,  BD,  ad  se  invicem  evadit  ratio  asqualitatis.  Quare  in  eo  casu 

illae  ties  vires  sunt  ut  AB,  AE,  EB,  in  quibus  prima  asquatur  summae  reliquarum.  Conci- 
piantur  rectas  parallelae  directioni  virium  ductas  per  omnium  trium  massarum  centra 
gravitatis,  quarum  massarum  earn,  quae  jacuerit  inter  reliquarum  binarum  parallelas  diximus 
mediam  :  ac  si  ducantur  in  quavis  alia  directione  data  rectae  ab  iis  massis  ad  illas  parallelas  ; 
erunt  ejusmodi  distantiae  ab  iis  parallelis,  ut  ipsas  AB,  EB,  ad  quas  erunt  singular  in  ratione 
data,  ob  datas  directiones.  Quare  pro  vinous  parallelis  habetur  hujusmodi  theorema  : 
Vires  parallels  matrices  binarum  quarunvis  ex  tribus  massis  sunt  inter  se  reciproce  ut  distantice 
a  directione  communi  transeunte  per  tertiam :  vires  autem  acceleratrices  prtzterea  in  ratione 
reciproca  massarum,  &  media  est  directionis  contraria  respectu  reliquarum,  ac  vis  media 
matrix  tequatur  reliquarum  summce,  utralibet  vero  extrema  differentia. 

Appiicatio     ratio-  52i.  Hoc   theorema   primo   quidem   exhibet   centrum   asquilibrii,   viribus   utcunque 

num  supenorum  ad     ,.       J          ..  ,  r     .,  *  n.  .     T»    /-i  /„ 

centrum  zquUibrii.  divergentibus,  vel  convergentibus.  bi  nimirum  sint  tres  massas  A,  B,  C  (&  nomine  massarum 
etiam  intelligi  possunt  singula  puncta),  quarum  binas,  ut  A,  &  B,  solicitentur  viribus 
motricibus  externis ;  poterunt  mutuis  viribus  illas  elidere,  ac  esse  in  asquilibrio,  &  eas 
elident  omnino,  mutatis,  quantum  libuerit,  parum  mutuis  distantiis ;  si  fuerint  ante 
applicationem  earum  virium  externarum  in  satis  validis  limiribus  cohassionis,  ac  vis  massas 
C  elidatur  fulcro  opposite  in  directione  DC,  vel  suspensione  contraria  :  dummodo  binae 
illas  vires  ductae  in  massas  habeant  conditiones  requisitas  in  superioribus,  ut  nimirum  ambae 
tendant  ad  idem  punctum,  vel  ab  eodem,  aut  si  fuerint  parallelae,  ambae  eandem  directione m 
habeant,  ubi  simul  ambae  ingrediantur,  vel  simul  ambae  evitent  triangulum  ABC  :  ubi 
vero  altera  ingrediatur  triangulum,  altera  evitet,  tendat  aliera  ad  punctum  concursus, 
altera  ad  partes  illi  oppositas  :  vel  si  fuerint  parallelae,  habeant  directiones  [149]  oppositas  : 
&  si  parallelas  fuerint ;  sint  inter  se,  ut  distantiae  a  directione  virium  transeunte  per  C  ; 
si  fuerint  convergentes,  sint  reciproce,  ut  sinus  angulorum,  quos  earum  directiones  continent 
cum  recta  ex  C  tendente  ad  earum  concursum,  vel  sint  in  ratione  reciproca  sinuum 
angulorum,  quos  continent  cum  rectis  AC,  BC,  &  ipsarum  rectarum  conjunctim. 


Dcterminatio    vis,  322.  Determinabitur  autem  admodum  facile  per  ipsa  theoremata  etiam  vis,   quam 

™  '    sustinebit  fulcrum  C,  quae  in  casu  parallelism!  aequabitur  summae,  vel  differentiae  reliquarum, 

prout  ibi  fuerit  media,  vel  extrema  :  &  in  casibus  reliquis  omnibus  aequabitur  summae 
pariter,  vel  differentiae  reliquarum  ad  suam  directionem  reductarum,  reliquis  binis  in 
resolutione  priorum  sociis  se  per  contrariam  directionem,  &  aequalitatem  elidentibus. 


Consideratio    mas-  *2*.  Habebitur  ieitur,  quidquid  pertinet  ad  aequilibrium  virium  agentium  in  eodem 

sarum  etiam  inter-      1J-i  •  •    n      -i  ••  i_       • 

mediarum.    qua:  piano,  &  connexarum  non  per  virgas  mnexiles  carentes  omni  vi  praster  cohaesionem,  uti 
connectant  massas  eas  vulgO  concipiunt,  sed  hisce  viribus  mutuis.     Et  Theoria  quidem  habebit  locum  turn 

viribus    externis   ,  .  3.  .,  ,.  A     ~n    /-<  •  ••  •  J*  J* 

prsditas,  &  positas  hie,  turn  in  sequentibus  ;  licet  massae  A,  B,  C  non  agant  in  se  invicem  immediate,  sed  sint 
in  aequiiibno.  ^{^  massa;  intermedise,  quae  ipsas  jungant.     Nam  si  inter  massam  B,  &  C  sint  alias  massae 

nullis  externis  viribus  agitatas,  &  positae  in  aequilibrio  cum  hisce  massis,  &  inter  se,  ac  prima, 
quae  venit  post  B,  agat  in  ipsam  vi  motrice  asquali  BP,  aget  &  B  in  ipsam  vi  asquali  :  quare 
debebit  ilia  ad  servandum  aequilibrium  urgeri  a  secunda,  quae  est  post  ipsam,  vi  aequali 
in  partes  contrarias.  Hinc  asquali  contraria  aget  tertia  in  secundam,  ut  secunda  in 
asquilibrio  sit,  &  ita  porro,  donee  deveniatur  ad  C,  ubi  habebitur  vis  motrix  asqualis  motrici, 
quae  erat  in  B,  &  erunt  vires  BP,  CV  acceleratrices  in  ratione  reciproca  massarum  B,  &  C, 
cum  vires  illas  motrices  asquales  sint  producta  ex  acceleratricibus  ductis  in  massas.  At  si 
circumquaque  sint  massas  quotcunque  cum  vacuis  quibuscunque,  ac  ubicunque  interjectis, 
quae  connectantur  cum  punctis  A,  B,  C,  affectis  illis  tribus  viribus  externis,  quarum  una 
concipitur  provenire  a  fulcro,  una  solet  appellari  potentia,  &  una  resistentia,  ac  vires  illas 
externas  QB,  HA  concipiantur  resolutae  singulas  in  binas  agentes  secundum  eas  rectas, 


A  THEORY  OF  NATURAL  PHILOSOPHY  247 

of  the  ratio  by  ED/AD,  as  AB.ED/AD.BD  is  to  AE/AD.  By  a  similar  argument  we 
obtain  also  that  sinRDA  is  to  sinEDE  as  AB.ED/AD.BD  is  to  BE/BD  ;  from  which  the 
whole  proposition  is  clear. 

320.  If  the  point  D  goes  off  to  infinity,  &  the  directions  of  the  forces  thus  become  A  more  simple 
parallel  to  one  another,  the  ratios  of  the  straight  lines  ED,  AD,  BD  finally  become  ratios  caTe^'para'uefem6 
of  equality.     Hence,  in  that  case,  the  three  forces  are  to  one  another  as  AB  to  AE  to  EB  ; 

&  the  first  of  these  is  equal  to  the  sum  of  the  other  two.  Imagine  straight  lines  drawn 
parallel  to  the  directions  of  the  forces,  through  the  centres  of  gravity  of  all  three  masses, 
&  let  that  one  of  the  masses  which  lies  between  the  parallels  drawn  through  the  other 
two  be  called  the  middle  mass  ;  then,  if  we  draw  in  any  given  direction  straight  lines  from 
the  masses  to  meet  the  parallels,  the  distances  from  the  parallels  measured  along  these  lines 
will  be  as  AB,  EB  ;  for  the  distances  bear  the  same  given  ratio  to  AB,  EB,  on  account  of 
the  given  directions.  Hence  for  parallel  forces  we  obtain  the  following  theorem.  Parallel 
motive  forces  for  any  two  out  of  three  masses  are  to  one  another  inversely  as  the  distances  from  a 
common  direction  -passing  through  the  third  ;  &  the  accelerations  have  in  addition  the  inverse 
ratio  of  the  masses.  The  middle  acceleration  is  in  an  opposite  direction  to  that  of  the  others  ; 
y  the  middle  motive  force  is  equal  to  the  sum  of  the  other  two,  whilst  either  outside  one  is 
equal  to  the  difference  of  the  other  two. 

321.  The  theorem  of  the  preceding  article  will  yield  the  centre  of  equilibrium  for  Application  of  the 
any  forces,  whether  diverging  or  converging.     For  instance,  if  A,  B,  C  are  three  masses  ^£t™  "^  *°*^ 
(&  in  the  term  masses,  single  points  can  also  be  understood  to  be  included),  of  which  two,  brium. 

A  &  B  say,  are  acted  upon  by  external  motive  forces ;  then  the  mass  will  be  able  to  eliminate 
these  by  means  of  mutual  forces,  &  remain  in  equilibrium,  &  then  to  eliminate  the  mutual 
forces  entirely  by  changing  slightly  their  mutual  distances,  as  required  ;  provided  that, 
before  the  application  of  those  external  forces,  they  were  in  positions  corresponding  to  a 
sufficiently  strong  limit  point  of  cohesion,  &  the  force  on  the  mass  C  was  cancelled  by  a 
fulcrum  opposite  to  the  direction  DC,  or  by  a  contrary  suspension  ;  &  so  long  as  the  two 
forces  multiplied  each  by  its  corresponding  mass  preserve  the  conditions  stated  as  requisite 
in  the  above,  namely,  that  both  tend  to  the  same  point  or  both  away  from  it,  or  if  they 
are  parallel  both  have  the  same  direction,  when  they  both  together  fall  within  the  triangle 
ABC,  or  both  tend  away  from  it ;  or  if,  on  the  other  hand,  when  one  of  them  falls  within  the 
triangle  &  the  other  away  from  it,  the  one  tends  to  the  point  of  intersection  &  the  other 
away  from  it,  or  if  they  are  parallel  have  opposite  directions.  Further,  if  they  are  parallel, 
they  are  to  one  another  as  the  distances  from  the  direction  of  forces  which  passes  through 
C  ;  if  they  are  convergent,  they  are  inversely  as  the  sines  of  the  angles  between  their  directions 
&  the  straight  line  through  C  to  their  point  of  intersection  ;  or  are  in  the  inverse  ratio 
of  the  sines  of  the  angles  between  their  directions  &  the  straight  lines  AC,  BC  &  the  ratio 
of  these  straight  lines  jointly. 

322.  It  is  moreover  quite  easy  by  means  of  the  theorems  to  determine  also  the  force  Determination    of 
on  the  fulcrum  placed  at  C  ;    this,  in  the  case  of  parallelism,  will  be  equal  to  the  sum  or  |u]ecruf^rce  on  the 
the  difference  of  the  other  two  forces  according  as  C  is  the  middle  or  one  of  the  outside 

masses.  In  all  other  cases,  it  will  be  equal  to  the  sum  or  difference  of  the  other  forces, 
in  a  similar  way,  if  these  are  reduced  to  the  direction  of  the  force  on  C,  the  remaining  pairs 
of  forces  that  are  associated  with  the  former  in  the  resolution  cancelling  one  another  on 
account  of  their  being  equal  &  opposite. 

323.  Hence  may  be  obtained  all  things  that  relate  to  the  equilibrium  of  forces  acting  investigation  of  the 
in  one  plane,  &  connected,  not  by  inflexible  rods  lacking  all  force  except  cohesion,  but  by  ^e  "iJJ*1^*,!^0 
these  mutual  forces.     The  Theory  holds  good  indeed,  not  only  here,  but  also  in  what  diate  masses  con- 
follows  ;  that  is  to  say,  although  the  bodies  A,  B,  C  may  not  act  upon  one  another  directly,  "l^"^^^  the^f 
yet  there  are  other  intermediate  masses  which  connect  them.     For,  if  between  A  &  B  ternai  forces  act,  & 
there  were  other  masses  not  influenced  by  any  external  forces,  &  placed  in  equilibrium  fr?um    '"    equUl" 
with  these  masses  &  with  one  another,  then  the  first  mass  which  comes  after  B  will  act 

upon  B  with  a  motive  force  equal  to  BP,  &  B  will  act  upon  it  with  an  equal  force  ;  hence, 
to  preserve  the  equilibrium,  this  mass  must  be  acted  upon  by  the  second,  the  one  which 
comes  next  after  it,  with  a  force  equal  &  opposite  to  this.  Hence  it  follows  that  the  third 
must  act  on  this  second  with  a  force  equal  &  opposite  to  that,  in  order  that  the  second 
may  be  in  equilibrium  ;  &  so  on,  until  we  come  to  C,  where  we  have  a  motive  force  equal 
to  that  acting  on  B  ;  &  the  accelerations  BP,  CV  will  be  in  the  inverse  ratio  of  the  masses 
B  &  C,  since  the  equal  motive  forces  are  proportional  to  the  products  of  the  accelerations 
into  the  masses.  Moreover,  if  in  any  positions  there  are  any  number  of  masses,  having 
any  empty  spaces  interspersed  anywhere,  &  these  are  in  connection  with  three  masses  A, 
B,  C,  which  are  under  the  influence  of  those  three  forces,  of  which  one  is  assumed  to  be 
produced  by  a  fulcrum,  one  is  usually  termed  the  power,  &  the  third  the  resistance  ;  & 
if  the  external  forces  BQ,  HA  are  considered  to  be  resolved  each  into  two  parts  acting  along 


248  PHILOSOPHISE  NATURALIS  THEORIA 


ilia  tria  puncta  conjungunt  ;  poterit  elisis  mutuo  reliquis  omnibus  aequilibrium 
constituentibus  deveniri  ad  vires  in  punctis  binis,  ut  A,  &  C,  acceleratrices  contrarias  viribus 
BP,  BR,  &  reciproce  proportionales  massis  ipsarum  respectu  massae  B  ;  licet  ipsae  proveniant 
a  massis  quibusvis  etiam  non  in  eadem  directione  sitis,  &  agentibus  in  latus  :  nam  per 
ejusmodi  resolutionem,  &  ejusmodi  virium  considerationem  adhuc  habetur  aequilibrium 
totius  systematis  affecti  in  illis  tribus  punctis  per  illas  tres  vires,  cum  assumantur  in  iis 
tantummodo  vires  motrices  contrarias,  &  sequales  :  unde  fit,  ut  etiam  illae,  qua;  praeterea 
ad  has  in  illis  considerandas  assumuntur,  &  per  quas  connectuntur  cum  reliquis  massis, 
se  mutuo  elidant. 

Qui  motus,  ubi  non  [1501324.    Quod   si  vires   eiusmodi  non   fuerint  in   ea   ratione  inter  se  ;    non  poterunt 

habeatur   aequili-    T>     o     \  •  -1-1     •  i  i-  •  • 

brium.  puncta  D,  oc  A  esse  in  aequuibno,  sed  consequetur  motus  secundum  directionem  ejus,  quae 

prevalet  :  ac  si  omnis  motus  puncti  C  fuerit  impeditus  ;  habebitur  conversio  circa  ipsum  C. 
Extensio  ad  aequi-  325.  Quod  si  non  in  tribus  tantummodo  massis  habeantur  vires  externae,  sed  in  pluribus  ; 

imrssanim0t&UIinde  li^bit  considerare  quanvis  aliam  massam  carentem  omni  externa  vi,  &  earn  concipere 
principium  generate  connexam  cum  singulis  reliquarum  massis,  &  massa  C  per  vires  mutuas,  ac  habebitur  itidem 
ratio  momentorunf  Theoria  pro  sequilibrio  omnium,  cum  positione  omnium  constanter  servata  etiam  sine 
ulla  figurse  mutatione,  quae  sensu  percipi  possit.  Quin  immo  si  singulae  vires  illae  externae 
resolvantur  in  duas,  quarum  altera  urgeat  in  directione  rectse  transeuntis  per  C,  ac  elidatur 
vi  proveniente  a  solo  puncto  C,  &  altera  agat  perpendiculariter  ad  ipsam,  ut  habeatur 
aequilibrium  in  singulis  ternariis  ;  oportebit  esse  singulas  vires  novae  massae  assumptae  ad 
vim  ejus,  cum  qua  conjungitur,  in  ratione  reciproca  distantiarum  ipsarum  massarum  a  C  ; 
cum  jam  sinus  anguli  recti  ubique  sit  idem.  Debebunt  autem  omnes  vires,  quae  in  massam 
assumptam  agunt  directionibus  contrariis,  se  mutuo  elidere  ad  habendum  aequilibrium. 
Quare  debebit  summa  omnium  productorum  earum  virium,  quae  urgent  conversione  in 
unam  plagam,  per  ipsarum  distantias  a  centre  conversionis,  aequari  summas  productorum 
earum,  quae  urgent  in  plagam  oppositam,  per  distantias  ipsarum,  ut  habeatur  aequilibrium  ; 
curnque  arcus  circulares  in  ea  conversione  descripti  dato  tempusculo  sint  illis  distantiis 
proportionales,  &  proportionales  sint  ipsis  arcubus  velocitates  ;  debebunt  singularum 
virium  agentium  in  unam  plagam  producta  per  velocitates,  quas  haberent  puncta,  quibus 
applicantur  secundum  suam  directionem,  si  vincerentur,  vel  contra,  si  vincerent,  simul 
sumpta  aequari  summae  ejusmodi  productorum  agentium  in  plagam  oppositam.  Atque 
inde  habetur  principium  pro  machinis  &  simplicibus,  &  compositis,  ac  notio  illius,  quod 
appellant  momentum  virium,  deducta  ex  eadem  Theoria. 

Appiicatio  ad  om-  326.  Casus  trium  tantummodo  massarum  exhibet  vectem,  cujus  brachia  sint  utcunque 

ma  vectmm  genera.  jnflexa>  Quod  si  tres  massae  jaceant  in  directum,  efformabunt  rectilineum  vectem,  qui 
quidem  applicatis  viribus  inflectetur  semper  nonnihil,  ut  &  in  superioribus  casibus  semper 
non  nihil  a  priore  positione  discedet  systema  novis  viribus  externis  affectum  ;  sed  is  discessus 
poterit  esse  utcunque  exiguus,  ut  supra  monui  :  si  limites  sint  satis  validi  ;  adeoque  poterit 
adhuc  vectis  esse  ad  sensum  rectilineus.  Turn  vero  vires  externae  debebunt  esse  unius 
directionis,  &  contrariae  direction!  vis  mediae,  &  binae  quaevis  ex  iis  erunt  ad  se  invicem 
reciproce,  ut  distantiae  a  tertia.  Inde  autem  oriuntur  tria  genera  vectium  :  si  fulcrum, 
vel  hypomochlium,  sit  in  medio  in  E,  vis  in  altero  extremo  A,  [151]  resistentia  in  altero 
B  ;  vis  ad  resistentiam  est,  ut  BE,  distantia  resistentiae  a  fulcro,  ad  AE  distantiam  vis  ab 
eodem  :  fulcrum  autem  sentiet  summam  virium.  Et  quod  de  hoc  vectis  genere  dicitur, 
id  omne  ad  libram  pariter  pertinet,  quae  ad  hoc  ipsum  vectis  genus  reducitur.  Si  fulcrum 
sit  in  altero  extremo,  ut  in  B,  vis  in  altero,  ut  in  A,  &  resistentia  in  medio,  ut  in  E  ;  vis 
ad  resistentiam  erit  in  ratione  distantiae  EB  ad  distantiam  majorem  AB,  cujus  idcirco 
momentum,  seu  energia,  augetur  in  ratione  suae  distantiae  AB  ad  EB,  ut  nimirum  possit 
tanto  majori  resistentiae  aequivalere.  Si  demum  fuerit  quidem  fulcrum  in  altero  extremo 
B,  &  resistentia  in  A,  vis  prior  in  E  ;  turn  e  contrario  erit  resistentia  ad  vim  in  majore 
ratione  AB  ad  EB,  decrescente  tantundem  hujus  energia,  seu  momento.  In  utroque 
autem  casu  fulcrum  sentiet  differentiam  virium. 


trime^'vectibus  327-  Qu°d  si  perticae  utcunque  inclinatae  applicetur  pondus  in  aliquo  puncto  E,  &  bini 

&  principium   pro  humeros  supponant  in  A,  &  B,  sentient  ponderis  partes  inaequales  in  ratione  reciproca 

statera  ;  cur  totum  djstantiarum  aD  ipso  ;    &  si  e  contrario  bina  pondera  suspendantur  in  A,  &  B  utcunque 

ponQus     consiQ.crG~  __,           ,         •*-,  .             .                          *       \      o     T>      *         * 

tur,   ut   coiiectum  inaequalia,  assumpto  autem  puncto  E,  cujus  distantiae  a  punctis  A,  &  J3  sint  in  ratione 

in  centre    ravitatis. 


in  centre  gravitatis. 


A  THEORY  OF  NATURAL  PHILOSOPHY  249 

the  lines  which  join  the  three  points ;  then  it  will  be  possible,  all  the  other  forces  constituting 
the  equilibrium  cancelling  one  another,  to  arrive  at  accelerations  for  the  two  points  A  &  C 
say,  in  opposite  directions  to  the  forces  BP,  BR,  &  inversely  proportional  to  their  masses 
with  regard  to  the  mass  B.  This  will  be  the  case,  even  although  they  may  proceed  from 
any  masses  not  lying  in  the  same  direction,  &  acting  to  one  side  ;  for,  by  means  of  resolution 
of  this  kind,  &  a  consideration  of  such  forces,  we  yet  have  equilibrium  of  the  whole  system 
affected  at  the  three  points  by  the  three  forces,  since  here  are  assumed  only  motive  forces 
such  as  are  equal  &  opposite.  Hence  it  follows  that  the  former,  which  are  assumed  in 
addition  for  the  consideration  of  the  latter  in  such  cases,  &  by  which  they  are  connected 
with  the  other  masses,  must  also  cancel  one  another. 

324.  But  if  such  forces  are  not  in  this  ratio  to  one  another,  the  points  B  &  A  cannot  The  nature  of  the 
be  in  equilibrium;  but  motion  would  follow  in  the  direction  of  that  which  preponderated  ;  "r°um^ d o^s^no t 
also  if  all  motion  of  the  point  C  were  prevented,  then  there  would  be  rotation  about   C.  obtain. 

325.  Now  if  we  have  external  forces  acting,  not  on  three  masses  only,  but  on  several,  Extension   to   the 
we  can  consider  any  one  mass  to  be  without  an  external  force,  &  suppose  that  this  mass  equllib"urr,?fc^ 

•  i        r     i  n  /~*    i  i    r  rt       i       f-ni  *ii    nuniDer  01   masses  , 

is  connected  to  each  of  the  others,  &  to  the  mass  (_,  by  mutual  forces  ;   &  the  Theory  will  &  thence  a  general 
hold  good  for  the  equilibrium  of  them  all,  with  the  position  of  them  all  constantly  maintained  Prin.clPle,     £°r 

<•  r  i-  i        T«        i  T     11     i  i   r  machines     &     the 

without  any  change  of  figure  so  tar  as  can  be  observed,     .further,  it  all  the  external  forces  ratio  of  moments. 

are  resolved  each  into  two  parts,  of  which  one  acts  along  the  straight  line  passing  through 

C,  &  is  cancelled  by  a  force  proceeding  from  C  alone,  &  the  other  acts  perpendicularly 

to  this  line,  so  that  equilibrium  is  obtained  for  each  set  of  three  ;   then  it  will  be  necessary 

that  each  of  the  forces  on  the  new  mass  chosen  will  be  to  the  force  of  that  to  which  it  is 

joined  in  the  inverse  ratio  of  these  masses  from  C,  since  now  the  sines  of  the  right  angles 

are  everywhere  the  same.     Also  all  the  forces  which  act  on  the  chosen  mass  in  opposite 

directions,  must  cancel  one  another  to  maintain  equilibrium.     Hence  the  sum  of  all  the  forces 

which  tend  to  produce  rotation  in  one  direction,  each  multiplied  by  its  distance  from  the 

centre  of  rotation,  must  be  equal  to  the  sum  of  the  products  of  the  forces  which  tend  to 

produce  rotation  in  the  opposite  direction,  multiplied  by  their  distances,  in  order  that 

equilibrium  may  be  maintained.     Since  the  circular  arcs  in  this  rotation  which  are  described 

in  any  interval  of  time  are  proportional  to  the  distances,  &  these  are  proportional  to  the 

velocities  in  the  arcs,  it  follows  that  the  products  of  each  of  the  forces  acting  in  one  direction 

by  the  velocities  which  correspond  to  the  points  to  which  they  are  applied,  in  the  direction 

of  the  forces  if  they  are  overcome,  &  in  the  opposite  direction  if  they  overcome,  all  together 

must  be  equal  to  the  sum  of  the  like  products  acting  in  the  other  direction.     Hence  is 

derived  a  principle  for  machines,  both  simple  &  complex ;   &  also  an  idea  of  what  is  called 

the  moment  of  forces ;    &  these  have  been  deduced  from  this  same  Theory. 

326.  The  case  of  three  masses  only  yields  the  case  of  the  lever,  whose  arms  are  curved  Application  to  ail 
in  any  manner.     But  if  the  three  masses  lie  in  one  straight  line,  they  will  form  a  rectilinear  kmds  o£  levers- 
lever  ;   now  this,  on  the  application  of  forces,  will  always  be  bent  to  some  degree ;  just  as, 

in  the  cases  above,  the  system  when  affected  by  fresh  external  forces  always  departed  from 
its  original  position  to  some  extent.  But  this  departure  is  exceedingly  slight  in  every  case, 
as  I  mentioned  above,  if  only  the  limit-points  are  sufficiently  strong  ;  &  thus  the  lever 
can  still  be  considered  as  sensibly  rectilinear.  In  this  case,  then,  the  external  forces  must 
be  in  the  same  direction,  &  in  an  opposite  direction  to  that  of  the  middle  force,  &  any  two 
of  them  must  be  to  one  another  in  the  inverse  ratio  of  their  distances  from  the  third.  Now 
from  this  there  arise  three  kinds  of  levers.  If  the  fulcrum,  or  lever-support,  is  in  the 
middle  at  E,  the  force  acting  on  one  end  A  &  the  resistance  at  the  other  end  B  ;  then  the 
ratio  of  the  force  to  the  resistance  is  as  BE,  the  distance  of  the  resistance  from  the  fulcrum, 
to  AE  the  distance  of  the  force  from  it ;  &  the  force  on  the  fulcrum  will  be  the  sum  of 
the  two.  What  is  said  about  this  kind  of  lever  applies  equally  well  to  the  balance,  which 
reduces  to  this  kind  of  lever.  If  the  fulcrum  should  be  at  one  end,  at  B  say,  the  force  at 
the  other,  A,  &  the  resistance  in  the  middle,  at  E ;  then  the  force  is  to  the  resistance  in 
the  ratio  of  the  distance  EB  to  the  greater  distance  AB  ;  &  therefore  the  moment,  or 
energy,  will  increase  in  the  ratio  of  the  distance  AB  to  EB,  so  that  indeed  it  may  be  able 
to  balance  a  much  greater  resistance  in  proportion.  Finally,  if  the  fulcrum  were  at  one 
end,  B,  the  resistance  at  A,  &  the  former  force  at  E ;  then,  on  the  contrary,  the  resistance 
is  to  the  force  in  the  greater  ratio  of  AB  to  EB,  thus  decreasing  its  energy  or  momentum 
in  the  same  proportion.  In  both  these  latter  cases  the  force  on  the  fulcrum  will  be  equal 
to  the  difference  of  the  forces.  Consequences  of 

327.  Now,  if  to  a  long  pole,  inclined  at  any  angle  to  the  horizontal,  a  weight  is  applied  *his   ^ocffine  °f 
at  any  point  E  ;   &  if  two  men  place  their  shoulders  under  the  pole  at  A  &  B  ;   then  they  dpte  ol  the6  steel. 
will  support  unequal  parts  of  the  weight,  in  the  inverse  ratio  of  their  distances  from  it.  ytrd'uTh?  ,reason 

/-i  i       •  r  ,  i         •    i  r  -,-,,  .    T.  .        T-I   •      why  the  whole  may 

Conversely,  if  two  unequal  weights  of  any  sort  are  suspended  from  A  &  B,  &  a  point  E  is  be  considered  as  if 
taken  whose  distances  from  the  points  A  &  B  are  in  the  inverse  ratio  of  the  weights,  &  so  collected    at  the 

centre  of  gravity. 


250 


PHILOSOPHISE  NATURALIS  THEORIA 


Theoriam  exhibere 
egregie  itidem  cen- 
trum oscillationis. 
Quid  ipsum  sit. 


Preparatio  ad  solu- 
tionem  problematis 
quaerentis  ipsum 
centrum. 


Solutio  problematis, 
ac  demonstratio. 


reciproca  ipsorum  ponderum,  adeoque  massarum,  quibus  pondera  proportionalia  sunt, 
quod  idcirco  erit  centrum  gravitatis ;  suspensa  per  id  punctum  pertica,  vel  supposito 
fulcro,  habebitur  aequilibrium,  &  in  E  habebitur  vis  aequalis  summae  ponderum.  Quin 
immo  si  pertica  sit  utcunque  inflexa,  &  pendeant  in  A,  &  B  pondera  ;  suspendatur  autem 
ipsa  pertica  per  C  ita,  ut  directio  verticalis  transeat  per  centrum  gravitatis ;  habebitur 
sequilibrium,  &  ibi  sentietur  vis  aequalis  summae  ponderum,  cum  ob  naturam  centri 
gravitatis  debeant  esse  singula  pondera,  seu  massae  ductae  in  suas  perpendiculares  distantias 
a  linea  verticali,  quam  etiam  vocant  lineam  directionis,  hinc,  &  inde  aequalia.  Nam  vires 
ponderum  sunt  parallelae,  &  in  iis  juxta  num.  320  satis  est  ad  aequilibrium,  si  vires  motrices 
sint  reciproce  proportionales  distantiis  a  directione  virium  transeunte  per  tertium  punctum  : 
sentietur  autem  in  suspensione  vis  aequalis  summae  ponderum.  Atque  inde  fluit,  quidquid 
vulgo  traditur  de  aequilibrio  solidorum,  ubi  linea  directionis  transit  per  basim,  sive  fulcrum, 
vel  per  punctum  suspensionis,  &  simul  illud  apparet,  cur  in  iis  casibus  haberi  possit  tota 
massa  tanquam  collecta  in  suo  centro  gravitatis,  &  habeatur  aequilibrium  impedito  ejus 
descensu  tantummodo.  Gravitas  omnium  punctorum  non  applicatur  ad  centrum  gravi- 
tatis, nee  ibi  ipsa  agit  per  sese  ;  sed  ejusmodi  esse  debent  distantiae  punctorum  totius 
systematis,  ut  inter  fulcrum,  &  punctum  ipsi  imminens  habeatur  vis  quaedam  aequalis 
summae  virium  omnium  parallelarum,  &  directa  ad  partes  oppositas  directionibus  illarum. 
[152]  328.  At  non  minus  feliciter  ex  eadem  Theoria,  &  ex  eodem  illo  theoremate, 
fluit  determinatio  centri  oscillationis.  Pendula  breviora  citius  oscillant,  remotiora  lentius. 
Quare  ubi  connexa  sunt  inter  se  plura  pondera,  aliud  propius  axi  oscillationis,  aliud  remotius 
ab  ipso,  oscillatio  neque  fiet  tarn  cito,  quam  requirunt  propiora,  neque  tarn  lente,  quam 
remotiora,  sed  actio  mutua  debebit  accelerare  haec,  retardare  ilia.  Erit  autem  aliquod 
punctum,  quod  nee  accelerabitur,  nee  retardabitur,  sed  oscillabit,  tanquam  si  esset  solum. 
Illud  dicitur  centrum  oscillationis.  Determinatio  illius  ab  Hugenio  primum  est  facta,  sed 
precario,  &  non  demonstrate  principle  :  turn  alii  alias  itidem  obliquas  inierunt  vias,  ac 
praecipuas  quasque  methodos  hue  usque  notas  persecutus  sum  in  Supplements  Stayanis 
§  4  lib.  3.  En  autem  ejus  determinationem  simplicissimam  ope  ejusdem  theorematis 
numeri  313. 

329.  Sint  plures  massae,  quarum  una  A  in  fig.  63,  mutuis  viribus  singulae  connexae  cum 
P,  cujus  motus  sit  impeditus  suspensione,  vel  fulcro,  &  cum  massa  Q  jacente  in  quavis 
recta  PQ,  cujus  massae  Q  motus  a  mutuo  nexu  nihil  turbetur,  quae  nimirum  sit  in  centro 
oscillationis.     Porro  hie  cum  massas  pone  in  punctis  spatii  A,  P,  Q,  intelligo  vel  puncta 
singula,  vel  quaevis  aggregata  punctorum,  quae  concipiantur,  ut  compenetrata  in  iis  punctis. 
Velocitati  jam  acquisitae  in  descensu  nihil  obstabit  is  nexus, 

cum  ea  sit  proportionalis  distantiae  a  puncto  suspensionis  P, 
nisi  quatenus  per  eum  nexum  retrahentur  omnes  massae  a 
recta  tangente  ad  arcum  circuli,  sustinente  puncto  ipso  sus- 
pensionis justa  num.  282  vim  mutuam  respondentem  iis  om- 
nibus viribus  centrifugis.  Resoluta  gravitate  in  duas  partes, 
quarum  altera  agat  secundum  rectam,  quas  jungit  massam  cum  A 
P,  altera  sit  ipsi  perpendicularis,  idem  punctum  P  sustinebit 
etiam  priorem  illam,  posterior  autem  determinabit  massas 
ad  motus  AN,  QM,  perpendiculares  ipsis  AP,  QP,  ac  pro- 
portionales per  num.  301  sinubus  angulorum  APR,  QPR, 
existente  PR  verticali.  Sed  nexus  coget  describere  arcus 
similes,  adeoque  proportionales  distantiis  a  P.  Quare  si  sit  AO 
spatium,  quod  vi  gravitatis  obliquae,  sed  ex  parte  impeditas  a 
nexu,  revera  percurrat  massa  A ;  quoniam  Q  non  turbatur, 
adeoque  percurrit  totum  suum  spatium  QM  ;  erit  QM  ad 
AO,  ut  QP  ad  AP.  Demum  actio  ex  A  in  Q  ad  actionem 
ex  Q  in  A  proportionalem  ON,  erit  ex  theoremate  numeri 
3i4utestQ  X  QPadA  X  AP,  &  omnes  ejusmodi  actiones 
ab  omnibus  massis  in  Q  debebunt  evanescere,  positivis  &  negativis  valoribus  se  mutuo 
elidentibus.  Ex  illis  tribus  proportionibus,  &  hac  aequalitate  res  omnis  sic  facillime 
expeditur. 

330.  Dicatur  QM  =  V,  sinus  APR  =  a,  sinus  QPR  =  q.     Erit  ex  prima  proportione 

XV. 


Q 


M 


FIG.  63. 


q  :  a  :  :  QM  =  V  :  AN  =—  X  V.     [153]  Ex  secunda  QP.  AP  :  :  QM  -  V.   AO= 


Sed  ex  tertia 


Quar.ON=(i--Qp)xV. 

Q  X  QP.  A  X  AP  :  :ON  =(  - -~  )  X  V. 

\  q       \j,"  ' 


__ 
'CTxQP' 


A  THEORY  OF  NATURAL  PHILOSOPHY  251 

of  the  masses  to  which  the  weights  are  proportional,  so  that  the  point  is  their  centre  of 
gravity ;  then,  if  the  pole  is  suspended  by  this  point,  or  a  fulcrum  is  placed  beneath  it, 
there  will  be  equilibrium,  &  the  force  at  E  will  be  equal  to  the  sum  of  the  two  weights. 
Further,  if  the  pole  were  bent  in  any  manner,  &  weights  were  suspended  at  A  &  B,  &  the 
pole  itself  were  suspended  at  C,  so  that  the  vertical  direction  passes  through  the  centre 
of  gravity  of  the  weights  ;  then  there  would  be  equilibrium,  &  there  would  be  a  force  at 
C  equal  to  the  sum  of  the  weights.  For,  on  account  of  the  nature  of  the  centre  of  gravity, 
each  of  the  weights,  or  masses,  multiplied  by  its  perpendicular  distance  from  the  vertical 
line,  which  is  also  called  the  line  of  direction,  must  be  equal  on  the  one  side  &  on  the  other. 
For  the  forces  of  the  weights  are  parallel ;  &  for  such,  according  to  Art.  320,  it  is  sufficient 
for  equilibrium,  if  the  motive  forces  are  proportional  inversely  to  the  distances  from  the 
direction  of  forces  passing  through  the  third  point ;  moreover  there  will  be  experienced 
at  the  point  of  suspension  a  force  equal  to  the  sum  of  the  weights.  Hence  is  derived  every- 
thing that  is  usually  taught  concerning  the  equilibrium  of  solids,  where  a  line  of  direction 
passes  through  the  base,  or  through  the  fulcrum,  or  through  the  point  of  suspension  ;  at 
the  same  time  we  get  a  clear  perception  of  the  reason  why  in  such  cases  the  whole  mass 
can  be  considered  as  if  it  were  condensed  at  its  centre  of  gravity,  &  equilibrium  can  be 
obtained  by  merely  preventing  the  descent  of  this  point.  The  gravity  of  all  the  points  is 
not  applied  at  the  centre  of  gravity,  nor  does  it  act  there  of  itself ;  but  the  distances  of 
the  points  of  the  whole  system  must  be  such  that  between  the  fulcrum  &  the  point  hanging 
just  over  it  there  must  be  a  certain  force  equal  to  the  sum  of  all  the  parallel  forces,  &  directed 
so  as  to  be  opposite  to  their  direction. 

328.  In  a  no  less  happy  manner  there  follows  from  this  same  Theory,  &  from  the  The  Theory  affords 
very  same  theorem,  the  determination  of  the  centre  of  oscillation.     Shorter  pendulums  an  excellent  expia- 

...  i        i  TT  i  i          •    i  nation     of    the 

oscillate  more  quickly,  &  longer  ones  more  slowly.  Hence  when  several  weights  are  centre  of  oscillation 
connected  together,  one  nearer  to  the  axis  of  oscillation,  &  another  more  remote  from  it,  as  well- 
the  oscillation  is  neither  so  fast  as  that  required  by  the  nearer,  nor  so  slow  as  that  required 
by  the  more  remote  ;  but  a  mutual  action  must  accelerate  the  one  &  retard  the  other. 
Moreover  there  will  be  one  point,  which  will  be  neither  accelerated  nor  retarded,  but 
will  oscillate  as  if  it  were  alone  ;  that  point  is  called  the  centre  of  oscillation.  Its  deter- 
mination was  first  made  by  Huygens,  but  from  a  principle  that  was  doubtful  &  unproved. 
After  him,  others  came  upon  it  indirectly,  some  in  one  way  &  some  in  another ;  &  I 
investigated  some  of  the  best  methods  then  known  in  the  Supplements  to  Stay's  Philosophy, 
§  4,  Bk.  3.  Now  I  present  you  with  an  exceedingly  simple  determination  of  it,  derived 
from  that  same  theorem  of  Art.  313. 

329.  Suppose  there  are  several  masses,  of  which  in  Fig.  63  one  is  at  A,  &  that  each  of  Preparation  for  the 
these  is  connected  to  P  by  mutual  forces ;  &  let  the  motion  of  P  be  prevented  by  suspension,  P°10Uye<nJl  Of°find^e 
or  by  a  fulcrum  ;   also  let  A  be  connected  with  a  mass  Q  lying  in  a  straight  line  PQ,  &  let  this  centre. 

the  motion  of  this  mass  Q  be  in  no  way  affected  by  the  mutual  connection,  as  will  happen 
if  Q  is  at  the  centre  of  oscillation.  Now,  when  I  place  masses  at  the  points  of  space  A, 
P,  Q,  I  intend  single  points  of  matter,  or  any  aggregates  of  such  points,  which  may  be 
considered  as  condensed  at  those  points  of  space.  The  connection  will  not  oppose  in  any 
way  the  velocity  already  acquired  in  descent,  since  it  is  proportional  to  the  distance  from 
the  point  of  suspension  P  ;  except  in  so  far  as  all  the  masses  are  pulled  out  of  the  tangent 
line  into  a  circular  arc  by  the  connection,  the  point  of  suspension  itself  being  under  the 
influence  of  a  mutual  force  corresponding  to  all  the  centrifugal  forces;  If  gravity  is 
resolved  into  two  parts,  one  of  which  acts  along  the  straight  line  joining  the  mass  to  P, 
&  the  other  perpendicular  to  it ;  then  the  point  P  will  sustain  the  former  of  these  as  well, 
but  the  latter  will  give  to  the  masses  the  motions  AN,  QM,  respectively  perpendicular 
to  AP,  QP,  &  proportional,  by  Art.  301,  to  the  sines  of  the  angles  APR,  QPR,  where  PR 
is  the  vertical.  But  the  connection  forces  them  to  describe  arcs  that  are  similar,  &  therefore 
proportional  to  the  distances  from  P.  Hence,  if  AO  is  the  space,  which  under  the  oblique 
force  of  gravity,  but  partly  hindered  by  the  connection,  the  mass  A  would  really  pass  over  ; 
then,  since  Q  is  not  affected,  &  will  thus  pass  over  the  whole  of  its  course  QM,  we  shall 
have  QM  to  AO  as  QP  to  AP.  Lastly,  the  action  of  A  on  Q  is  to  the  action  of  Q  on  A, 
(which  is  proportional  to  ON),  as  Q  X  QP  is  to  A  X  AP,  by  the  theorem  of  Art.  314; 
&  all  such  actions  from  all  the  masses  upon  Q  must  vanish,  the  positive  &  negative  values 
cancelling  one  another.  From  the  three  proportions  &  this  equality  the  whole  question 
is  worked  out  in  the  easiest  possible  way. 

330.  Suppose  QM  =  V,  the  sine  of  APR  =  a,  the  sine  of  QPR  =  q.     Then,  since  Solution   of   the 
from   the   first   proportion,  q  :  a  =  QM  :  AN,  therefore   AN  =  a.V/q  ;    &,   since   from  demonstration 
the   second    proportion,   QP  :  AP  =  QM  :    AO,    therefore   AO  =  AP.V/QP.     Hence 

ON=  (a/q  —  AP/QP).V.  But,  from  the  third  proportion,  Q  X  QP  is  to  A  X  AP  as 
ON  is  to  the  action  of  A  on  Q.  Therefore  the  action  on  Q  due  to  the  connection  with  A 


252  PHILOSOPHIC  NATURALIS  THEORIA 

quas  erit  actio  in  Q  ex  nexu  cum  A.  At  eodem  pacto  si  esset  alibi  alia  massa  B  itidem 
connexa  cum  P,  &  Q,  actio  in  Q  inde  orta  haberetur,  positis  B,  b  loco  A,  a  ;  &  ita  porro 
in  quibusquis  massis  C,  D,  &c.  Omnes  autem  isti  valores  positi  =  o,  dividi  possent  per 

Q— — Qp,  utique  commune  omnibus,  &  deberent  e  valoribus  conclusis  intra  parentheses  ii, 


qui  sunt  positivi,  aequales  esse  negativis.     Quare  habebitur 

a  X  A  x  AP  +  b  x  B  x  BP        A  x  AP2  +  B  x  BP2  &c. 

QP 


&  inde  OP  —  ^  v        «•  x  AT   -f-  j>  x  or   etc. 

K  rx  Ax  AP +  b  xinTBP  &c. 
Evoiutjo  casus  pon.  331.  Sint  jam  primo  omnes  massse  in  eadem  recta  linea  cum  puncto  suspensionis 

derum  jacentium  in    T>     s  -11   ...•       •     r\       o  i        r\r»n  i  • 

eadem    recta  cum  "i  &  cum  centro  oscillationis  Q  ;  &  angulus  QPR  aequabitur  cuivis  ex  angulis  APR,  ac  ejus 

puncto  suspension.     ...  A   X  AP2  -I-  B    V  BP2    &r 

is.  sinus  g  smgulis  smubus  a,  b  &c.    Quare  pro  eo  casu  formula  evadit  -j— - — — 

A  x  AP  +  B  x  BP   &c.  ' 

quse  est  ipsa  formula  Hugeniana  pro  ponderibus  jacentibus  recta  transeunte  per  centrum 
suspensionis. 

Et  casus  jacentium  332.  Quod  si  jaceant  extra  ejusmodi  rectam  in  piano  FOR  perpendiculari  ad  axem 

rotationis  transeuntem  per  P ;  sit  G  centrum  commune  gravitatis  omnium  massarum, 
ducanturque  perpendicula  AA',  GG',  QQ'  ad  PR,  &  erit  ut  radius  =  i  ad  a,  ita  AP  ad 
AA'  =  a  x  AP  ;  &  eodem  pacto  QQ'  =  q  x  QP,  GG'  =  g  x  GP.  Substitutis  AA' 
pro  a  X  AP  &  eodem  pacto  BB'  (quam  Figura  non  exprimit)  pro  b  X  BP 

j      ™>             A  x  AP2  +  B  x  BP2  &c     c   ,    .  ,. 

&c.    evadat  QP    =  q  X^ ~A~A^~JTB — pp/fl     •     bed  si  summa   massarum   dicatur  M, 


est  per  num.  245  ex  natura  centri  gravitatis,  A  X  AA'  +  B  X  BB'  &c.  =  M  X  GG'  = 
M  X  g  X  GP.     Habebitur  igitur  valor  QP  radii  nihil  turbati  in  ea  inclinatione 

q     A  x  AP2  +  B  x  BP2  &c. 

g  M  x  GP 


initium      appiica-  [154]  333.    Is    valor    erit    variabilis    pro    varia    inclinatione    ob    valores  sinuum    q,  &  g 

ones Sina<iatuSsCi Jon"  variatos,  nisi  QP  transeat  per  G,  quo  casu  sit  q  =  g  ;  &  quidem  ubi  G  accedit  in  infinitum 

eodem  piano!11"    *  ad  PR,  decrcscente  g  in  infinitum,  si  PQ  non  transeat  per  G,  manente  finito  q,  valor  - 

excrescit  in  infinitum  ;  contra  vero  appellente  QP  ad  PR,  evadit  q  =  o,  &  g  remanet  aliquid, 

adeoque  ?.  evanescit.     Id  vero  accidit,  quia  in  appulsu  G  ad  verticalem  tntum  systema 

vim  acceleratricem  in  infinitum  imminuit,  &  lentissime  acceleratur  ;  adeoque  ut  radius 
PQ  adhuc  obliquus  sit  ipsi  in  ea  particula  oscillationis  infinitesima  isochronus,  nimirum 
jeque  parum  acceleratus,  debet  in  infinitum  produci.  Contra  vero  appellente  PQ  ad  PR 
ipsius  acceleratio  minima  esse  debet,  dum  adhuc  acceleratio  radii  PG  obliqui  est  in 
immensum  major,  quam  ipsa ;  adeoque  brevitate  sua  ipse  radius  compensare  debet 
accelerationis  imminutionem. 

Finis  ejusdemcum  334.  Quare  ut  habeatur  pendulum  simplex  constantis  longitudinis,  &  in  quacunque 

formula  generaii.      inciinatione  isochronum  composito,  debet  radius  PQ  ita  assumi,  ut  transeat  per  centrum 

gravitatis    G,    quo    unico    casu    fit    constanter    q  =  g,    &    formula    evadit    constans 

OP  —       X  x    -•    quae  est  formula  generalis  pro  oscillationibus  in  latus 

M  xGP 

massarum  quotcumque,  &  quomodocunque  collocatarum  in  eodem  piano  perpendiculari 
ad  axem  rotationis,  qui  casus  generaliter  continet  casum  massarum  jacentium  in  eadem 
recta  transeunte  per  punctum  suspensionis,  quern  prius  eruimus. 

Coroliarium     pro  335.  Inde    autem    pro    hujusmodi    casibus    plura    corollaria    deducuntur.     Inprimis 

positione  centri  _atet  .  gravitas  centrum  debere  jacere  in  recta,  qua  a  centra  suspensionis  ducitur  -per  centrum 
gravitatis  ex  eadem  oscillationis,  uti  demonstratum  est  num.  334.  Sed  &  debet  jacere  ad  eandem  partem  cum 
PUnCt°  *PSO  centro  oscillationis.  Nam  utcunque  mutetur  situs  massarum  per  illud  planum, 
manentibus  puncto  suspensionis  P,  &  centro  gravitatis  G,  signum  valoris  quadrati  cujusvis 
AP,  BP  manebit  semper  idem.  Quare  formula  valoris  sui  signum  mutare  non  poterit ; 


A  THEORY  OF  NATURAL  PHILOSOPHY  253 

//?  v  A   v  AP          A   v  AP2\  V 

will  be  (-  np       )  x  n      r>P"     In  tne  same  manner,  if  there  were  another 

^  0  vj-t  vi  X  v2-T 

mass  somewhere  else,  also  connected  with  P  &  Q,  the  action  on  Q  arising  from  its  presence 
would  be  obtained,  if  B  &  b  were  substituted  for  A  &  a  ;  &  so  on  for  any  masses  C,  D, 
&c.  Now,  putting  all  these  values  together  equal  to  zero,  they  can  be  divided  through 
by  V/(Q  x  QP),  which  is  common  to  every  one  of  them  ;  &  those  of  the  values  included 
in  the  brackets  that  are  positive  must  be  equal  to  those  that  are  negative.  Hence  we  have 

(*xAxAP+£xBxBP  +  &c.)/?  -  (A  x  AP2  -f  B  x  BP2  +  &c.)/QP  ; 

A  x  AP2  +  B  x  BP2  +  &c. 

and  hence  QP  =  a.  -    -  i=  -  =  -  =-!  —  ==  - 
a  x  A  x  AP  +  b  X  B  X  BP  +  &c. 

331.  Suppose  now,  first  of  all,  that  all  the  masses  lie  in  one  straight  line  with  the  point  Derivation  of  the 

of  suspension  P,  &  so  with  the  point  of  oscillation  Q  ;    then  the  angle  QPR  will  be  equal  hanging0  ^"fh'e 

to  any  one  of  the  angles  like  APR,  &  its  sine  q  will  be  equal  to  any  one  of  the  sines  a,  b,  same  straight  line 

&c.     Hence  for  this  case  the  formula  reduces  to  Tus  Jensen  P°int  °f 

A  x  AP2  +  B  BP2  +  &c. 


A  x  AP  +  B  x  BP   +  &c.  ' 

&  this  is  the  selfsame  formula  found  by  Huygens  for  weights  lying  in  the  straight  line  passing 
through  the  centre  of  suspension. 

332.  But  if  the  masses  lie  outside  of  any  such  line,  in  the  plane  FOR,  perpendicular  The  case  of  when 

,JJ.  .  t    T\  i         f-~t  •       i  r  •         the  masses  are  not 

to  the  axis  of  rotation  passing  through  P,  suppose  that  G  is  the  common  centre  of  gravity  On  this  line. 
of  all  the  masses,  &  let  perpendiculars  AA',  GG',  QQ'  be  drawn  to  PR.     Then,  since  the 
radius  (=  i)  :  a  =  AP  :  AA',  therefore  AA'  =  a  X  AP  :    &  in  a  similar  manner,  QQ'  = 
q  X  QP,  &  GG'  =  g  X  GP.     Now,  if  AA'  is    substituted  for  a  X  AP,  &  similarly  BB' 
(not  shown  in  the  figure)  for  b  X  BP,  &  so  on  ;   the  formula  will  become 

OP=      A  x  AP2  +  B  x  BP2  +  &c. 
?>A  x  AA'  +  B  x  BB'  +  &c. 

But,  if  the  sum  of  the  masses  is  denoted  by  M,  then,  by  Art.  245,  from  the  nature  of  the 
centre  of  gravity,  we  have  A  X  AA'  +  B  X  BB'  -f  &c.  =  M  X  GG'  =  M  X  g  X  GP  ; 
&  therefore  we  obtain  the  value  of  the  radius  QP,  in  a  form  that  is  independent  of  the 
inclination,  namely, 

q      A 
' 


g'  M  x  GP 

.  The  value  obtained  will  vary  with  various  inclinations,  owing  to  the  varying  Commencement  of 

i  r    i         •  t^-n  i  1^1-         i  •   i_  T    J      J        -U         tne   application   to 

values  of  the  sines  q  &  g,  unless  QP  passes  through  G  ;  in  which  case  q  =  g.     Indeed,  when  oscillations  to  one 

G  approaches  indefinitely  n'ear  to  PR,  &  g  thus  decrease  indefinitely,  if  PQ  does  not  pass  side  of  bodies  lying 

through  G,  thus  leaving  q  finite,  the  value  of  q/g  will  increase  indefinitely.     On  the  other 

hand,  when  QP  coincides  with  PR,  q  —  O,  &  g  will  remain  finite  ;  &  thus  q/g  will  vanish. 

This  indeed  is  just  what  does  happen  ;    for,  when  G  approaches  the  vertical  the  whole 

system  diminishes  the  accelerating    force    indefinitely,  &  it  is  accelerated  exceedingly 

slowly  ;   thus,  in  order  that  the  radius  PQ  whilst  still  oblique  may  be  isochronous  during 

that  infinitesimally  small  part  of  the  oscillation,  that  is  to  say,  may  be  accelerated  by  an 

equally  small  amount,  it  must  be    prolonged  indefinitely.     On  the  other  hand,  as  PQ 

approaches  PR,  its  acceleration  must  be  very  small,  whilst  the  acceleration  of  the  radius 

PG  which  is  still  oblique  is  immensely  greater  in  comparison  with  it  ;    &  thus  the  radius 

PQ  must  by  its  shortness  compensate  for  the  diminution  of  the  acceleration. 

334.  Hence,  in  order  to  obtain  a  simple  pendulum  of  constant  length,  isochronous  Conclusion  of  the 

•      i.        .  -11  •  11  i  j-        -nr\  L  i  T.    ^  •      same,   with  a  gene- 

at  any  inclination  with  the  composite  pendulum,  the  radius  PQ  must  be  so  taken  that  it  rai  formula. 
passes  through  the  centre  of  gravity  G,  in  which  case  alone  q  —  g,  &  the  formula  reduces 
to  a  constant  value  for  QP,  which 

_  A  x  AP2  +  B  X  BP2  +  &c. 

M  xGP 

This  is  a  general  formula  for  oscillations  to  one  side  of  any  number  of  masses,  disposed  in 
any  way  whatever  in  the  same  plane,  the  plane  being  perpendicular  to  the  axis  of  rotation  ; 
&  this  case  contains  in  general  the  case  of  masses  lying  in  the  same  straight  line  through 
the  point  of  suspension,  which  we  have  already  solved. 

335.  Now  for  cases  of  this  sort  many  corollaries  can  be  derived  from  the  theorem  Corollary  with  re- 
proved  above.     First  of  all,  it  is  clear  that  :  —  The  centre  of  gravity  must  lie  in  the  straight  centres  of  oscillation 
line  joining  the  centres  of  oscillation  &  suspension  ;   this  has  been  proved  in  Art.  335.     But  &  gravity  on   the 
also  it  must  lie  on  the  same  side  of  the  -point  of  suspension  as  does  the  centre  of  oscillation.     For 

however  the  positions  of  the  masses  are  changed  in  the  plane,  so  long  as  the  positions  of 
the  points  of  suspension  P  &  of  the  centre  of  gravity  G  remain  unaltered,  the  sign  of  the 
value  of  any  square,  such  as  AP,  BP,  will  remain  the  same.  Hence  the  formula  cannot 


254  PHILOSOPHI/E  NATURALIS  THEORIA 

adeoque  si  in  uno  aliquo  casu  jaceat  Q  respectu  P  ad  eandem  plagam,  ad  quam  jacet  G  ; 

debebit  jacere  semper.     Jacet  autem  ad  eandem  plagam  in  casu,  in  quo  concipiatur,  omnes 

massas  abire  in  ipsum  centrum  gravitatis,  quo  casu  pendulum  evadit  simplex,  &  centrum 

oscillationis  cadit  in  ipsum  centrum  gravitatis,  in  quo  sunt  massae.     Jacebit  igitur  semper 

ad  eandem  partem  cum  G. 

[155]  336.  Deinde  debet  centrum  gravitatis  jaccre  inter  punctum 
bhia'reiiqiTa  ex"iis  suspensionis,  y  centrum  oscillationis.  Sint  enim  in  fig.  64 
punctis.  puncta  A,  P,  G,  Q  eadem,  ac  in  fig.  63,  ducanturque  AG, 

AQ,   &  Aa  perpendicularis   ad   PQ  ;    summa   autem   omnium 

massarum  ductarum  in   suas    distantias    a  recta    quapiam,    vel 

piano,  vel  in  earum  quadrata,  designetur  praefixa  litera  J  soli 

termino   pertinente   ad   massam   A,    ut   contractiores   evadant 

f  A  x  AP2 
demonstrationes.     Erit  ex  formula  inventa  PQ  = 


M  xGP 

Porro  est  AG2  =  AP2  +  GP2  —  2  GP  X  Pa,  adeoque 
AP2  =  AG2  —  GP2  +  2  GP  X  Pa, 

&  J.A  x  GPZ  est    M  x  GP2, 

ob  GP  constantem  ;  ac  J.A  x  Pa  =  M  X  GP,  cum  Pa  sit 
sequalis  distantise  massae  a  piano  perpendiculari  rectae  QP 
transeunte  per  P,  &  eorum  productorum  summa  sequetur  distan- 
tias centri  gravitatis  ductae  in  summam  massarum  ;  adeoque 
J.A  x  2  GP  X  Pa  erit  =  2  M  X  GP2. 

0         J.A  x  AP2        f.A  x  AG2  -  M  x  GP2  +  2  M  x  GP2        J.A  X  AG2+np 

MxGP  MxGP  MxG 

f  A   v 
Erit  igitur  PQ  major,  quam  PG,  excessu  GQ=  } 


~ 

IVl     / 

Valor  constans  pro.  337.  £x  iflo  excessu  facile  constat,  mutato  utcunque  puncto  suspensions,  rectangulum 

ducti  ex  bmis  dis.        i     L-    •      j-  ••  •  ,     •  ^        r     ...     .      .r  r  °  ~, 

lantiis  centri  gravi-  sub  bmis  distantns  centri  gravitatis  ab  ipso,  &  a  centro  oscillationis  fore  constans.     Cum 

tatis  ab  iisdem.  f  A    V   AO2  f  A    V   ACi* 

enim  sit  QG  =  Tr       r-P  '  erit  GQ  X  GP  =        4:pL>  quod  productum  est  constans, 

1V1    /\  VJA  1VJ. 

&  habetur  hujusmodi  elegans  theorema  :  singula:  masste  ducantur  in  quadrata  suarum 
distantiarum  a  centro  gravitatis  communi,  y  dividatur  omnium  ejusmodi  productorum  summa 
•per  summam  massarum,  ac  habebitur  productum  sub  binis  distantiis  centri  gravitatis  a  centro 
suspensionis  y  a  centro  oscillationis. 

Manente  puncto  338.  Inde  autem  primo  eruitur  illud  ;  manente  puncto  suspensionis,  &  centro  gravitatis, 

centronS1g°ravitatif  debere  ctiam  centrum  oscillationis  manere  nibil  mutatum  ;  utcunque  totum  sy  sterna,   servata 

manere      centrum  respectiva  omnium  massarum  distantia,   y  positione  ad  se  invicem  convertatur  intra   idem 

planum  circa  ipsum  gravitatis  centrum  ;   nam  ilia  GP  inventa  eo  pacto  pendet  tantummodo 

a  distantiis,  quas  singular  massae  habent  a  centro  gravitatis. 

Centrum      osciiia-  330,.  Sed  &  illud  sponte  consequitur  :    Centrum  oscillationis,  y  centrum  suspensionis 

suspensionlsUIred™  reciprocari  ita,  ut,  si  fiat  suspensio  per  id  punctum,  quod  fuerat  centrum  oscillationis  ;   evadat 

rocari-  oscillationis  [156]  centrum  illud,  quod  fierat  punctum  suspensionis  ;    y  alterius  distantia  a 

centro  gravitatis  mutata,  mutetur  y  alterius  distantia  in  eadem  rations  reciproca.     Cum  enim 

earum  distantiarum  rectangulum  debcat  esse  constans  ;  si  pro  secunda  ponatur  valor,  quern 

habuerat  prima  ;   debet  pro  prima  obvenire  valor,  quern  habuerat  secunda,  &  altera  debet 

sequari  quantitati  constanti  divisae  per  alteram. 

Altera   ex  iis  dis-  340.  Consequitur  etiam  illud  :    Altera  ex  Us  binis  distantiis  evanescente,  abibit  altera 

cente  'abire  aUeram  *'K  infinitum,  nisi  omnes  masses  in  unico  puncto  sint  simul  compenetratce.     Nam  sine  ejusmodi 

in  infinitum.  compenetratione  summa   omnium  productorum  ex   massis,  &  quadratis   distantiarum   a 

centro  gravitatis,  remanet  semper  finita  quantitas  :    adeoque    remanet    finita  etiam,  si 

dividatur  per  summam  massarum,  &  quotus,  manente  diviso  finite,  crescit  in  infinitum  ; 

si  divisor  in  infinitum  decrescat. 

Suspensione    facta  341.  Hinc  vero  iterum  deducitur  :    Suspensione  facia  per  ipsum   centrum  gravitatis 

tatis0  imUum  haberi  nu^um  motum  consequi.     Evanescit  enim  in  eo  casu  distantia  centri  gravitatis  a  puncto 
motum.  suspensionis,   adeoque   distantia    centri    oscillationis    crescit    in    infinitum,    &    celeritas 

oscillationis  evadit  nulla. 

Quae  distantia  cen-  342.  Quoniam  utraque  distantia  simul  evanescere  non  potest,  potest  autem  centrum 

omniurn^'lmn'ima  oscillationis  abire  in    infinitum  ;    nulla  erit  maxima  e  longitudinibus  penduli  simplicis 

pro  data  positione  isochroni  pendulo  facto  per  suspensionem  dati  systematis  ;    sed  aliqua  debet  esse  minima, 

datarum  T^maxi^  Suspensione   quadam   inducente   omnium   celerrimam   dati   systematis   oscillationem.     Ea 

mam  haberi  nuiiam.  vero  minima  debet  esse,  ubi  illae  binas  distantiae  aequantur  inter  se  :  ibi  enim  evadit  minima 

earum    summa,  ubi    altera    crescente,  &    altera    decrescente,  incrementa   prius  minora 

decrementis,  incipiunt  esse  majora,  adeoque  ubi  ea  aequantur  inter  se.     Quoniam  autem 

illae  binae  distantiae  mutantur  in  eadem  ratione,  utut  reciproca  ;    incrementum  alterius 


A  THEORY  OF  NATURAL  PHILOSOPHY  255 

change  the  sign  of  its  value ;  &  thus,  if  in  any  one  case,  Q  lies  on  the  same  side  of  P  as 
G  does,  it  must  always  lie  on  the  same  side.  Now  they  lie  on  the  same  side  for  the  case 
in  which  it  is  supposed  that  all  the  masses  go  to  their  common  centre  of  gravity  ;  for  in 
this  case  the  pendulum  becomes  a  simple  pendulum,  &  the  centre  of  oscillation  coincides 
with  the  centre  of  gravity,  at  which  all  the  masses  are  placed.  Hence  it  will  always  fall 
on  the  same  side  of  the  centre  of  suspension  as  G  does. 

336.  Next,  the  centre  of  gravity  must  lie  intermediate  between  the  centre  of  suspension  of  the  three  points, 
W  the  centre  of  oscillation.     For,  in  Fig.  64,  let  the  points  A,  P,  G,  O  be  the  same  points  as  *hte  c^e  °|  e™- 
in  Fig.  63  ;   &  let  AG,  AQ,  &  Aa  be  drawn  perpendicular  to  PQ.     Then,  the  sum  of  all  tweer^The  'other 
the  masses,  each  multiplied  into  its  distance  from  some  chosen  straight  line  or  plane,  or  two- 

into  their  squares,  may  be  designated  by  the  letter  J  prefixed  to  the  term  involving  the 
mass  A  alone,  so  as  to  make  the  proofs  shorter.  If  this  is  done,  the  formula  found  will 
become  PQ  =  J.A  x  AP2  /M  x  GP.  Now  AG3  =  AP2  +  GP2  -  aGP  X  Pa,  & 
therefore  AP2  =  AG2  -  GP2  +  2GP  X  Pa  ;  &  J.A  X  GP2  =  M  X  GP2,  since  GP  is 
constant ;  also  /.A  X  7a  —  M  X  GP,  since  Pa  is  equal  to  the  distance  of  the  mass  A 
from  the  plane  perpendicular  to  the  straight  line  QP,  passing  through  P,  &  thus  the  sum 
of  these  products  will  be  equal  to  the  distance  of  the  centre  of  gravity  multiplied 
by  the  sum  of  the  masses  ;  hence  J.A  X  2GP  X  Pa  =  2M  X  GP2.  Therefore 
I A  x  AP2/M  x  GP  -  J-(A  X  AG2  -  M  x  GP2  +  zM  x  GP2)  _  f.A  X  AG2  Qp 

M  xGP  M  xGP 

Hence  PQ  will  be  greater  than  PG  ;  &  the  excess  GQ  will  be  equal  to  J.A  xAG2/M  xGP. 

337.  From  the  value  of  this  excess,  it  is  readily  seen  that,  however  the    point    of  The  value  of  the 
suspension  may  be  changed,  the  rectangle  contained  by  the  two  distances  of  the  centre  distances  °of   the 
of  gravity  from   it   &  from   the    centre    of  oscillation,   will  be  constant.       For,   since  centre   of  gravity 
QG  =  J.A  x  AG2/M  x  GP,  it  follows  that  GQ  X  GP  =J.A  X  AG2/M  ;  &  this  product  SS^woS^E 
is  constant.     Hence  we  have  the  following  elegant  theorem  : — //  each  of  the  masses  is  multi- 
plied by  the  square  of  its  distance  from  the  common  centre  of  gravity,  y  the  sum  of  all  these 

products  is  divided  by  the  sum  of  the  masses,  then  the  result  obtained  will  be  the  product  of  the 
two  distances  of  the  centre  of  gravity  from  the  centres  of  suspension  tff  oscillation. 

338.  Now,  from  this  theorem,  we  can  derive  first  of  all  the  following  theorem.     //  H  the    centre    of 
the  centre  of  gravity  W  the  centre  of  suspension  remain  unchanged,  then  also  the  centre  of  centre^of1  gravity 
oscillation  must  remain  quite  unchanged  ;  no  matter  how  the  whole  system  is  rotated  about  the  remain  unchanged, 
centre  of  gravity,  in  the  same  plane,  so  long  as  the  mutual  distances  of  all  the  masses  &  their  centre  of    osciiia- 
position  with  regard  to  one  another  are  preserved.     For,  the  value  of  GP  found  in  the  manner  tion. 

above  depends  solely  on  the  distances  of  the  several  masses  from  their  centre  of  gravity. 

330.  But  there  is  another  theorem  that 'aKo  follows  immediately.     The  centre  of  The  centre  of  osci!- 

.,jJ7       ,,  nii  i         •  j         j     j  •        lation  &  the  centre 

oscillation  C5  the  centre  of  suspension  are  mutually  related  to  one  another  in  such  a  fashion  Of  suspension   are 
that,  if  the  suspension  is  made  from  the  point  which  formerly  was  the  centre  of  oscillation,  then  reversible. 
the  new  centre  of  oscillation  will  prove  to  be  that  point  which  was  formerly  the  centre  of  suspension ; 
y  if  the  distance  of  either  of  them  from  the  centre  of  gravity  is  changed  the  distance  of  the 
other  will  be  also  changed  in  the  same  ratio  inversely.     For,  since  the  rectangle  contained 
by  their  distances  remains  constant,  if  for  the  second  there  is  substituted  that  which  the 
first  had,  then  for  the  first  there  must  be  obtained  the  value  which  the  second  formerly 
had  ;    &  either  of  the  two  is  equal  to  the  constant  quantity  divided  by  the  other. 

340.  It  also  follows  that,  if  either  of  the  distances  vanishes,  the  other  must  become  infinite,  If  one  of  the  djs. 
unless  all  the  masses  are  condensed  at  a  single  point.     For,  unless  there  is  condensation  of  other  will   |^c'ome 
this  kind,  the  sum  of  all  the  products  formed  from  the  masses  &  the  squares  of  their  distances  infinite. 

from  their  centre  of  gravity  will  always  remain  a  finite  quantity  ;  &  thus  it  will  still  remain 
finite  if  it  is  divided  by  the  sum  of  the  masses,  &  the  quotient,  still  left  finite  after  division, 
will  increase  indefinitely,  if  its  divisor  decreases  indefinitely. 

341.  Hence,  again,  it  can  be  deduced  that  if  the  suspension  is  made  from  the  centre  of  «  the   suspension 

.  .  '    .          ,.'  .      ,  ,  .    'is   made   from    the 

gravity,  no  motion  will  ensue.  For,  in  this  case,  the  distance  of  the  centre  ot  gravity  centre  of  gravity, 
from  the  centre  of  suspension  vanishes  and  so  the  distance  of  the  centre  of  oscillation  there  is  no  motion, 
increases  indefinitely,  &  therefore  the  speed  of  the  oscillation  becomes  zero. 

342.  Since  both  distances  cannot  vanish  together,  but  the  centre  of  oscillation  can  To  find  the  least 

aT       ./-.  ,  •  11  i_        *         •        i  j    i  distance  of  the 

go  oft  to  infinity,  there  cannot  be  a  maximum  among  the  lengths  of  a  simple  pendulum  centre  of  oscillation 
isochronous  with  the  pendulum  made  by  the  suspension  of  the  given  system;  but  there  for  a  given  position 

.  .    .  F  .      '  ,    ,         .   c  i  .   i        -ii     •         of  the  masses  with 

must  be  a  minimum,  since  there  must  be  one  suspensidn  of  the  given  system  which  will  give  regarti  to 
the  greatest  speed  of  oscillation.  Indeed,  this  least  value  must  occur,  when  the  two  distances 
are  equal  to  one  another  ;  for  their  sum  will  be  least  when,  as  the  one  increases  &  the  other 
decreases,  the  increments,  which  were  before  less  than  the  decrements,  now  begin  to  be 
greater  than  the  latter  ;  &  thus,  at  the  time  when  they  are  equal  to  one  another.^  Moreover 
since  the  two  distances  change  in  the  same  ratio,  although  inversely,  the  infinitesimal 


PHILOSOPHIC  NATURALIS  THEORIA 


Superiora  habere 
locum  t  an  t  u  m- 
modo,  ubi  omnes 
massae  sint  in  eo- 
dem  piano  perpen- 
dicular! ad  axem 
rotationis  :  transi- 
tus  ad  centrum 
percussionis. 


Praeparatio  ad  in- 
veniendum  cen- 
trum percussionis 
massarum  jacen- 
tium  in  e  a  d  e  m 
recta. 


Calculus   cum    ejus 
determinatione. 


infinitesimum  erit  ad  alterius  decrementum  in  ratione  ipsarum,  nee  ea  aequari  poterunt 
inter  se,  nisi  ubi  ipsas  distantias  inter  se  aequales  fiant.  Turn  vero  illarum  productum 
evadit  utriuslibet  quadratum,  &  longitude  penduli  simplicis  isochroni  aequatur  eorum 
summse  ;  ac  proinde  habetur  hujusmodi  theorema  :  Singulee  masses  ducantur  in  quadrata  suarum 
distantiarum  a  centro  gravitatis,  ac  productorum  summa  dividatur  per  summam  massarum  : 
y  dupla  radix  quadrata  quoti  exhibebit  minimam  penduli  simplicis  isochroni  longitudinem. 
Vel  Geometrice  sic  :  Pro  quavis  massa  capiatur  recta,  ques  ad  distantiam  cujusvis  masses  a 
centro  gravitatis  sit  in  ratione  subduplicata  ejusdem  masses  ad  massarum  summam  :  inveniatur 
recta,  cujus  quadratum  eequetur  quadratis  omnium  ejusmodi  rectarum  simul :  &  ipsius  duplum 
dabit  quczsitum  longitudinem  mediam,  quce  brevissimam  pr&stet  oscillationem. 

343.  Haec  quidem  omnia  locum  habent,  ubi  omnes  massae  sint  in  unico  piano  perpen- 
diculari  ad  axem  rotationis,  ut  ni-[i57]-mirum  singulae  massae  possint  connecti  cum  centro 
suspensionis,  &  centro  oscillationis.     At  ubi  in  diversis  sunt  planis,  vel  in  piano  non  per- 
pendiculari  ad  axem  rotationis,  oportet  singulas  massas  connectere  cum  binis  punctis  axis, 
&  cum  centro  oscillationis,  ubi  jam  occurrit  systema    quatuor    massarum    in  se  mutuo 
agentium  (?)  ;    &  relatio  virium,  quae  in  latus  agant  extra  planum,  in  quo  tres  e  massis 
jaceant,  quae  perquisitio  est  operosior,  sed  multo    foecundior,  &  ad  problemata  plurima 
rite  solvenda  magni  usus  ;   sed  quae  hucusque  protuli,  speciminis  loco  abunde  sunt ;  mirum 
enim,  quo  in  hujusmodi  Theoria  promovenda,  &  ad  Mechanicam  applicanda  progredi 
liceat.     Sic  etiam  in  determinando  centro  percussionis,  virgam  tantummodo  rectilineam 
considerabo,  speciminis  loco  futuram,  sive  massas  in  eadem  recta  linea  sitas,  &    mutuis 
actionibus  inter  se  connexas. 

344.  Sint    in   fig.  65  massae  A,  B,  C,  D  connexae  inter  se   in  recta 
quadam,  quae    concipiatur    revoluta  circa  punctum  P  in  ea  situm,    & 
quaeratur  in  eadem  recta  punctum  quoddam  Q,  cujus  motu  impedito 
debeat   impediri    omnis    motus  earumdem  massarum    per    mutuas    ac- 
tiones ;    quod   punctum   appellatur   centrum  percussionis.     Quoniam   sys- 
tema totum  gyrat  circa  P,  singulae  massae  habebunt  velocitates  Aa,  B£ 
&c.   proportionales  distantiis  a  puncto  P,   adeoque   singularum    motus, 
qui  per  mutua*  vires  motrices    extingui    debent,   poterunt   exprimi  per 
A  X  AP,  B  X  BP  &c.      Quare  vires  motrices  in  iis  debebunt  esse  pro- 
portionales iis  motibus.      Concipiantur    singulae  connexae    cum    punctis 
P,  &  Q,  &   quoniam   velocitas    puncti    P    erat  nulla  ;    ibi  omnium  ac- 
tionum  summa  debebit  esse  =  o  :  summa  autem  earum,  quae   habentur 
in  Q,  elidetur  a  vi  externa  percussionem  sustinente. 

345.  Quoniam    actiones    debent    esse   perpendiculares    eidem    rectae 
jungenti    massas,    erit   per   theorema   numeri   314,   ut   PQ   ad  AQ,   ita 


actio   in   A  =  A    X  AP,   ad  actionem    in  P  = 


A  X 


X  AQ> 


sive   ob 


AQ  =  PQ  —  AP,  erit  ea  actio  [158] 
pacto  actio  in  P  ex  nexu  cum  B  erit 


xPQ-AxAfr      £odem 


T)    *. 


PQ 


FIG.  65. 


__  "R 


2 

'  &  ita  Porro'     Iis  omnlbus 


Determinatio  vis 
percussionis  in  ipso 
centro. 


"pfT" 

positis=o.  divisor  communis  PQ  abit,  &  omnia  positiva  aequantur  negativis.  Erit 
igitur  A  x  AP  x  PQ+  B  x  BP  x  PQ  &c.  =  A  x  AP2  +  BxBP2  &c.  ;  quare 
PQ  =  A  X  AP2  +  B  X  BP2  &c.^  quse  formuja  est  eadem,  ac  formula  centri  oscilla- 

J\   X  .A.A    ~T~  -^   X  IJX   oCC. 

tionis,  ac  habetur  hujusmodi  theorema  :  Distantia  centri  percussionis  a  puncto  conver- 
sionis  eequatur  distantly  centri  oscillationis  a  puncto  suspensionis  ;  adeoque  hie  locum  habent 
in  hoc  casu,  quaecunque  de  centro  oscillationis  superius  dicta  sunt. 

346.  Quod    si    quis    quserat    vim    percussionis    in    Q,    hie    habebit 


QP  .  AP  :  :  A  xAP. 


<lU3e   eT'lt 


ex    nexu    cum  A-     Eodem  pacto  in- 


.     A  X  AP2  +  B  X  BP2 
venientur  vires   ex  reliquis  :     adeoque    summa    virium    erit  '  &c., 

(q)  Systema  binarwm  massarum  cum  binis  punctis  connexarum,  W  inter  se,  sed  adhuc  in  eodem  piano  jacentium, 
persecutus  fueram  ante  aliquot  annos  ;  quod  sibi  a  me  communicatum  exhibuit  in  siia  Synopsi  Physicae  Generalis 
P.  Benvenutus,  ut  ibidem  ipse  innuit.  Id  inde  excerptum  habetur  hie  in  Supplements  §  5. 

Habetur  autem  post  idem  supplementum  y  Epistola,  quam  delatus  Florentiam  scripsi  ad  P.  Scherferum,  dum  hoc 
ipsum  opus  relictum  Vienna  ante  tres  menses  jam  ibidem  imprimeretur,  qua  quidem  adjecta  est  in  ipsa  prima  editione 
in  -fine  operis.  Ibi  W  theoriam  trium  massarum  extendi  ad  casum  massarum  quatuor  ita  ;  ut  inde  generaliter  deduct 
fossit  W  equilibrium,  &  centrum  oscillationis,  &  centrum  percussionis,  pro  massis  quotcunque,  W  utcunque  dispositis. 


A  THEORY  OF  NATURAL  PHILOSOPHY  257 

increment  of  the  one  will  be  to  the  infinitesimal  decrement  of  the  other  in  the  ratio  of 
the  distances  themselves ;  &  the  former  cannot  be  equal  to  one  another,  unless  the  distances 
themselves  are  equal  to  one  another.  In  this  case  their  product  becomes  the  square  of 
either  of  them,  &  the  length  of  the  simple  isochronous  pendulum  will  be  equal  to  their 
sum.  Hence  we  have  the  following  theorem  : — //  each  mass  is  multiplied  by  the  square  of 
the  distance  from  the  centre  of  gravity,  y  the  sum  of  all  such  products  is  divided  by  the  sum 
of  the  masses  ;  then,  twice  the  square  root  of  the  quotient  will  give  the  least  length  of  a  simple 
isochronous  pendulum.  This  may  be  expressed  geometrically  as  follows  : — For  each  mass, 
take  a  straight  line,  which  is  to  the  distance  of  that  mass  from  the  centre  of  gravity  in  the 
subduplicate  ratio  of  the  mass  to  the  sum  of  all  the  masses  ;  find  a  straight  line  whose  square 
is  equal  to  the  sum  of  the  squares  on  all  the  straight  lines  so  found  ;  then  the  double  of  this 
straight  line  will  give  the  required  mean  length,  which  will  afford  the  quickest  oscillation. 

343.  These  theorems  hold  good  when  all  the  masses  are  in  a  single  plane  perpendicular  The  theorems  given 
to  the  axis  of  rotation,  so  that  each  of  the  masses  can  be  connected  with  the  point  of  abo^e  ,only  ,,h?,!d 

,  ......  _  .  .  .        ,.„  .  ..    .       good  when  all  the 

suspension  &  the  centre  ot  osculation.  But,  when  they  are  m  ditterent  planes,  or  all  in  masses  are  in  the 
a  plane  that  is  not  perpendicular  to  the  axis  of  rotation,  it  is  necessary  to  connect  each  *;am<f  plane  perpen- 

-r ,  .   ,  .r      ,  •     „        •  i      i  r  -11      •  dicular  to  the  axis 

of  the  masses  with  a  pair  or  points  on  the  axis  &  with  the  centre  oi  oscillation  :  &  we  thus  Of  rotation ;  let  us 
have  the  case  of  a  system  of  four  masses  acting  upon  one  another  (q),  &  the  relation  between  pass  on  to  the  centre 

if  1-1  •  i  fit-          i'ii  r      -i  i>  *        rrn  •      ot  percussion. 

the  forces  which  act  to  one  side,  out  ot  the  plane  in  which  three  or  the  masses  lie.  Ihis 
investigation  is  much  more  laborious,  but  also  far  more  fertile,  &  of  great  use  for  the  correct 
solution  of  a  large  number  of  problems.  However,  I  have  already  given  enough  as 
examples  ;  for  it  is  wonderful  how  far  one  can  go  in  developing  a  Theory  of  this  kind, 
&  in  applying  it  to  Mechanics.  So  also  in  determining  the  centre  of  percussion,  I  shall 
only  consider  a  rectilinear  rod,  which  will  serve  as  an  example,  or  masses  in  the  same  straight 
line,  connected  together  by  mutual  actions. 

344.  In  Fig.  65,  let  A,B,C,D  be  masses  connected  together,  lying  in  one  straight  line,  preparation    for 
which  is  supposed  to  be  rotated  about  a  point  P  situated  in  it ;   it  is  required  to  find  in  findijlg  the.  centre 

.    ,  rf.  .~  .  -  .  r  ..  ,',  1^11  •  r   of    percussion    for 

this  straight  line  a  point  Q  such  that,  if  its  motion  is  prevented,  then  the  whole  motion  of  masses  lying  in  the 

the  masses  is  also  prevented  through  the  mutual  actions.     This  point  is  called  the  centre  same  straight  line. 

of  percussion.     Now,  since  the  whole  system  rotates  round  P,  each  of  the  masses  will  have 

velocities,  such  as  Aa,  B£,  &c.,  proportional  to  their  distances  from  the  point  P ;   &  thus 

the  motions  of  each,  which  have  to  be  destroyed  by  the  mutual  motive  forces^can  be 

represented    by  A  X  AP,  B  x  BP,  &c.      Hence,  the    motive    forces  on  them  must  be 

proportional  to  these  motions.     Suppose  each  of  the  masses  to  be  connected  with  P  & 

Q  ;    then,  since  the  velocity  of  the  point  P  is  zero,  at  P  the  sum  of  all  the  actions  must 

be  equal  to  zero  ;    moreover,  the  sum  of  those  that  act  at  Q  is  cancelled  by  the  external 

force  sustaining  the  percussion. 

345.  Since  the  actions  must  be  perpendicular  to  the  straight  line  joining  the  masses,  The  calculation 
we  shall  have,  by  Art.  314,  PQ  to  AQ  as  the  action  on  A,  which  is  equal  to  A  X  AP,  is  Kfion^ 

to  the  action  on  P  ;  hence  the  latter  is  equal  to  A  X  AP  X  AQ/PQ,  or,  since  AQ  =  PQ  —  AP,  centre, 
this  action  will  be  equal  to  (A  X  AP  x  PQ— A  X  AP2)/PQ.  In  the  same  way,  the 
action  on  P  due  to  the  connection  with  B  is  equal  to  (B  x  BP  X  PQ  —  B  X  BP2)/PQ, 
&  so  on.  If  all  these  together  are  put  equal  to  zero,  the  common  divisor 
PQ  goes  out,  &  all  the  positives  will  be  equal  to  the  negatives.  Therefore 
A  x  AP  X  PQ  +  B  x  BP  x  PQ  +  &c.  =  A  x  AP2  +  B  x  BP2  +  &c.  Hence 

PQ  =  -j-  ]  — ',  which  is  the  same  formula  as  the  formula  for  the 

A  X  -A..T    ~\~  K  X  xSJr     ~\-  oCC. 

centre  of  oscillation.  Thus  we  have  the  following  theorem  : — The  distance  of  the  centre 
of  percussion  from  the  point  of  rotation  is  equal  to  the  distance  of  the  centre  of  oscillation  from 
the  centre  of  suspension.  Hence  all  that  has  been  said  above  concerning  the  centre  of 
oscillation  holds  good  also  for  the  centre  of  percussion. 

346.  Now,  if  the  force  of  percussion  at  Q  is  required,  we  have  QP  is  to  AP  as  A  X  AP  Determination 

1 1        r  >~\    i  i  •  •   i      A  i  •     i  •  i  A         A  T>»  rn/~\     the     force    °*    Per' 

is  to  the  force  on  Q  due  to  the  connection  with  A  ;  hence  this  latter  is  equal  to  A  xAPyrQ.  cussion  at  the 
In  the  same  way  we  can  find  the  forces  due  to  the  rest ;  and  thus  the  sum  of  all  centre  of  percus- 
the  forces  will  be  (A  X  AP2  +  B  X  BP2  +  &c.)/PQ.  Now,  since  PQ  is  equal  to  s 

(q)  /  investigated  the  system  of  two  masses  connected  with  two  faints  W  with  one  another,  yet  all  lying  in  the 
same  -plane,  several  years  ago  :  y,  when  I  had  communicated  the  matter  to  Father  Benvenuto,  he  expounded  it  in  his 
Synopsis  Physicje  Generalis,  mentioning  that  he  had  obtained  it  from  me.  It  is  also  included  in  this  work,  abstracted 
from  the  above,  as  Supplement  5. 

Moreover,  after  this  supplement,  it  is  also  contained  in  a  letter,  which  1  wrote  'to  Father  Scherffer  when  I 
reached  Florence,  whilst  this  work,  which  I  had  left  in  his  hands  at  Vienna  three  months  before,  was  in  the  press  there  ; 
W  it  was  added  to  the  first  edition  at  the  end  of  the  work.  In  it  I  have  also  extended  the  theory  of  three  masses  to  the 
case  of  fow  masses,  in  such  a  manner  that  from  it  it  is  possible  to  deduce,  in  a  perfectly  general  way,  the  equilibrium, 
the  centre  of  oscillation,  y  the  centre  of  percussion  for  any  number  of  masses  disposed  in  any  manner  whatever. 

S 


of 


258 


PHILOSOPHIC  NATURALIS  THEORIA 


sive      ob 


PQ  =    A  X  AP2  +  B  x  BP2  &c. 


summa  ilia  erit  A  X  AP  +  B  X  BP  &c.  ; 


Omitti  hie  multa 
quae  adhanc  Theo- 
riam  pertinerent, 
ad  quam  pertinet 
universa  Mcchanica. 


A 

! 


Pressio  fluidorum 
si  puncta  sint  in 
recta  vertical!. 


Eadem  p  u  n  c  t  i  s 
utcunque  dispersis, 
&  cum  omnibus 
directionibus  agens. 


Ax  AP  -f  B  x  BP&c. 

nimirum  ejusmodi  vis  erit  sequalis  summse  virium,  quse  requiruntur  ad  sistendos  omnes 
motus  massarum  A,  B,  &c.,  cum  illis  diversis  velocitatibus  progredientium,  videlicet  ejusmodi, 
quas  in  massa  percussionem  excipiente  possit  producere  quantitatem  motus  sequalem 
toti  motui,  qui  sistitur  in  massis  omnibus,  quod  congruit  cum  lege  actionis,  &  reactionis 
aequalium,  &  cum  conservatione  ejusdem  quantitatis  motus  in  eandem  plagam,  de  quibus 
egimus  num.  265,  &  264. 

347.  Haberent  hie  locum  alia  sane  multa,  quse  pertinent  ad  summas  virium,  quibus 
agunt  massse,  compositarum  e  viribus,  quibus  agunt  puncta,  vel  a  Newtono,  vel  ab  aliis 
demonstrata,  &  magni  usus  in  Mechanica,  &  Physica  :    hujusmodi  sunt  ea  omnia,  quse 
Newtonus  habet  sectione  12,  &  13  libri  I  Princip.  de  attractionibus  corporum  sphaericorum, 
&  non  sphaericorum,  quae  componantur  ex  attractionibus  particularum  ;    ubi  habentur 
praeclarissima  theoremata  tarn  pro  viribus  quibuscunque  generaliter,  quam  pro  certis  virium 
legibus,  ut  illud,  quod   pertinet    ad    rationem    reciprocam    duplicatam    distantiarum,  in 
qua  globus   globum    trahit,    tanquam    si    omnis  materia   esset    compenetrata   in    centris 
eorundem  ;    punctum  intra  [159]  orbem  sphaericum,  vel  ellipticum  vacuum  nullas  vires 
sentit,  elisis  contrariis ;    intra  globos  plenos  punctum  habet  vim  directe  proportionalem 
distantiae  a  centro  ;   unde  fit,  ut  in  particulis  exiguis  ejusmodi  vires  fere  evanescant,  &  ad 
hoc,  ut  vires  adhuc  etiam  in  iis  sint  admodum  sensibiles,  debeant  decrescere  in  ratione 
multo   majore,    quam    reciproca    duplicata    distantiarum.     Hujusmodi   etiam   sunt,    quse 
Mac-Laurinus  tradit  de  sphaeroide  elliptico  potissimum,  quae  Clairautius  de  attractionibus 
pro    tubulis    capillaribus,  quae    D'Alembertus,  Eulerus,  aliique    pluribus    in    locis 
persecuti  sunt ;   quin  omnis  Mechanica,  quse  agit  vel  de  asquilibrio,  vel  de  moti- 

bus,  seclusa  omni  impulsione,  hue  pertinet,  &  ad  diversos  arcus  reduci  potest  curvae 
nostrse,  qui  possunt  esse  quantumlibet  multi,  habere  quascunque  amplitudines,  sive 
distantias  limitum,  &  areas  quae  sint  inter  se  in  ratione  quacunque,  ac  ad  curvas 
quascunque  ibi  accedere,  quantum  libuerit ;  sed  res  in  immensum  abiret,  &  satis 
est,  ea  omnia  innuisse. 

348.  Addam  nonnulla  tantummodo,  quae  generaliter  pertinent  ad  pressionem, 
&  velocitatem  fluidorum.     Tendant  directione  quacunque  AB  puncta  disposita  in 
eadem  recta  in  fig.  66  vi  quadam   externa   respectu  systematis  eorum  punctorum, 
cujus   actionem   mutuis    viribus  elidant  ea  puncta,  &  sint  in  sequilibrio.      Inter 
primum  punctum  A,  &  secundum  ipsi  proximum  debebit  esse  vis  repulsiva,  quae 
sequetur  vi  externse  puncti  A.     Quare  urgebitur  punctum  secundum  hac  vi  repul- 
siva, &  praeterea  vi  externa  sua.      Hinc  vis  repulsiva  inter  secundum,  &  tertium 
punctum  debebit  aequari  vi  huic  utrique,  adeoque  erit  aequalis  summse  virium  ex- 
ternarum  puncti  primi,  &  secundi.     Adjecta  igitur  sua  vi  externa  tendet  deorsum 
cum  vi  sequali  summae  virium  externarum  omnium  trium  ;    &  ita  porro  progred- 
iendo  usque  ad  B,  quodvis  punctum  urgebitur  deorsum  vi  aequali  summae  virium  FIG-  66- 
externarum  omnium  superiorum  punctorum. 

349.  Quod  si  non  in  directum  disposita  sint, sed  utcunque  dispersa  per  parallelepipedum, 
cujus  basim  perpendicularem  directioni  vis  externae  exprimat  recta  FHin  fig.  67,  &  FEGH 
faciem  ipsi  parallelam  ;   adhuc  facile  demonstrari  potest 

componendo,  vel  resolvendo  vires ;  sed  &  per  se  patet,  A  C  G 

vires  repulsivas,  quas  debebit  ipsa  basis  exercere  in  par- 

ticulas  sibi  propinquas,  &  ad  quas  vis  ejus  mutua  perti- 

nebit,  fore  aequales  summae  omnium  superiorum  virium 

externarum  ;  atque  id  erit  commune  tarn  solidis,  quam 

fluidis.      At  quoniam  in  fluidis  particulae  possunt  ferri 

directione  quacunque,  quod  unde  proveniat,  videbimus 

in  tertia  parte  ;  quaevis  particula,  ut  ibidem  videbimus, 

in  omnem  plagam  urgebitur  viribus  aequalibus,  &  urgebit  L 

sibi  proximas,  quse  pressionem  in  alias  propagabunt  ita, 

ut,  quse  sint  in  eodem  piano  LI,  parallelo  FH,  in  cujus 

directione    [160]    nulla    vis    externa    agit,  vires  ubique 

eaedem  sint.     Quamobrem  quaevis  particula  sita  ubicun- 

que  in  ea  recta  in  N,  habebit  eandem  vim  tarn  versus 

planum  EF,  quam  versus  planum  EG,  &  versus  FH,  quam 

habet  particula  collocata  in  eadem  linea  in  MK  etiam, 

ubi  addantur  parietes  AM,  CK  parallel!  FE,  cum  planis 

LM,  KI,    parallelis    FH,    nimirum  vi,    quse    respondet 

altitudini  MA  :    ac  particula  sita  in  O  prope  basim  FH  urgebitur,  ut  quaquaversum,  ita 

&  versus  ipsam,  iisdem  viribus,  quibus  particula  sha  in  BD  sub    AC.     Ipsam  urgebunt 


B 


H 


A  THEORY  OF  NATURAL  PHILOSOPHY  259 

(A  x  AP*+B  x  BP2  +  &c.)/(A  x  AP  +  B  x  BP  +  &c.),  this  sum  will  be  equal  to 
A  X  AP  -f-B  X  BP  -f-  &c.  That  is,  the  whole  force  will  be  equal  to  the  sum  of  the  forces, 
which  are  required  to  stop  all  the  motions  of  the  masses  A,  B,  &c.,  which  are  proceeding 
with  their  several  different  velocities ;  in  other  words,  a  force  which,  acting  on  the  mass 
receiving  percussion,  can  produce  a  quantity  of  motion  equal  to  the  whole  motion  existing 
in  all  the  masses ;  and  this  agrees  with  the  law  of  equal  action  &  reaction,  &  with  the  con- 
servation of  the  same  quantity  of  motion  for  the  same  direction,  with  which  I  dealt  in 
Art.  265,  &  264. 

347.  Many  other  things  indeed  should  find  a  place  here,  such  as  relate  to  the  sums  Man.y  things  per. 
of  forces,  with  which  masses  act,  these  being  compounded   from  the  forces  with  which  Theory  must  here 
points  act ;   such  as  have  been  proved  by  Newton  &  others ;   &  things  that  are  of  great  use  be  omitted ;  for  the 
in  Mechanics  &  Physics.     Of  this  kind  are  all  those  which  Newton  has  in  the  I2th  &  I3th  rTics  pertains toCthts 
sections  of  The  First  Book  of  the  Principia  concerning  the  attractions  of  spherical  bodies,  Theory. 

&  non-spherical  bodies,  such  as  are  compounded  from  the  attractions  of  their  particles. 
Here  we  have  some  most  wonderful  theorems,  not  only  for  forces  in  general,  but  also  for 
certain  laws  of  forces  like  that  relating  to  the  inverse  square  of  the  distances,  where  a  sphere 
attracts  another  sphere  as  if  the  whole  of  its  matter  were  condensed  at  the  centre  of  each 
of  them  :  the  theorem  that  a  point  within  a  spherical  or  elliptic  hollow  shell  is  under  the 
action  of  no  force,  equal  &  opposite  forces  cancelling  one  another  ;  the  theorem  that  within 
solid  spheres  a  point  is  under  the  action  of  a  force  proportional  to  the  distance  from  the 
centre  directly.  From  this  it  follows  that  in  exceedingly  small  particles  of  this  kind  the 
forces  must  almost  vanish  ;  &  in  order  that  the  forces  even  then  may  be  quite  sensible, 
they  must  decrease  in  a  much  greater  ratio  than  that  of  the  inverse  square  of  the  distances. 
Also  we  have  theorems  such  as  Maclaurin  enunciated  with  regard  to  the  elliptic  spheroid 
especially,  &  those  which  Clairaut  gave  with  regard  to  attractions  in  the  case  of  capillary 
tubes,  &  those  which  D'Alembert,  Euler,  &  others  have  investigated  in  many  places.  Nay, 
the  whole  ot  Mechanics,  which  deals  with  equilibrium,  or  motions,  impulse  being  ex- 
cluded, belongs  here  :  the  whole  of  it  can  be  reduced  to  different  arcs  of  our  curve  ;  &  these 
may  be  as  many  in  number  as  you  please,  they  can  have  any  amplitudes,  or  distances 
between  the  limit-points,  any  areas,  which  may  be  in  any  ratio  whatever  to  one  another, 
&  can  approach  as  nearly  as  you  please  to  any  given  curves.  But  the  matter  would 
become  endless,  &  it  is  quite  sufficient  for  me  to  have  given  all  those  that  I  have  given. 

348.  I  will  add  a  few  things  only  that  in  general  deal  with  pressure  &  velocity   of  Pressure  of  fluids 
fluids.     Suppose  we  have  a  set  of  points,  in  Fig.  66,  lying  in  a  straight  line,  extended  in  any  ^e^tha Vertical 
direction  AB,  under  the  action  of  some  force  external  to  the  system  of  points ;   &  suppose  line. 

that  the  action  of  this  external  force  is  cancelled  by  the  mutual  forces  between  the  points, 
&  that  the  latter  are  in  equilibrium.  Then  between  the  first  point  A  &  the  next  to  it 
there  must  be  a  repulsive  force  which  is  equal  to  the  external  force  on  the  point  A. 
Then  the  second  point  will  be  under  the  action  of  this  repulsive  force  in  addition  to  the 
external  force  on  it.  Hence  the  repulsive  force  between  the  second  &  third  points  must 
be  equal  to  both  of  these ;  &,  further,  it  will  be  equal  to  the  sum  of  the  external  forces 
on  the  first  &  second  points.  Hence,  adding  the  external  force  on  the  third  point,  it 
will  tend  downwards  with  a  force  equal  to  the  sum  of  the  external  forces  on  all  three  ; 
&  so  on,  until  we  reach  B,  any  point  will  be  under  the  action  of  a  force  equal  to  the  sum 
of  the  external  forces  on  all  the  points  lying  above  it. 

349.  Now  if  the  points  are  not  all  situated  in  a  straight  line,  but  dispersed  anyhow  The  same  for  points 
throughout  a  parallelepiped,  &  if,  in  Fig.  67,  FH  denotes  the  base  of  the  parallelepiped,  ^annei?  &"  acting 
which  is  perpendicular  to  the  direction  of  the  external  force,  &  FEGH  is  a  face  parallel  in  all  directions, 
thereto ;  then,  it  can  yet  easily  be  proved,  either  by  composition  or  by  resolution  of  forces, 

indeed  it  is  self-evident,  that  the  repulsive  forces,  which  the  base  exerts  on  the  particles  next 
to  it,  &  to  which  its  mutual  force  will  pertain,  must  be  equal  to  the  sum  of  the  external  forces 
on  all  points  above  it  :  &  this  will  hold  good  for  solids  as  well  as  for  fluids.  But,  since  in 
fluids  the  particles  can  move  in  any  direction  (we  will  leave  the  cause  of  this  to  be  seen  in 
the  third  part),  any  particle  (as  we  shall  also  see  there)  will  be  urged  in  any  direction  with 
equal  forces  :  &  each  will  act  on  the  next  to  it  &  propagate  the  pressure  to  the  others  in 
such  a  manner  that  the  forces  on  those  points  which  lie  in  the  same  plane  LI,  parallel  to  the 
base  FH,  in  which  direction  there  is  no  external  force  acting,  will  be  everywhere  the  same. 
Hence,  every  particle  situated  anywhere  in  the  straight  line,  at  N  say,  will  have  the  same  force 
towards  the  plane  EF  as  towards  the  plane  EG,  &  towards  FH  ;  the  same  also  as  there 
is  on  a  particle  situated  in  the  same  straight  line  in  MK  also,  where  the  partitions  AM, 
CK  are  added  parallel  to  FE,  together  with  the  planes  LM,  KI  parallel  to  FH,  namely, 


one  equal  to  a  force  corresponding  to  the  altitude  MA.  And  a  particle  situated  close  to  the 
base  FH,  at  O  say,  will  be  urged  in  all  directions  &  towards  FH  with  the  same  forces  as 
a  particle  situated  in  BD  which  is  below  AC.  All  the  particles  lying  in  the  same  horizontal 


260  PHILOSOPHI/E  NATURALIS  THEORIA 

particulae  in  eodem  piano  horizontali  jacentes,  &  accedet  ad  omnes  fluidi,  &  baseos  particu- 
las,  donee  vi  contraria  elidatur  vis  ejus  tota  ab  ejusmodi  pressione  derivata.  Quamobrem 
basis  FH  a  fluido  tanto  minore  FLMACKIH  sentiet  pressionem,  quam  sentiret  a  toto 
fluido  FEGH  :  superficies  autem  LM  sentiet  a  particulis  N  vim  aaqualem  vi  massae  LEAM, 
accedentibus  ad  ipsam  particulis,  donee  vis  mutua  repulsiva  ei  vi  aequetur. 

fd'i  ponderffieri  ?5°'  Hinc  autem  Patet'  cur  in  fluidis  nostris  gravitate    praaditis  basis  FH  sentiat 

ssit  ingens  pres-  pressionem  tanto  majorem  massae  fluidse  incumbentis  pondere,  &  cur  pondere  perquam 
-  exiguo  fluidi  AMKC  elevetur  pondus  collocatum  supra  LM  etiam  immane,  ubi  repagulum 

LM  sit  ejusmodi,  ut  pressioni  fluidi  parere  possit,  quemadmodum  sunt  coriacea.  At 
totum  yas  FLMACKIH  bilanci  impositum  habebit  pondus  aequale  ponderi  suo,  &  fluidi 
content!  tantummodo  ;  nam  superficies  vasis  LM,  KI  horizontalis  vi  repulsiva  mutua 
urgebit  sursum,  quantum  urget  deorsum  puncta  omnia  N  versus  O,  &  ilia  pressio  tantundem 
imminuit  vim,  quam  in  bilancem  exercet  vas,  ac  tota  vis  ipsius  habebitur  dempta  pressione 
sursum  superficiei  LM,  KI  a  pressione  fundi  FH  facta  deorsum  :  &  pariter  se  mutuo  elident 
vires  ^exercitae  in  parietes  oppositos.  Atque  haec  Theoria  poterit  applicari  facile  aliis  etiam 
figuris  quibuscunque.  Respondebit  semper  pressio  superficiei,  &  toti  ponderi  fluidi,  quod 
habeat  basim  illi  superficiei  asqualem,  &  altitudinem  ejusmodi,  quae  usque  ad  supremam 
superficiem  pertinet  inde  accepta  in  directione  illius  externae  vis. 


sone  351-  Quod  s.i  ™es  particularum  repulsivae  sint  ejusmodi,  ut  ad  eas  multum  augendas 

sensibili  unde  requiratur  mutatio  distantiae,  quae  ad  distantiam  totam  habeat  rationem  sensibilcm  ;    turn 
provemat   m   hac  vero  compressio  massae  erit  sensibilis,  &  densitas  in  diversis 

altitudinibusadmodum  diversa:  sed  iniisdemhorizontalibus 

planis  eadem.    Si  vero  mutatio  sufficiat,  quae  rationem  habet 

prorsus  insensibilem  ad  totam  distantiam  ;  turn  vero  com- 

pressio   sensibilis    nulla    erit,  &  massa    in  fundo   eandem 

habebit  ad  sensum  densitatem,   quam  prope   superficiem 

supremam.     Id  pendet  a  lege  virium  mutua  inter  particu- 

las,  &  a  curva,  qua;  illam  expri-[i6i]-mit.     Exprimat  in 

fig.  68  AD  distantiam  quandam,  &  assumpta  BD  ad  AB  in 

quacunque  ratione  utcunque  parva,  vel   utcunque   sensi- 

bili, capiantur  rectae  perpendiculares  DE,    BF   itidem  in 

quacunque  ratione  minoris  inaequalitatis  utcunque  magna  :  FlG  6g 

poterit  utique  arcus  MN  curvse  exprimentis  mutuas  par- 

ticularum vires  transire  per  ilia  puncta  F,  F,  &  exhibere   quodcunque   pressionis   incre- 

mentum    cum  quacunque   pressione  utcunque  magna,  vel  utcunque  insensibili. 
Compressio     aeris  3^2.  Compressionem  ingentem  experimur  in  acre,  quae  in  eo  est  proportionalis  vi 

a  qua  vi  provemat  :  JJ.  .     *-,  .  r.  .      '  j.  r  _ 

aquae     compressio  compnmenti.     Pro  eo  casu  demonstravit  Newtonus  Prmc.  Lib.  3.  prop.  23,  vim  particularum 
cur     ad     sensum  repulsivam  mutuam  debere  esse  in  ratione  reciproca  simplici  distantiarum.     Quare  in  iis 

nulla:  unde  muta-     ,.r  ...  .  ,. 

tio  in  vapores  tam  distantiis,  quas  nabere  possunt  particulas  aeris  perseverantis  cum  ejusmodi  propnetate, 
&  formam  aliam  non  inducentis  (nam  &  aerem  posse  e  volatili  fieri  fixum,  Newtonus  innuit, 
ac  Halesius  inprimis  uberrime  demonstravit),  oportet,  arcus  MN  accedat  ad  formam  arcus 
hyperbolae  conicae  Apollonianae.  At  in  aqua  compressio  sensibilis  habetur  nulla,  utcunque 
magnis  ponderibus  comprimatur.  Inde  aliqui  inferunt,  ipsam  elastica  vi  carere,  sed 
perperam  ;  quin  immo  vires  habere  debet  ingentes  distantiis  utcunque  parum  imminutis  ; 
quanquam  esedem  particulse  debent  esse  prope  limites,  nam  &  distraction!  resistit  aqua. 
Infinita  sunt  curvarum  genera,  quae  possunt  rei  satisfacere,  &  satis  est,  si  arcus  EF  directionem 
habeat  fere  perpendicularem  axi  AC.  Si  curvam  cognitam  adhibere  libeat  ;  satis  est,  ut 
arcus  EF  accedat  plurimum  ad  logisticam,  cujus  subtangens  sit  perquam  exigua  respectu 
distantise  AD.  Demonstratur  passim,  subtangentem  logisticae  ad  intervallum  ordinatarum 
exhibens  rationem  duplam  esse  proxime  ut  14  ad  10  ;  &  eadem  subtangens  ad  intervallum, 
quod  exhibeat  ordinatas  in  quacunque  magna  ratione  inaequalitatis,  habet  in  omnibus 
logistic  is  rationem  eandem.  Si  igitur  minuatur  subtangens  logisticae,  quantum  libuerit  ; 
minuetur  utique  in  eadem  ratione  intervallum  BD  respondens  cuicunque  rationi  ordina- 
tarum BF,  DE,  &  accedet  ad  aequalitatem,  quantum  libuerit,  ratio  AB  ad  AD,  a  qua  pendet 
compressio  ;  &  cujus  ratio  reciproca  triplicata  est  ratio  densitatum,  cum  spatia  similia  sint 
in  ratione  triplicata  laterum  homologorum,  &  massa  compressa  possit  cum  eadem  nova 
densitate  redigi  ad  formam  similem.  Quare  poterit  haberi  incrementum  vis  comprimentis 


A  THEORY  OF  NATURAL  PHILOSOPHY  261 

plane  will  act  upon  it  &  it  will  approach  all  the  particles  of  the  fluid  &  the  base,  until  the 
whole  of  its  force  is  cancelled  by  a  contrary  force  derived  from  pressure  of  this  kind. 
Hence  the  base  FH  would  be  subject,  from  the  much  smaller  amount  of  fluid  FLMACKIH, 
to  the  same  pressure  as  it  would  be  subject  to  from  the  whole  fluid  FEGH  ;  &  the  surface 
LM  would  be  subject  to  a  force  from  the  particles  like  N  equal  to  the  force  of  the  mass 
LEAM,  these  particles  tending  to  approach  LM,  until  the  mutual  repulsive  force  is  equal 
to  this  pressure. 

350.  Further,  from  this  the  reason  is  evident,  why  the  base  FH  should  be  subject,  Hence    the  reason 
in  our  fluids  possessed  of  gravity,  to  a  pressure  so  much  greater  than  the  weight  of  the  why  ™  a  very  small 

n    -i  i  11-1          to    -j     vi        ATI/TTT-/-.      i  •   i       amount     of      fluid 

superincumbent  fluid  ;    &  why  by  a  very  small  weight  of  fluid,  like  AMKC,  the  weight  there  can  exist  a 

collected  above  LM  can  be  upheld,  even  though  this  is  immensely  great,  when  the  restraint  verv  great  pressure. 

LM  is  of  such  a  nature  that  it  can    submit  to    the  pressure  of   the  fluid,  leather  for 

example.     But  if  the  whole  vessel  FLMACKIH  is  placed  on  a  balance  it  will  only  have  a 

weight  equal  to  its  own  weight  plus  that  of    the  fluid  contained.     For,  the  horizontal 

surface  LM,  KI  of  the  vessel  will  urge  it  upwards  with  its  mutual  repulsive  force,  just 

the  same  amount  as  all  the  points  N  will  urge  it  downwards  towards  O,  &  this  pressure 

will  to  the  same  extent  diminish  the  force  which  the  vessel  exerts  upon  the  balance  ;    & 

the  whole  force  will  be  obtained  by  taking  away  the  pressure  upwards  on  the  surface  LM, 

KI  from  the  pressure  produced  downwards  on  the  base  FH.     In  the  same  way  the  forces 

exerted  on  the  partitions  will  mutually  cancel  one  another.     The  Theory  can  also  easily 

be  applied  to  any  other  figures  whatever.     The    pressure    on    the    surface  will    always 

correspond  to  the  whole  weight  of  the  fluid  having  for  its  base  an  area  equal  to  the  surface, 

&  for  its  height  that  which  belongs  to  the  highest  surface  from  it  measured  in  the  direction 

of  the  external  force. 

351.  Now  if  the  repulsive  forces  of  the  particles  are  of  such  a  kind  that,  in  order  to  The   source  of 
increase  them  to  any  sensible  extent,  a  change  of  distance  is  required,  which  bears  a  sensible  Pr.efsure  .f°r  flulds 

,     ,      '..  '  &         .  .  *       •«    1  MI  i       with  sensible    com- 

ratio  to  the  whole  distance  ;   then  the  compression  of  the  mass  will  also  be  sensible,  &  the  pression  according 

density  at  different  heights  will  be  quite  different  ;    nevertheless,  they  will  still  be  the  to  thls  Theory- 

same  throughout  the  same  horizontal  planes.     However,  if  a  change,  which  bears  to  the 

whole  distance  a  ratio  that  is  quite  insensible,  is  sufficient,  then  the  mass  at  the  bottom 

will  have  approximately  the  same  density  as  near  the  top  surface.     This  depends  on  the 

mutual  law  of  forces  between  the  particles,  &  on  the  curve  which  represents  this  law.     In 

Fig.  68,  let  AD  be  any  distance,  &  suppose  that  BD  is  taken  in  AB  produced,  bearing  to 

AB  any  ratio  however  small,  or  however  sensible  ;    ta*ke  the  perpendicular  straight  lines 

DE,  BF,  also  in  any  ratio  of  less  inequality  however  great.     In  all  cases,  it  will  be  possible 

for  the  arc  MN  of  the  curve  representing  the  mutual  forces  of  the  particles  to  pass  through 

the  points  E  &  F,  &  to  represent  any  increment  of  pressure,  together  with  any  pressure 

however  great,  or  however  insensible,  it  may  be. 

352.  We  find  that  in  air  there  is  great  compression,  &  that  this  is  proportional  to  The     force     that 
the  compressing  force.     For  this  case,  Newton  proved,  in  prop.  3,  of  the  Third  Book  of  '  * 


air 


his  Principia,  that  the  mutual  repulsive  force  between  the  particles  must  be  inversely  the  reason  for  the 
proportional  to  the  first  power  of  the  distance.     Hence,  for  these  distances,  which  the  ™.??Pm?rtlllbll'tJr?,! 

'if-  i  •  •  •   i  »i-»»i«i  •      i  i  water  ,    i 

particles  of  air  can  have  as  it  persists  with  a  property  of  this  kind,  &  does  not  induce  another  of   the  change  in 

form  (for  Newton  remarked  that  an  air  could  from  being  volatile  become  fixed,  &  Hales  elastlc   va 

especially  gave  a  very  full  proof  of  this),  the  arc  MN  must  approach  the  form  of  an  arc 

of  the  rectangular  hyperbola.     But  in  water  there  is  no  sensible  compression,  however 

great  the  compressing  weights  may  be.     Hence  some  infer  that  it  lacks  elastic  force  ;  but 

that  is  not  the  case  ;    nay  rather,  there  are  bound  to  be  immense  forces  if  the  distances 

are  diminished  ever  so  slightly  ;    although  the  particles  must  be  nea'r  limit-points,  for 

water  also  resists  separation.     There  are  infinitely  many  classes  of  curves  which  would 

satisfy  the  conditions  ;  &  it  is  sufficient  if  the  arc  EF  has  a  direction  that  is  nearly  perpen- 

dicular to  the  axis  AC.     If  it  is  desired  to  employ  some  known  curve,  it  is  sufficient  to 

know  that  the  arc  EF  approximates  closely  to  the  logistic  curve  whose  subtangent  is  very 

small  compared  with  the  distance  AD.     Now  it  is  proved  that  the  subtangent  of    the 

logistic  curve  is  to  the  interval  corresponding  to  a  double  ratio  between  the  ordinates 

very  nearly  as  14  is  to  10  ;  &  the  subtangent  is  to  the  interval,  corresponding  to  a  ratio  of 

inequality  between  the  ordinates  of  any  magnitude,  in  the  same  ratio  for  all  logistic  curves. 

If  therefore  the  subtangent  of  the  logistic  curve  is  diminished  indefinitely,  in  every  case  there 

is  a  diminution  in  the  same  ratio  of  the  interval  BD  corresponding  to   any'  ratio  of  the 

ordinates  BF,  DE,  &  the  ratio  of  AB  to  AD,  upon  which  depends  the  compression,  will 

approach  indefinitely  near  to  equality.     Now  the  ratio  of    the  densities  is  the  inverse 

triplicate  of  this  ratio  :   for  similar  parts  of  space  are  in  the  triplicate  ratio  of  homologous 

lengths,  &  the  mass  when  compressed  can  be  reduced  to  similar  form  having  the  same 

new  density.     Thus,  we  can  have  the  increment  of  the  compressing  force,  increased  in 


262  PHILOSOPHIC   NATURALIS  THEORIA 

in  quacunque  ingenti  ratione  auctae  cum  compressione  utcunque  exigua,  &  ratione  densi- 
tatum  utcunque  accedente  ad  aequalitatem.  Verum  ubi  ordinata  ED  jam  satis  exigua 
fuerit,  debet  curva  recedere  plurimum  ab  arcu  logisticae,  ad  quern  accesserat,  &  qui  in 
infinitum  protenditur  ex  parte  eadem,  ac  debet  accedere  ad  axem  AC,  &  ipsum  secare, 
ut  habeantur  deinde  vires  attractivae,  quae  ingentes  etiam  esse  possunt ;  turn  post  exiguum 
intervallum  debet  haberi  alius  arcus  [162]  repulsivus,  recedens  plurimum  ab  axe,  qui 
exhibeat  vires  illas  repulsivas  ingentes,  quas  habent  particulse  aquese,  ubi  in  vapores  abierunt 
per  fermentationem,  vel  calorem. 

Ubi  pressio  propor-  353.  In  casu  densitatis  non  immutatae  ad  sensum,  &  virium  illarum  parallelarum 

tionaiis    aititudmi,  gequalium   uti  eas  in  gravitate  nostra  concipimus,  pressiones  erunt  ut  bases,  &  altitudines  ; 

&  unde.  .  to.  ..  .   .  r  ..    ..'  ,  .  '.  .  ' 

nam  numerus  particularum  panbus  altitudimbus  respondens  ent  aequans,  adeoque  in 
diversis  altitudinibus  erit  in  earum  ratione  ;  virium  autem  aequalium  summae  erunt  ut 
particularum  numeri.  Atque  id  experimur  in  omnibus  homogeneis  fluidis,  ut  in  Mercuric, 
&  aqua. 

Quomodo  fiat  ac.  354*  Ubi  facto  foramine  liber  exitus  relinquitur  ejusmodi  massae  particulis,  erumpent 

ceieratio  in  effluxu.  ipsae  velocitatibus,  quas  acquirent,  &  quae  respondebunt  viribus,  quibus  urgentur,  &  spatio, 
quo  indigent,  ut  recedant  a  particulis  se  insequentibus ;  donee  vis  mutua  repulsiva  jam 
nulla  sit.  Prima  particula  relicta  libera  statim  incipit  moveri  vi  ilia  repulsiva,  qua 
premebatur  a  particulis  proximis  :  utcunque  parum  ilia  recesserit,  jam  secunda  illi  proxima 
magis  distat  ab  ea,  quam  a  tertia,  adeoque  movetur  in  eandem  plagam,  differentia  virium 
accelerante  motum  ;  &  eodem  pacto  aliae  post  alias  ita,  ut  tempusculo  utcunque  exiguo 
omnes  aliquem  motum  habeant,  sed  initio  eo  minorem,  quo  posteriores  sunt.  Eo  pacto 
discedunt  a  se  invicem,  &  semper  minuitur  vis  accelerans  motum,  donee  ea  evadat  nulla  ; 
quin  immo  etiam  aliquanto  plus  asquo  a  se  invicem  deinde  recedunt  particulae,  &  jam 
attractivis  viribus  retrahuntur,  accedentes  iterum,  non  quod  retro  redeant,  sed  quod 
anteriores  moveantur  jam  aliquanto  minus  velociter,  quam  posteriores ;  turn  iterum  aucta 
vi  repulsiva  incipiunt  accelerari  magis,  &  recedere,  ubi  &  oscillationes  habentur  quaedam 
hinc,  &  inde. 

Unde   velocitas  355.  Velocitates,  quae  remanent  post  exiguum  quoddam  deter minatum  spatium,  in 

duplicate16 aititudT  cluo  v*res  mutU3e>  ve^  nullas  jam  sunt,  vel  aeque  augentur,  &  minuuntur,  pendent  ab  area 
nis.  curvae,  cujus  axis  partes  exprimant  non  distantias.  a  proxima  particula,  sed  tota  spatia  ab 

initio  motus  percursa,  &  ordinatae  in  singulis  punctis  axis  exprimant  vires,  quas  in  iis  habebat 
particula.  Velocitates  in  effluxu  aquae  experimur  in  ratione  subduplicata  altitudinum, 
adeoque  subduplicata  virium  comprimentium. 
Id  haberi  debet,  si  id  spatium  sit  ejusdem 
longitudinis,  &  vires  in  singulis  punctis  res- 
pondentibus  ejus  spatii  sint  in  ratione  primae 
illius  vis.  Turn  enim  areae  totae  erunt  ut  ipsae 
vires  initiales,  &  proinde  velocitatum  quadrata, 
ut  ipsae  vires.  Infinita  sunt  curvarum  genera, 
quae  rem  exhibere  possunt ;  verum  id  ipsum 
ad  sensum  exhibere  potest  etiam  arcus  al- 
terius  logisticae  cujuspiam  amplioris  ilia,  quae 
exhibuit  distantias  singularum  particularum. 

Sit    ea   in   fig.  69   MFIN.     Tota    ejus    area  FIG.  69: 

infinita  ad    partes  CN    asymptotica    a  quavis 

ordinata  [163]  sequatur  producto  sub  ipsa  ordinata,  &  subtangente  constanti.  Quare 
ubi  ordinata  ED  jam  est  perquam  exigua  respectu  ordinatarum  BE,  HI  tota  area 
CDEN  respectu  CBFN  insensibilis  erit,  &  areae  CBFN,  CHIN  integrae  accipi  poterunt 
pro  areis  FBDE,  IHDE,  qua;  idcirco  erunt,  ut  vires  initiales  BF,  HI. 

Quid     requiritur,  35°"-  Inde  quidem  habebuntur  quadrata  celeritatum  proportionalia  pressionibus,  sive 

ut     velocitas     sit  altitudinibus.    Ut  autem  velocitas  absoluta  sit  aequalis  illi,  quam  particula  acquireret  cadendo 

habetjfr  ca'dendo  *  superficie  suprema,  quod  in  aqua  experimur  ad  sensum;    debet  praeterea  tota  ejusmodi 

per  aititudinem.      area  32quari  rectangulo  facto  sub  recta  exprimente  vim  gravitatis^  unius  particulss,  sive  vis 

repulsive,  quam  in  se  mutuo  exercent  binae  particulae,  quae  se  primo  repellunt,  sustinente 

inferiore  gravitatem  superioris,  &  sub  tota  altitudine.     Deberet  eo  casu  esse  totum  pondus 

BF  ad  illam  vim,  ut  est  altitude  tota  fluidi  ad  subtangentem  logisticae,  si  FE  est  ipsius 

logistics    arcus.     Est  autem  pondus    BF    ad   gravitatem  primae  particulae,  ut_  numerus 

particularum   in   ea   altitudine   ad   unitatem,   adeoque   ut_  eadem   ilia   tota   altitudo    ad 

distantiam  primarum    particularum.      Quare  subtangens    illius    logisticas  deberet  aequan 


A  THEORY  OF  NATURAL  PHILOSOPHY  263 

any  very  great  ratio  in  conjunction  with  a  compression  that  is  small  to  any  extent,  &  a 
ratio  of  densities  which  approaches  indefinitely  near  to  equality.  But  when  the  ordinate 
ED  is  sufficiently  small,  the  curve  must  depart  considerably  from  an  arc  of  the  logistic 
curve,  to  which  it  formerly  approximated,  &  which  proceeded  to  infinity  in  the  same 
direction  ;  it  must  approach  the  axis  AC,  &  cut  it,  in  order  that  attractive  forces  may 
be  obtained,  which  may  also  become  very  great.  Then,  after  a  small  interval,  we  must 
have  another  repulsive  arc,  receding  far  from  the  axis,  to  represent  those  very  great 
repulsive  forces,  which  the  particles  of  water  have,  when  they  pass  into  vapour  through 
fermentation  or  heat. 

353.  In  the  case  of  the  density  not  being  sensibly  changed,  &  of  those  equal  parallel  Where  the  pressure 
forces,  such  as  we  suppose  our  gravity  to  be,  the  pressures  will  be  proportional  to  the  bases  ihe^hrTude0"*1  the 
&  the  altitudes.     For,  the  number  of  particles  corresponding  to  equal  altitudes  will  be  reason  for  this, 
equal,  &  therefore,  in  different  altitudes,  the  numbers  will  be  proportional  to  the  altitudes ; 

moreover  the  sums  of  the  equal  forces  will  be  proportional  to  the  numbers  of  particles. 
We  find  this  to  be  the  case  in  all  homogeneous  fluids,  such  as  mercury  &  water. 

354.  When,  on  making  an  opening,  a  free  exit  is  left  for  the  particles  of  a  mass,  they  HOW    acceleration 
burst  forth  with  the  velocities  which  they  acquire  &  which  correspond  to  the  forces  urging  *"  efflux  arises- 
them,  &  to  the  space  to  which  it  is  necessary  for  them  to  recede  from  those  particles  that 

follow,  before  the  mutual  repulsive  force  becomes  zero.  The  first  particle,  when  left  free, 
immediately  begins  to  move  under  the  action  of  the  repulsive  force  by  which  it  is  pressed 
by  the  particles  next  to  it.  As  soon  as  it  has  moved  ever  so  little,  the  second  particle  next 
to  it  becomes  more  distant  from  it  than  from  the  third,  &  thus  moves  in  the  same  direction 
as  the  difference  of  the  forces  accelerates  the  motion.  Similarly,  one  after  the  other  they 
acquire  motion  in  such  a  manner  that  in  any  little  interval  of  time,  no  matter  how  brief, 
all  of  them  will  have  some  motion  ;  this  motion  at  the  commencement  is  so  much  the  less, 
the  farther  back  the  particles  are.  In  this  way  they  separate  from  one  another,  &  the  force 
accelerating  the  motion  ever  becomes  less  until  finally  it  vanishes.  Nay  rather,  to  speak 
more  correctly,  the  particles  still  recede  from  one  another,  &  come  under  the  action  of 
attractive  forces,  &  approach  one  another  ;  not  indeed  that  they  retrace  their  paths,  but 
because  the  more  forward  particles  are  now  moving  with  somewhat  less  velocity  than 
those  behind  ;  then  once  more  the  repulsive  force  is  increased  &  they  begin  to  be  accelerated 
more  than  those  behind  &  to  recede  from  them  ;  &  so  oscillations  to  &  fro  are  obtained. 

355.  The  velocities  that  are  left  after  any  determinate  interval  of  space,  in  which  the  Why  the  velocity 
mutual  forces  are  either  nothing  or  are  equally  increased  &   diminished,  depend  on  the  the^ub^u^Hcate 
area  of  the  curve,  of  which  parts  of  the  axis  represent  not  the  distances  from  the  next  of  the  height, 
particle,  but  the  whole  spaces  travelled  from  the  beginning  of  the  motion,  &  the  ordinates 

at  each  point  of  the  axis  represent  the  forces  which  the  particle  had  at  those  points.  It 
is  found  that  the  velocities  of  effluent  water  are  in  the  subduplicate  ratio  of  the  altitudes, 
&  thus  in  the  subduplicate  ratio  of  the  compressing  forces.  Now  this  is  what  must  be 
obtained,  if  the  space  is  of  the  same  length,  &  the  forces  at  each  corresponding  point  of 
that  space  are  in  the  ratio  of  that  first  force.  For,  then  the  total  areas  will  be  as  the  initial 
forces,  &  hence  the  squares  of  the  velocities  will  be  as  the  forces.  There  are  an  infinite 
number  of  classes  of  curves  which  will  serve  to  represent  the  case  ;  but  this  also  can  be 
represented  by  the  arc  of  another  logistic  curve  more  ample  than  that  which  represented 
the  distances  of  the  single  particles.  Let  MFIN  be  such  a  curve,  in  Fig.  69.  The  whole 
area,  indefinitely  produced  in  the  direction  of  C  &  N,  which  are  asymptotic,  measured 
from  any  ordinate,  will  be  equal  to  the  product  of  that  ordinate  &  the  constant  subtangent. 
Therefore  when  the  ordinate  ED  is  now  very  small  with  respect  to  the  ordinates  BF,  HI, 
the  whole  area  CDEN  will  be  insensible  with  respect  to  the  area  CBFN  ;  &  thus  the  whole 
areas  CBFN,  CHIN  can  be  taken  instead  of  the  areas  FBDE,  IHDE  ;  &  therefore  these 
are  to  one  another  as  the  initial  forces  BF,  HI. 

356.  From  this,  then,  we  have  that  the  squares  of  the  velocities  are  proportional  to  what  is  required 
the  pressures,  or  the  altitudes.     Now,  in  order  that  the  absolute  velocity  may  be  equal  so  that  the  velocity 

.,  ,,  ..,.,,.  ,  ,  f  •      r  J    snail    be    equal    to 

to  that  which  the  particle  would  acquire  in  falling  from  the  upper  surface,  as  is  found  that    acquired    in 

to  be  approximately  the  case  for  water,  we  must  have,  in  addition,  that  the  whole  of  such  failing  from   the 

area  must  be  equal  to  the  rectangle  formed  by  multiplying  the  straight  line  representing 

the  force  of  gravity  on  one  particle  (or  the  repulsive  force  which  a  pair  of  particles  mutually 

exert  upon  one  another,  when    they  first  repel  one  another,  the  lower  sustaining  the 

gravity  of  the  one  above)  by  the   whole  altitude.      In  this  case,  the   whole  weight  BF 

would  be  bound  to  be  to  the  force  as  the  whole  altitude  of  the  fluid  is  to  the  subtangent 

of  the  logistic  curve,  if  FE  is  an  arc  of  the  logistic  curve.      Moreover,  the^  weight  BF 

is  to  the  gravity  of  the   first   particle  as  the   number  of   particles  in  the  altitude   is   to 

unity  ;   &  thus  in  the  ratio  of  the  altitude  to  the  distance  between  the  primary  particles. 

Hence  the  subtangent  of  the  logistic  curve  would  have  to  be  equal  to  the  distance  between 


264  PHILOSOPHIC  NATURALIS  THEORIA 

illi  distantise  primarum  particularum,  quae  quidem  subtangens  erit  itidem  idcirco  perquam 

exigua. 

Tentandum   an  in  357.  An  in  omnibus  fluidis  habeatur    ejusmodi   absoluta  velocitas  &  an    quadrata 

aotidat S  Transitus  vel°citatum  *n  effluxu  respondeant  altitudinibus ;    per  experimenta  videndum   est,   ut 
ad  partem  tertiam.  constet,  an  curvse  virium  in  omnibus  sequantur  superiores  leges,  an  diversas.     Sed  ego  jam 

ab  applicatione  ad  Mechanicam  ad  applicationem  ad  Physicam  gradum  feci,  quam  uberius 

in  tertia  Parte  persequar.     Haec  interea  speciminis  loco  sint  satis  ad  immensam  quandam 

hujusce  campi  foecunditatem  indicandam  utcunque. 


A  THEORY  OF  NATURAL  PHILOSOPHY  265 

the  primary  particles ;    &  thus  the  subtangent  must  also  be  itself    very   small   on  this 
account. 

357.  Whether  such  an  absolute  velocity  exists  in  all  fluids,  &  whether  the  squares  of  it  must  be  tested 
the  velocities  with  which  they  issue  correspond   to   the  altitudes,  must  be  investigated  wheth?r  t*lsflh.^p' 

11-  »         i         •  1111  i  f  r   11  pens   in   all   fluids. 

experimentally  ;  m  order  that  it  may  be  shown  whether  the  curves  of  forces  follow  the  laws  we  will  now  pass 
given  above,  or  different  ones.     But  now  I  will  pass  on  from  the  application  to  Mechanics  on   to    the   third 
to  the  application  to  Physics,  which  I  will  follow  out  more  fully  in  the  third  part.     These 
things,  in  the  meanwhile,  may  be  sufficient  in  some  sort  to   indicate  an  immense  fertility 
in  this  field  of  knowledge. 


[164]  PARS  III 
Applicatio    Theories  ad  Physicaih 

Agendum  hie primo  358.  In    secunda    hujusce    Operis    parte,    dum    Theoriam    meam    applicarem    ad 

prietatSus'bcorpor"  Mechanicam,  multa  identidem  immiscui,  quae  application!  ad  Physicam  sterncrcnt  viam, 

um,  turn  de  discrim-  &  vero  etiam  ad  eandem  pertinerent  ;   at  hie,  quae  pertinent  ad  ipsam  Physicam,  ordinatius 

species"  *       Va"aS  Persequar  ;    &  primo  quidem  de  generalibus  agam  proprietatibus  corporum,  quas  omnes 

omnino  exhibet  ilia  lex  virium,  quam  initio  primae  partis  exposui ;   turn  ex  eadem  prsecipua 

discrimina    deducam,  quae    inter   diversas  observamus    corporum   species,  &  mutationes, 

quae  ipsis  accidunt,  alterationes,  atque  transformations  evolvam. 

Enumeratio  earum,  359.  Primum  igitur  agam   de   Impenetrabilitatc,   de  Extensione,   de   Eigurabilitate, 

&Co?dobUS  &8etUr'  de  Mole>  Massa,  &  Densitate,  dc  Inertia,  de  Mobilitate,  de  Continuitate  motuum,  de 
/Equalitate  Actionis  &  Reactionis,  de  Divisibilitate,  &  Componibilitate,  quam  ego  divisi- 
bilitati  in  infinitum  substiluo,  de  Immutabilitate  primorum  materiae  elementorum,  de 
Gravitate,  de  Cohaesione,  quas  quidem  generalia  sunt.  Turn  agam  de  Varietate  Naturae, 
&  particularibus  proprietatibus  corporum,  nimirum  de  varietate  particularum,  &  massarum 
multiplici,  de  Solidis,  &  Fluidis,  de  Elasticis,  &  Mollibus,  de  Principiis  Chemicarum 
Operationum,  ubi  de  Dissolutione,  Praecipitatione,  Adhaesione,  &  Coalescentia,  de  Fermen- 
tatione,  &  emissione  Vaporum,  de  Igne,  &  emissione  Luminis ;  ac  ipsis  praecipuis  Lutninis 
proprietatibus,  de  Odore,  de  Sapore,  de  Sono,  de  Electricitate,  de  Magnetismo  itidem 
aliquid  innuam  sub  finem ;  ac  demum  ad  generaliora  regressus,  quid  Alterationes, 
Corruptiones,  Transformationes  mihi  sint,  explicabo.  Verum  in  horum  pluribus  rem 
a  mea  Theoria  deducam  tantummodo  ad  communia  principia,  ex  quibus  peculiares 
singulorum  tractatus  pendent ;  ac  alicubi  methodum  indicabo  tantummodo,  quae  ad 
rei  perquisitionem  aptissima  mihi  videatur. 

impenetrabiiitas  360.  Impenetrabiiitas  corporum  a   mea  Theoria  omnino  sponte  fluit ;    si  enim  in 

Theoria  '  haC  mmimis  distantiis  agunt  vires  repulsivae,  quae  iis  in  infinitum  imminutis  crescant  in  infinitum 
ita,  ut  pares  sint  extinguendae  cuilibet  velocitati  utcunque  magnae,  utique  non  potest 
ulla  finita  vis,  aut  velocitas  efncere,  ut  distantia  duorum  punctorum  evanescat,  quod 
requiritur  ad  compenetrationem  ;  sed  ad  id  praestandum  infinita  Divina  virtus,  quae 
infinitam  vim  exerceat,  vel  infinitam  producat  velocitatem,  sola  sufficit. 

Aliud  impenetra-  [165]  361.  Praeter  hoc  impenetrabilitatis  genus,  quod  a  viribus  repulsivis  oritur,  est 
priumhuk/rheona".  &  aliud,  quod  provenit  ab  inextensione  punctorum,  &  quod  evolvi  in  dissertationibus 
De  Spatio,  W  Tempore,  quas  ex  Stayanis  Supplementis  hue  transtuli,  &  habetur  hie  in  fine 
Supplementorum  §  i,  &  2.  Ibi  enim  ex  eo,  quod  in  spatio  continue  numerus  punctorum 
loci  sit  infinities  infinitus,  &  numerus  punctorum  materiae  finitus,  erui  illud  :  nullum 
punctum  materiae  occupare  unquam  punctum  loci,  non  solum  illud,  quod  tune  occupat 
aliud  materiae  punctum,  sed  nee  illud,  quod  vel  ipsum,  vel  ullum  aliud  materiae  punctum 
occupavit  unquam.  Probatio  inde  petitur,  quod  si  ex  casibus  ejusdem  generis  una  classis 
infinities  plures  contineat,  quam  altera,  infinities  improbabilius  sit,  casum  aliquem,  de 
quo  ignoremus,  ad  utram  classem  pertineat,  pertinere  ad  secundam,  quam  ad  primam. 
Ex  hoc  autem  principio  id  etiam  immediate  consequitur  ;  si  enim  una  massa  projiciatur 
contra  alteram,  &  ab  omnibus  viribus  repulsivis  abstrahamus  animum  ;  numerus  projec- 
tionum,  quae  aliquod  punctum  massae  projectae  dirigant  per  rectam  transeuntem  per 
aliquod  punctum  massae,  contra  quam  projicitur,  est  utique  finitus ;  cum  numerus 
punctorum  in  utraque  massa  finitus  sit ;  at  numerus  projectionum,  quae  dirigant  puncta 
omnia  per  rectas  nulli  secundse  massae  puncto  occurrentes,  est  infinities  infinitus,  ob  puncta 
spatii  in  quovis  piano  infinities  infinita.  Quamobrem,  habita  etiam  ratione  infinitorum 
continui  temporis  momentorum,  est  infinities  improbabilior  primus  casus  secundo  ;  &  in 
quacunque  projectione  massae  contra  massam  nullus  habebitur  immediatus  occursus  puncti 
materiae  cum  altero  puncto  materiae,  adeoque  nulla  compenetratio,  etiam  independenter 
a  viribus  repulsivis. 


266 


PART  III 

Application  of  the    Theory  to  Physics 

358.  In  the  second  part  of  this  work,  in  applying  my  Theory  to  Mechanics,  I  brought  We  wUi  first  of  all 
in  also  at  the  same  time  many  things  which  opened  the  road  for  an  application  to  Physics,  generaf^properttes 
&  really  even  belonged  to  the  latter.     In  this  part  I  will  investigate  in  a  more  ordered  of  bodies,  &  then 
manner  those  things  that  belong  to  Physics.     First  of  all,  I  will  deal  with  general  properties  between^he^venu 
of  bodies ;  &  these  will  be  given  by  that  same  law  of  forces  that  I  enunciated  at  the  beginning  species. 

of  the  first  part.  After  that,  from  the  same  law  I  will  derive  the  most  important  of  the 
distinctions  that  we  observe  between  the  different  species  of  bodies,  &  I  will  discuss  the 
changes,  alterations  &  transformations  that  happen  to  them. 

359.  First,  therefore,  I  will  deal  with  Impenetrability,  Extension,  Figurability,  Volume,  Enumeration   of 
Mass,  Density,  Inertia,  Mobility,  Continuity  of    Motions,  the    Equality    of    Action    &  deai^with"  &°the 
Reaction,  Divisibility,  &  Componibility  (for  which  I  substitute  infinite  divisibility),  the  order  in  which  they 
Immutability  of  the  primary  elements  of  matter,  gravity,  &  Cohesion  ;  all  these  are  general  wm  ^  taken- 
properties.     Then  I  will  consider  the  Variety  of  Nature,  &  special  properties  of  bodies ; 

such,  for  instance,  as  the  manifold  variety  of  particles  &  masses,  Solids  &  Fluids,  Elastic, 
&  Soft  bodies ;  the  principles  of  chemical  operations,  such  as  Solution,  Precipitation, 
Adhesion  &  Coalescence,  Fermentation,  &  emission  of  Vapours,  Fire  &  the  emission  of 
Light ;  also  about  the  principal  properties  of  Light,  Smell,  Taste,  Sound,  Electricity 
&  Magnetism,  I  will  say  a  few  words  towards  the  end.  Finally,  coming  back  to  more 
general  matters,  I  will  explain  my  idea  of  the  nature  of  alterations,  corruptions  &  trans- 
formations. Now  in  most  of  these,  I  shall  derive  the  whole  matter  from  my  Theory 
alone,  &  reduce  it  to  those  common  principles,  upon  which  depends  the  special  treatment 
for  each ;  in  certain  cases  I  shall  only  indicate  the  method,  which  seems  to  me  to  be  the 
most  fit  for  a  further  investigation  of  the  matter. 

360.  The  Impenetrability  of  bodies  comes  naturally  from  my  Theory.     For,  if  repulsive  The  origin  of  im- 
forces   act   at   very  small  distances,  &  these  forces   increase  indefinitely  as  the  distances  co'rfi^g^fo  this 
decrease,  so  that  they  are  capable  of  destroying  any  velocity  however  large  ;    then  there  Theory. 

never  can  be  any  finite  force,  or  velocity,  that  can  make  the  distance  between  two  points 
vanish,  as  is  required  for  compenetration.  To  do  this,  an  infinite  Divine  virtue,  exercising 
an  infinite  force,  or  creating  an  infinite  velocity,  would  alone  suffice. 

361.  Besides  this  kind  of  Impenetrability,  which  arises  from  repulsive   forces,  there  Another    kind    of 
is  also  another  kind,  which  comes  from  the  inextension  of  the  points ;    this  I  discussed  in  p^cTfiar  to  'this 
the  dissertations  De  Spatio,  y  Tempore,  which  I  have  abstracted  from   the   Supplement  Theory. 

to  Stay's  Philosophy,  &  set  at  the  end  of  this  work  as  Supplements,  §§1,2.  From  the  fact 
that  the  number  of  points  of  position  in  a  continuous  space  may  be  infinitely  infinite,  whilst 
the  number  of  points  of  matter  may  be  finite,  I  derive  the  following  principle  ;  namely, 
that  no  point  of  matter  can  ever  occupy  either  a  point  of  position  which  is  at  the  time 
occupied  by  another  point  of  matter,  or  one  which  any  other  point  of  matter  has  ever 
occupied  before.  The  proof  is  derived  from  the  argument  that,  if  of  cases  of  the  same 
nature  one  class  of  them  contains  infinitely  more  than  another,  then  it  is  infinitely  more 
improbable  that  a  certain  case,  concerning  which  we  are  in  doubt  as  to  which  class  it  belongs, 
belongs  to  the  second  class  rather  than  to  the  first.  It  also  follows  immediately  from  this 
principle  ;  if  one  mass  is  projected  towards  another,  &  we  disallow  a  directive  mind  in  all 
repulsive  forces,  the  number  of  the  ways  of  projection,  which  direct  any  point  of  the 
projected  mass  along  a  straight  line  passing  through  any  point  of  the  mass  against  which 
it  is  projected,  is  finite  ;  for  the  number  of  points  in  each  of  the  masses  is  finite.  But 
the  number  of  ways  of  projection,  which  direct  all  points  along  straight  lines  that  pass 
through  no  point  of  the  second  mass,  is  infinitely  infinite  because  the  number  of  points 
of  space  in  any  plane  is  infinitely  infinite.  Therefore,  even  when  the  infinite  number 
of  moments  in  continuous  time  is  taken  into  account,  the  first  case  is  infinitely  more 
improbable  than  the  second.  Hence,  in  any  projection  whatever  of  mass  against  mass 
there  is  no  direct  encounter  of  one  point  of  matter  with  another  point  of  matter ;  & 
thus  there  can  be  no  compenetration,  even  apart  from  the  idea  of  repulsive  forces. 

267 


268  PHILOSOPHIC  NATURALIS  THEORIA 

sine  viribus  repui.  362.  Si  vires  repulsivae  non  adessent ;  omnis  massa  libere  transiret  per  aliam  quanvis 

comSpenetrationeem  massam,  ut  lux  per  vitra,  &  gemmas  transit,  ut  oleum  per  marmora  insinuatur  ;    atque  id 

apparentem.    Quid  semper  fieret  sine  ulla  vera  compenetratione.     Vires,  quffi  ad  aliquod  intervallum  extend- 

tkifii^l6  ye/"  quo.  untur    sat^s    magnae,   impediunt   ejusmodi    liberum   commeatum.     Porro   hie    duo    casus 

dam,  potissimum  si  distinguendi  sunt ;    alter,  in  quo  curva  virium  non  habeat  ullum  arcum  asymptoticum 

toti.*         asymP-  cum  asymptoto  perpendicular!  ad  axem,  praeter  ilium  primum,  quem  exhibet  figura  i, 

cujus  asymptotus  est  in   origine  abscissarum  ;    alter,  in  quo    adsint  alii   ejusmodi   arcus 

asymptotici.     In  hoc  secundo  casu  si  sit  aliqua  asymptotus  ad  aliquam  distantiam  ab  origine 

abscissarum,  quae  habeat  arcum  citra  se   attractivum,  ultra  repulsivum  cum  area  infinita, 

ut  juxta  num.   188  puncta  posita  in  minore  distantia  non  possint  acquirere  distantiam 

majorem,  nee,  quae  in  majore  sunt,  minorem ;    turn  vero  particula  composita  ex  punctis 

in  minore  distantia  positis,  esset  prorsus  impenetrabilis  a  particula  posita  in  majore  distantia 

ab  ipsa,  nee  ulla  finita  velocitate  posset  cum  ilia  commisceri,  &  in  ejus  locum  irrumpere  ; 

&  si  duae  habeantur  [166]  asymptoti  ejusmodi  satis  proximae,  quarum  citerior  habeat  ulterius 

crus  repulsivum,  ulterior  citerius  attractivum  cum  areis  infinitis,  turn  duo  puncta  collocata 

in  distantia  a  se  invicem  intermedia  inter  distantias  earum  asymptotorum,  nee  possent 

ulla  finita  vi,  aut  velocitate  acquirere  distantiam  minorem,  quam  sit  distantia  asymptoti 

citerioris,  nee  majorem,  quam  sit  ulterioris ;  &  cum  eae  duae  asymptoti  possint  esse  utcunque 

sibi  invicem  proximae  ;   ilia  puncta  possent  esse  necessitata  ad  non  mutandam  distantiam 

intervallo  utcunque  parvo.     Si  jam  in  uno  piano  sit  series  continua  triangulorum  aequi- 

laterorum  habentium  eas  distantias  pro  lateribus,  &  in  singulis  angulis  poneretur  quicunque 

numerus  punctorum  ad  distantiam  inter  se  satis  minorem  ea,  qua  distent  illae  duae  asymptoti, 

vel  etiam  puncta  singula  ;    fieret  utique  velum  quoddam  indissoluble,  quod  tamen  esset 

plicatile  in  quavis  e  rectis  continentibus  triangulorum  latera,  &  posset  etiam  plicari  in 

gyrum  more  veterum  voluminum. 


Soiidum  indissolu-  363.  Si  autem  sit  solidum  compositum  ex  ejusmodi  velis,  quorum  alia  ita  essent  aliis 

bUe.  &  impermea-  }mpOSita,  ut  punctum  quodlibet  superioris  veli  terminaret  pyramidem  regularem  habentem 
pro  basi  unum  e  triangulis  veli  inferioris,  &  in  singulis  angulis  collocarentur  puncta,  vel 
massae  punctorum ;  id  esset  solidissimum,  &  ne  plicatile  quidem  ;  etiamsi  crassitude 
unicam  pyramidum  seriem  admitteret.  Possent  autem  esse  dispersa  inter  latera  illius 
veli,  vel  hujus  muri,  puncta  quotcunque,  nee  eorum  ullum  posset  inde  egredi  ad  distantiam 
a  punctis  positis  in  angulis  veli,  vel  muri,  majorem  ilia  distantia  ulterioris  asymptoti.  Quod 
si  praeterea  ultra  asymptotum  ulteriorem  haberetur  area  repulsiva  infinita  ;  nulla  externa 
puncta  possent  perrumpere  nee  murum,  nee  velum  ipsum,  vel  per  vacua  spatiola  transire, 
utcunque  magna  cum  velocitate  advenirent ;  cum  nullum  in  triangulo  aequilatero  sit 
punctum,  quod  ab  aliquo  ex  angulis  non  distet  minus,  quam  per  latus  ipsius  trianguli. 

Alia   ratio   acqui-  364.  Quod  si  ejusmodi  binae  asymptoti  inter  se  proximae  sint  in  ingenti  distantia  a 

bmtatem,I&Pn°xum  principio  abscissarum,  &  in  distantia  media  inter  earum  binas  distantias  ab  ipso  initio 
per  asymptotes  ponantur  in  cuspidibus  trianguli  aequilateri  tria  puncta  materise,  turn  in  cuspide  pyramidis 
regularis  habentis  id  triangulum  aequilaterum  pro  basi  ponantur  quotcunque  puncta,  quae 
inter  se  minus  distent,  quam  pro  distantia  illarum  asymptotorum  ;  massula  constans  hisce 
punctis  erit  indissolubilis ;  cum  nee  ullum  ex  iis  punctis  possit  acquirere  distantiam  a 
reliquis,  nee  reliqua  inter  se  distantiam  minorem  distantia  asymptoti  citerioris,  &  majorem 
distantia  ulterioris,  &  ipsa  haec  particula  impenetrabilis  a  quovis  puncto  externo  materiae, 
cum  nullum  ad  reliqua  ilia  tria  puncta  possit  ita  accedere,  si  distat  magis,  vel  recedere,  si 
minus,  ut  acquirat  distantiam,  quam  habent  puncta  ejus  massae.  Ejusmodi  massis  ita 
cohibitis  per  terna  puncta  ad  maximas  distantias  sita  posset  integer  constare  Mundus, 
qui  ha-[l67]-beret  in  suis  illis  massulis,  seu  primigeniis  particulis  impenetrabilitatem 
continuam  prorsus  insuperabilem,  sine  ulla  extensione  continua,  &  indissolubilitatem 
itidem  insuperabilem  etiam  sine  ullo  mutuo  nexu  inter  earum  puncta,  per  solum  nexum, 
quem  haberent  singula  cum  illis  tribus  punctis  remotis. 


in  us  &  aliis  casi.  365.  In  omnibus  hisce  casibus  habetur  in  massa  non  continua  vis  ita  continua,  ut 

ttnuaesineeco^tinuo  nu^a  ne  apparens  quidem  compenetratio,  &  permixtio  haberi  possit  aeque,  ac  in  communi 

faciente     vim,    &  sententia  de  continua  impenetrabilis  materiae  extensione.     Quod  autem  in  illo  velo,  vel 

meSabUitas.imper"  muro  exhibuit  triangulorum,  &  pyramidum  series,  idem  obtineri  potest  per  figuras  alias 


A  THEORY  OF  NATURAL  PHILOSOPHY 


269 


362.  If  there  were  no  repulsive  forces,  every  mass  would  pass  freely  through  every  other 
mass,  as  light  passes  through  glass  &  crystals,  &  as  oil  insinuates  itself  into  marble  ;    but 
such  a  thing  as  this  would  always  happen  without  any  true  compenetration.     Forces,  which 
extend  to  an  interval  that  is  sufficiently  large  for  the  purpose,  prevent  free  passage  of  that 
kind.     Further  there  are  here  two  cases  to  be  distinguished ;    one,  in  which  the  curve  of 
forces  has  not  any  asymptotic  arc  with  an  asymptote  perpendicular  to  the  axis,  except  the 
first,  as  is  shown  in  Fig.  I,  where  the  asymptote  occurs  at  the  origin  of  abscissae  ;  the  other, 
in  which  there  are  other  such  asymptotic  arcs.     In  the  second  case,  if  there  is  an  asymptote 
at  some  distance  from  the  origin  of  abscissae,  which  has  an  attractive  arc  on  the  near  side  of 
it,  &  on  the  far  side  a  repulsive  arc  with  an  infinite  area  corresponding  to  it,  so  that,  as 
was  shown  in  Art.  188,  points  situated  at  a  less  distance  cannot  acquire  a  greater,  &  those 
at  a  greater  distance  cannot  acquire  a  less ;  then  particles  that  are  made  up  of  points  situated 
at  the  less  distance  would  be  quite  impenetrable  by  a  particle  situated  at  a  greater  distance 
from  it ;  nor  could  any  finite  velocity  force  it  to  mingle  with  it  or  invade  its  position  ;  and  if 
there  are  two  asymptotes  of  the  kind  sufficiently  near  together,  of  which  the  nearer  to  the 
origin  has  its  further  branch  repulsive,  &  the  further  has  its  nearer  branch  attractive,  the 
corresponding  areas  being  infinite,  then  two  points  situated  at  a  distance  from  one  another 
that  is  intermediate  between  the  distances  of  these  asymptotes,  cannot  with  any  finite 
force  or  velocity  acquire  a  distance  less  than  that  of  the  nearer  asymptote  or  greater  than 
that  of  the   further   asymptote.      Now  since   these  two   asymptotes  may  be  indefinitely 
near  to  one  another,  the  two  points  may  be  forced  to  keep  their  distance  unchanged  within 
an  interval  of  any  smallness  whatever.     Suppose  now  that  we  have  in  a  plane  a  continuous 
series  of  equilateral  triangles  having  these  distances  as  sides,  &  that  at  each  of  the  angles 
there  are  placed  any  number  of  points  at  a  distance  from  one  another  sufficiently  less  than 
that  of  the  distance  between  the  two  asymptotes,  or  even  single  points ;  then,  in  every 
case,  we  should  have  a  kind  of  unbreakable  skin,  which  however  could  be  folded  along  any 
of  the  straight  lines  containing  sides  of  the  triangles,  or  could  even  be  folded  in  spirals 
after  the  manner  of  ancient  manuscripts. 

363.  Moreover,  if  we  have  a  solid  composed  of  such  skins,  one  imposed  upon  the  other 
in  such  a  manner  that  any  point  of  an  upper  skin  should  terminate  a  regular  pyramid  having 
for  its  base  one  of  the  equilateral  triangles  of  the  skin  beneath,  &  in  each  of  them  points 
were  situated,  or  masses  of  points ;  then  that  would  have  very  great  solidity,  &  would  not 
be  even  capable  of  being  folded,  even  if  its  thickness  only  admitted  of  a  single  series  of 
pyramids.     Further,  any  number  of  points  could  be  scattered  between  the  sides  of  the  former 
skin,  or  the  wall  of  the  latter,  &  none  of  these  could  get  out  of  this  position  to  a  distance  from 
the  points  situated  at  the  angles  of  the  skin,  or  of  the  wall,  greater  than  the  distance  of  the 
further  asymptote.     Now  if,  in  addition  to  these,  there  happened  to  be  beyond  the  further 
asymptote  a  corresponding  infinite  repulsive  area,  no  external  points  could  break  into  the 
skin  or  wall,  nor  could  they  pass  through  empty  spaces  in  it,  no  matter  how  great  the  velocity 
with  which  they  approached  it.     For,  there  is  no   point   within   an  equilateral   triangle 
that  is  at  a  less  distance  from  the  angular  points  than  a  side  of  the  triangle. 

364.  Again,  if  there  are  two  asymptotes  very  near  one  another,  at  a  great  distance 
from  the  origin  of  abscissae,  &  at  a  distance  intermediate  between  their  two  distances  from 
the  origin  there  are  placed  three  points  of  matter  at  the  vertices  of  an  equilateral  triangle, 
&  then  at  the  vertex  of  a  regular  pyramid  having  for  its  base  that  equilateral  triangle  there 
are  placed  any  number  of  points,  which  are  at  a  less  distance  from  one  another  than  that 
between  the  two  asymptotes,  the  little  mass  made  up  of  these  points  will  be  unbreakable. 
For,  none  of  these  points  can  acquire  from  the  rest,  nor  the  rest  from  one  another,  a  distance 
less  than  the   distance  of  the  nearer   asymptote,  nor   greater  than    that   of   the  further 
asymptote.     This  particle  will  also  be  impenetrable  by  any  external  point  of  matter  ;  for  no 
point  can  possibly  approach  those  other  three  points  so  nearly,  if  the  distance  is  greater,  or 
recede   from  them  so   far,   if   the   distance   is  less,   as   to  acquire  the   same   distance   as 
that  between  the  several  points  of  the  mass.     The  whole  Universe  may  be  made  up  of 
masses  of  this  kind  restrained  by  sets  of  three  points  situated  at  very  great  distances ;    & 
it  would  have  in  the  little  masses  forming  it,  or  in  the  primary  particles,  a  continuous 
impenetrability  that  was  quite  insuperable,  without  any  continuous  extension  ;    it  would 
also  have  an  insuperable  unbreakableness  without  any  mutual  connection  between  the 
points  forming  it,  simply  owing  to  the  connection  existing  between  each  of  its  points  with 
the  three  remote  points. 

365.  In  all   these  cases  there  is  obtained  for  a  non-continuous  mass  a  force  that  is 
continuous  in  such  sort  that  there  is  not  even  apparent  compenetration  ;   &  commingling 
can  be  had  just  as  well  as  with  the  usual  idea  of  continuous  extension  of  impenetrable 
matter.     Moreover,  what  has  been  represented  by  the  skin  or  wall  of  a  series  of  triangles 
or  pyramids,  can  be  obtained  by  means  of  very  many  other  figures ;   &  it  can  be  obtained 


Without  repulsive 
forces  there  must 
be  apparent  com- 
penetration. What 
these  forces  may 
give  us  in  particles, 
&  a  sort  of  skin, 
especially  if  there 
are  asymptotes. 


An  unbreakable  & 
impermeable  solid. 


Another  way  in 
which  impenetra- 
bility may  be  ac- 
quired, &  the  con- 
nection  with  asym- 
ptotes  that  a  re 
remote  from  the 
origin  of  abscissae. 


In  these  &  other 
cases,  we  have 
continuous  resist- 
ance without  im- 
agining a  continu- 
ous force,  &  also 
absolute  impene- 
trability. 


270  PHILOSOPHIC  NATURALIS  THEORIA 

quamplurimas,  &  id  multo  pluribus  abhuc  modis  obtineretur  ;  si  non  in  unica,  sed  in 
pluribus  distantiis  essent  ejusmodi  asymptotica  repagula  cum  impenetrabilitate  continua 
per  non  continuam  punctorum  dispersorum  dispositionem. 

Sineasymptotoom.  366.  At  in  primo  illo  casu,  in  quo  nulla  habetur  ejusmodi  asymptotus  praeter  primam, 

nfeaWks^fore91!^  res  ^onge  a^°  niodo  sc  haberet.     Patet  in  co  casu  illud,  si  velocitas  imprimi  possit  massae 

aiiis    si    iis   satis  cuipiam  satis  magna  ;   fore,  ut  ea  transeat  per  massam  quancunque  sine  ulla  perturbatione 

m^gnas  ^veioatas  suarum  part}umj  &  sine  ulla  partium  alterius ;    nam  vires,  ut  agant,  &  motum  aliquem 

Exempium  giobuii  finitum  sensibilem  gignant,   indigent  continue  tempore,   quo  imminuto  in  immensum, 

n"es  transeuntis.g"  ut*  imminuitur,  si  velocitas  in  immensum  augeatur,  imminuitur  itidem  in  immensum 

earum  effectus.     Rei  ideam  exhibebit  globulus  ferreus,  qui  debeat  transire  per  planum, 

in  quo  dispersae  sint  hac,  iliac  plurimae  massae  magneticae  vim  habentes  validam  satis.     Si 

is  globus  cum  velocitate  non  ita  ingenti  projiciatur  per  directionem  etiam,  quae  in  nullam 

massam  debeat  incurrere  ;   progredi  ultra  illas  massas  non  poterit ;   sed  ejus  motus  sistetur 

ab  illarum  attractionibus.     At  si  velocitas  sit  satis  magna,  ut  actiones  virium  magneticarum 

satis  exiguo  tempore  durare  possint,   praetervolabit  utique,   nullo  sensibili  damno  ejus 

velocitati  illato. 

Diversi  effectus  re-  367.  Quin  immo  ibi  considerandum  &  illud  ;    si  velocitas  eius  fuerit  exigua,  ipsum 

late    ad   magnetes      i    i    "     '  r     •••       •     •  .  '..  .     J  *., 

pro  diversa  veioci.  globum  taciic  sisti,  exiguo  motu  a  vi  mutua  aequall,  seu  reactione,  impresso  magnetibus, 
tate  ejus  giobuii.  quo  per  solam  plani  fractionem,  &  mutuas  eorum  vires  impedito,  exigua  in  eorum  position- 
ibus  mutatio  fiat.  Si  velocitas  impressa  aliquantulum  creverit ;  turn  mutatio  in  positione 
magnetum  major  fiet,  &  adhuc  sistetur  giobuii  motus ;  sed  si  velocitas  fuerit  multo  major, 
globulus  autem  transeat  satis  prope  aliquas  e  massis  magnetifcis ;  ab  actione  mutua  inter 
ipsum,  &  eas  massas  communicabitur  satis  ingens  motus  iis  ipsis  massis,  quo  possint  etiam 
ipsum  non  nihil  retardatum,  sed  adhuc  progredientem  sequi,  avulsae,  a  caeteris,  quae  ob 
actiones  in  majore  distantia  minores,  &  brevitatem  temporis,  remaneant  ad  sensum  immotae, 
&  nihil  turbatae.  Sed  si  velo-[i68]-citas  ipsa  adhuc  augeretur,  quantum  est  opus,  eo 
deveniri  posset ;  ut  massa  utcunque  proxima  in  giobuii  transitu  nullum  sensibilem  motum 
auferret  illi,  &  ipsa  sibi  acquireret. 

inde  faciiis  expiica.  368.  Porro  ejusmodi  exemplum  intueri  licet,  ubi  globus  aliquis  contra  obstaculum 

no  phsenomeni.  quo      vj  •••  j       •         •  i      •  11  •  n      vrr  •      • 

globus  sciopeto  ex-  aliquod  projicitur,  quod,  si  satis  magnam  velocitatem  habet,  concuti  totum,  &  diitnngit 
piosus    perforat  ac  eo  majorem  effectum   edit,  quo  maior  est  velocitas,  ut  in  muris  arcium  accidit,  qui 

plana    mobilia,  nee    .  i    i  •  A         i  •         i      •  i    •  j  i  • 

movet :  cur  lumini  tormentarns  globis  impetuntur.  At  ubi  velocitas  ad  mgentem  quandam  magmtudmem 
data  tanta  veioci-  devenerit  ;  nisi  satis  solida  sit  compages  obstaculi,  sive  vires  cohaesionis  satis  validae  ;  jam 
non  major  effectus  fit,  sed  potius  minor,  foramine  tantum  excavate,  quod  aequetur  ipsi 
globo.  Id  experimur  ;  si  globus  ferreus  explodatur  sciopeto  contra  portam  ligneam, 
quae  licet  semiaperta  sit,  &  summam  habeat  super  suis  cardinibus  mobilitatem  ;  tamen 
nihil  prorsus  commovetur  ;  sed  excavatur  tantummodo  foramen  aequale  ad  sensum  diametro 
globi,  quod  in  mea  Theoria  multo  facilius  utique  intelligitur,  quam  si  continue  nexu  partes 
perfecte  solidae  inter  se  complicarentur,  &  conjungerentur.  Nimirum,  ut  in  superiore 
magnetum  casu,  particulae  globi  secum  abripiunt  particulas  ligni,  ad  quas  accesserunt 
magis,  quam  ipsae  ad  sibi  proximas  accederent,  &  brevitas  temporis  non  permisit  viribus 
illis,  a  quibus  distantium  ligni  punctorum  nexus  praestabatur,  ut  in  iis  motus  sensibilis 
haberetur,  qui  nexum  cum  aliis  sibi  proximis  a  vi  mutua  ortum  dissolveret,  aut  illis,  & 
toti  portae  satis  sensibilem  motum  communicaret.  Quod  si  velocitas  satis  adhuc  augeri 
posset ;  "  ne  iis  quidem  avulsis  massa  per  massam  transvolaret,  nulla  sensibili  mutatione 
facta,  &  sine  vera  compenetratione  haberetur  ilia  apparens  compenetratio,  quam  habet 
lumen,  dum  per  homogeneum  spatium  liberrimo  rectilineo  motu  progreditur ;  quam 
ipsam  fortasse  ob  causam  Divinus  Naturae  Opifex  tarn  immanem  luci  velocitatem  voluit 
imprimi,  quantam  in  ea  nobis  ostendunt  eclipses  Jovis  satellitum,  &  annua  fixarum  aberratio, 
ex  quibus  Rcemerus,  &  Bradleyus  deprehenderunt,  lumen  semiquadrante  horae  percurrere 
distantiam  aequalem  distantiae  Solis  a  Terra,  sive  plura  milliariorum  millia  singulis  arteriae 
pulsibus. 


Cur   in  cinere  re-  ^^    Ac  eodem  pacto,  ubi  herbarum  forma  in  cinere  cum  tenuissimis  filamentis  remanet 

forma*  plants  avo*  intacta,  avolantibus  oleosis  partibus  omnibus  sine  ulla  laasione  structurae  illarum,  id  quidem 

lante  parte  volatili  admodum  facile  intelligitur,  qui  fiat  :    ibi  nova  vis  excitata  ingentem  velocitatem  parit 

brevi  tempore,  quae  omnem  alium  effectum  impediat  virium  mutuarum  inter  olea,  & 


A  THEORY  OF  NATURAL  PHILOSOPHY  271 

in  a  much  greater  number  of  ways  as  well,  if  not  only  at  one,  but  at  many  distances,  there 
were  these  asymptotic  restraints,  resulting  in  continuous  impenetrability  through  a  non- 
continuous  disposition  of  scattered  points. 

366.  Now,  in  the  first  case,  where  there  is  no  such  asymptote  besides  the  first,  there  if  there  were  no 
would  be  a  far  different  result.     In  this  case,  it  is  evident  that,  if  a  sufficiently  great  velocity  substances'  *  "would 
can  be  given  to  any  mass,  it  would  pass  through  any  other  mass  without  any  perturbation  be    permeable    by 
of  its  own  parts,  or  of  the  parts  of  the  other.     For,  the  forces  have  no  continuous  time  sufficiently  hgre  a*t 
in  which  to  act  &  produce  any  finite  sensible  motion  ;    since  if  this  time  is  diminished  velocity    is   given 
immensely  (as  it  will  be  diminished,  if  the  velocity  is  immensely  increased),  the  effect  of  an^orT'eio^'pass- 
the  forces  is  also  diminished  immensely.     We  can  illustrate  the  idea  by  the  example  of  an  ing  between  mag- 
iron  ball,  which  is  required  to  pass  across  a  plane,  in  which  lie  scattered  in  all  positions  nets- 

a  great  number  of  magnetic  masses  possessed  of  considerable  force.  If  the  ball  is  not 
projected  with  a  certain  very  great  velocity,  even  if  its  direction  is  such  that  it  is  not  bound 
to  meet  any  of  the  masses,  yet  it  will  not  go  beyond  those  masses  ;  but  its  motion  will  be 
checked  by  their  attractions.  But  if  the  velocity  is  great  enough,  so  that  the  actions  of 
the  magnetic  forces  only  last  for  a  sufficiently  short  interval  of  time,  then  it  will  certainly 
get  through  &  beyond  them  without  suffering  any  sensible  loss  of  velocity. 

367.  Lastly,  there  is  to  be  considered  also  this  point  ;   if  the  velocity  of  the  ball  were  Relatively  diverse 
very  small,  the  ball  might  easily  be  brought  to  rest,  a  slight  motion  due  to  an  equal  mutual  to  th^magnetsfdue 
force  or  reaction  being  communicated  to  the  magnets  ;    but  this  latter  being  prevented  to  diverse  velocities 
merely  by  the  friction  of  the  plane,  the  change  in  their  positions  would  be  very  small.  c 

Then  if  the  impressed  velocity  were  increased  somewhat,  the  change  in  the  positions  of 
the  magnets  would  become  greater,  &  still  the  ball  might  be  brought  to  rest.  But  if  the 
velocity  was  much  greater,  the  ball  may  also  pass  near  enough  to  some  of  the  magnetic 
masses  ;  &  by  the  mutual  action  between  it  &  the  masses  there  will  be  communicated  to 
the  masses  a  sufficiently  great  motion,  to  enable  them  to  follow  it  as  it  goes  on  with  its 
velocity  somewhat  retarded  ;  they  will  be  torn  from  the  rest,  which  owing  to  the  smaller 
action  corresponding  to  a  greater  distance,  &  the  shortness  of  the  time,  remain  approximately 
motionless,  &  in  no  wise  disturbed.  If  the  velocity  is  still  further  increased,  to  the  necessary 
extent,  it  could  become  such  that  a  mass,  no  matter  how  near  it  was  to  the  path  of  the 
ball,  would  communicate  no  velocity  to  it,  nor  acquire  any  from  it. 

368.  Further,  an  example  of  this  sort  of  thing  can  be  seen  in  the  case  where  a  ball  is  Hence  an  easy 

.,.  -ft  i      •       •         «•   •        i  ••  ii_io     explanation  of  the 

projected  against  an  obstacle  ;   if  the  velocity  is  sufficiently  great,  it  agitates  the  whole  &  phenomenon  in 
breaks  it  to  pieces  ;  &  the  effect  produced  is  the  greater,  the  greater  the  velocity,  as  is  the  ™hich  a  bal1 

r  r  111-1  iii-n  i  i  t      •  i  from 


a  cannon 


r  r  111-1  iii-n  i  i  t      •  i 

case  for  the  walls  of  forts  bombarded  with  cannon-balls.     But  when  the  velocity  reaches  a  perforate   a   mov- 
certain  very  great  magnitude,  unless  the  fabric  of  the  obstacle  is  sufficiently  solid  or  the  able  plane  without 

*        i       •  <•?•!  i  -11  i  rr  i  i  moving  it  ;    &  why 

forces  of  cohesion  sufficiently  great,  there  will  now  be  no  greater  effect,  rather  a  less,  a  such  a  great 
hole  only  being  made,  equal  to  the  size  of  the  ball.  Let  us  consider  this  ;  suppose  an  iron  Jel°tcity  is  given  to 
ball  is  fired  from  a  gun  against  a  wooden  door,  &  this  door  is  partly  open,  &  it  has  the  utmost 
mobility  to  swing  on  its  hinges  ;  nevertheless,  it  will  not  be  moved  in  the  slightest.  Merely 
a  hole,  approximately  equal  to  the  size  of  the  ball,  will  be  made.  Now  this  is  far  more  easily 
understood  according  to  my  Theory,  than  if  we  assume  that  there  are  perfectly  solid  parts 
united  &  joined  together  by  a  continuous  connection.  Indeed,  as  in  the  case  of  the  magnets 
given  above,  the  particles  of  the  ball  carry  off  with  them  particles  of  the  wood,  which  they 
have  approached  more  closely  than  these  particles  have  approached  to  the  particles  of 
wood  next  to  them  ;  &  the  shortness  of  the  time  does  not  allow  the  forces,  by  which  the 
connection  between  the  distances  of  the  points  of  the  wood  is  maintained,  to  give  to  the 
particles  a  sensible  motion  in  the  latter,  which  would  dissolve  the  connection  with  others 
next  to  them  arising  from  the  mutual  force,  or  in  the  former,  which  would  also  communicate 
a  sufficiently  sensible  motion  in  the  whole  door.  But  if  the  velocity  is  still  further  increased 
to  a  sufficient  extent,  not  even  the  latter  particles  are  torn  away,  &  one  mass  will  pass 
through  the  other,  without  any  sensible  change  being  made.  Thus,  without  real 
compenetration,  we  should  have  that  apparent  compenetration  that  we  have  in  the  case  of 
light,  as  it  passes  through  a  homogeneous  space  with  a  perfectly  free  rectilinear  motion. 
Perchance  that  is  the  reason  why  the  Divine  Founder  of  Nature  willed  that  so  enormous  a 
velocity  should  be  given  to  light  ;  how  great  this  is  we  gather  from  the  eclipses  of  Jupiter's 
satellites,  &  from  the  annual  aberration  of  the  fixed  stars.  From  which  Roemer  &  Bradley 
worked  out  the  fact  that  light  took  an  eighth  of  an  hour  to  pass  over  the  distance  from  the  Sun 
to  the  Earth,  or  many  thousands  of  miles  in  a  single  beat  of  the  pulse.  The  reason  Why 

369.  In  the  same  way,  when  the  form  of  stalks  remain  intact  in  the  ash  with  their  in  the  ash  there 
finest  fibres,  after  that  the  oleose  parts  have  all  been  driven  off  without  any  breaking  down  th™aiforr^imf"the 
of  their  structure,  what  happens  can  be  quite  easily  understood.  Here,  a  new  force  being  plant  after  that  the 
excited  produces  in  a  brief  space  of  time  a  mighty  velocity,  which  prevents  all  that  other  ^en^riverToff  by 
effect  arising  from  the  mutual  forces  between  the  oily  &  the  ashy  parts  ;  the  oily  particles  the  action  of  fire. 


272  PHILOSOPHIC  NATURALIS  THEORIA 

cineres^  oleaginosis  particulis  inter  terreas  cum  hac  apparenti  compenetratione  liberrime 
avolantibus  sine  ullo  immediate  impactu,  £  incursu. 

.  37°-  Quod  si  ita  res  habet  ;  liceret  utique  nobis  per  occlusas  ingredi  portas,  £  per 
retur,  si  possums  durissima  transvolarc  murorum  sc-[i69]-pta  sine  ullo  obstaculo, £  sine  ulla  vera  compene- 
ve!odtate1mPnsaetis  trati°ne,  nimirum  satis  magnam  velocitatem  nobis  ipsis  possemus  imprimere,  quod  si 
magnam.  Natura  nobis  permisisset,  &  velocitates  corporum,  quae  habemus  prae  manibus,  ac  nostrorum 

digitorum  celeritates  solerent  esse  satis  magnae ;  apparentibus  ejusmodi  continuis 
compenetrationibus  assueti,  nullam  impenetrabilitatis  haberemus  ideam,  quam  mediocritati 
nostrarum  virium,  &  velocitatum,  ac  experimentis  hujus  generis  a  sinu  materno,  &  prima 
infantia  usque  adeo  frequentibus,  &  perpetuo  repetitis  debemus  omnem. 

371'  ^x  imPenetrabilitate  oritur  extensio.  Ea  sita  est  in  eo,  quod  alise  partes  sint 
extra  alias  :  id  autem  necessario  haberi  debet  ;  si  plura  puncta  idem  spatii  punctum  simul 
occupare  non  possint.  Et  quidem  si  nihil  aliunde  sciremus  de  distributione  punctorum 
materias ;  ex  regulis  probabilitatis  constaret  nobis,  dispersa  esse  per  spatium  extensum 
in  longum,  latum,  &  profundum,  atque  ita  constaret,  ut  de  eo  dubitare  omnino  non  liceret, 
adeoque  haberemus  extensionem  in  longum,  latum,  &  profundum  ex  eadem  etiam  sola 
Theoria  deductam.  Nam  in  quovis  piano  pro  quavis  recta  linea  infinita  sunt  curvarum 
genera,  quae  eadem  directione  egressae  e  dato  puncto  extenduntur  in  longum,  &  latum 
respectu  ejusdem  rectae,  &  pro  quavis  ex  ejusmodi  curvis  infinitse  sunt  curvae,  quae  ex  illo 
puncto  egressae  habeant  etiam  tertiam  dimensionem  per  distantiam  ab  ipso.  Quare  sunt 
infinities  plures  casus  positionum  cum  tribus  dimensionibus,  quam  cum  duabus  solis,  vel 
unica,  &  idcirco  infinities  major  est  probabilitas  pro  uno  ex  iis,  quam  pro  uno  ex  his,  & 
probabilitas  absolute  infinita  omnem  eximit  dubitationem  de  casu  infinite  improbabili, 
utut  absolute  possibili.  Quin  immo  si  res  rite  consideretur,  &  numeri  casuum  inter  se 
conferantur  ;  inveniemus,  esse  infinite  improbabile,  uspiam  jacere  prorsus  accurate  in 
directum  plura,  quam  duo  puncta,  &  accurate  in  eodem  piano  plura,  quam  tria. 


Extensum  ejusmodi  372.  Haec  quidem  extensio  non  est  mathematice,  sed  physice  tantum  continua  :    at 

mathematics6'  con"  de  prsejudicio,  ex  quo  ideam  omnino  continuae  extensionis  ab  infantia  nobis  efformavimus, 
tinuum  :  real  em  satjs  dictum  est  in  prima  Parte  a  num.  158  ;  ubi  etiam  vidimus,  contra  meam  Theoriam 
consutat"  ^°  *  non  posse  afferri  argumenta,  quae  contra  Zenonistas  olim  sunt  facta,  £  nunc  contra 
Leibnitianos  militant,  quibus  probatur,  extensum  ab  inextenso  fieri  non  posse.  Nam 
illi  inextensa  contigua  ponunt,  ut  mathematicum  continuum  efforment,  quod  fieri  non 
potest,  cum  inextensa  contigua  debeant  compenetrari,  dum  ego  inextensa  admitto  a  se 
invicem  disjuncta.  Nee  vero  illud  vim  ullam  contra  me  habet,  quod  nonnulli  adhibent, 
dicentes,  hujusmodi  extensionem  nullam  esse,  cum  constet  punctis  penitus  inexten-[i7o]-sis, 
&  vacuo  spatio,  quod  est  purum  nihil.  Constat  per  me  non  solis  punctis,  sed  punctis 
habentibus  relationes  distantiarum  a  se  invicem  :  eae  relationes  in  mea  Theoria  non 
constituuntur  a  spatio  vacuo  intermedio,  quod  spatium  nihil  est  actu  existens,  sed  est 
aliquid  solum  possibile  a  nobis  indefinite  conceptum,  nimirum  est  possibilitas  realium 
modorum  localium  existendi  cognita  a  nobis  secludentibus  mente  omnem  hiatum,  uti 
exposui,  in  prima  Parte  num.  142,  £  fusius  in  ea  dissertatione  De  Spatio  £  Tempore, 
quam  hie  ad  calcem  adjicio  ;  constituuntur  a  realibus  existendi  modis,  qui  realem  utique 
relationem  inducunt  realiter,  £  non  imaginarie  tantum  diversam  in  diversis  distantiis. 
Porro  si  quis  dicat,  puncta  inextensa,  £  hosce  existendi  modos  inextensos  non  posse  con- 
stituere  extensum  aliquid ;  reponam  facile,  non  posse  constituere  extensum  mathematice 
continuum,  sed  posse  extensum  physice  continuum,  quale  ego  unicum  admitto,  £ 
positivis  argumentis  evinco,  nullo  argumento  favente  alteri  mathematice  continue  extenso, 
quod  potius  etiam  independenter  a  meis  argumentis  difficultates  habet  quamplurimas. 
Id  extensum,  quod  admitto,  est  ejusmodi,  ut  puncta  materis  alia  sint  extra  alia,  ac 
distantias  habeant  aliquas  inter  se,  nee  omnia  jaceant  in  eadem  recta,  nee  in  eodem  piano 
omnia,  sint  vero  multa  ita  proxima,  ut  eorum  intervalla  omnem  sensum  effugiant.  In  eo 
sita  est  extensio,  quam  admitto,  quae  erit  reale  quidpiam,  non  imaginarium,  £  erit  physice 
continua. 


A  THEORY  OF  NATURAL  PHILOSOPHY  273 

fly  off  between  the  earthy  particles  with  this  apparent  compenetration,  in  the  freest  manner, 
without  any  immediate  impulse  or  collision. 

370.  But  if  this  were  the  case,  we  could  walk  through  shut  doors,  or  pass  through  the  Apparent  compene- 
hardest  walled  enclosures  without  any  resistance,  &  without  any  real  compenetration;  wouw  be' obtained 
that  is  to  say,  if  we  could  impress  upon  ourselves  a  sufficiently  great  velocity.     Now  if  if  we  were  able  to 
Nature  allowed  us  this,  &  the  velocities  of  bodies  which  are  around  us,  &  the  speed  of  our  v'efocTt^^r^at 
fingers  were  usually  sufficiently  great,  we,  being  accustomed  to  such  continuous   apparent  enough. 
compenetration,    should  have    no  idea   of   impenetrability.     We   owe  the  whole  idea  of 
impenetrability  to  the  mediocrity  of  our  forces  &  velocities,  &  to  experiences  of  this  kind, 

which  have  happened  to  us  from  the  time  we  were  born,  during  infancy  &  up  till  the  present 
time,  frequently  &  continually  repeated. 

371.  From    impenetrability  there  arises  extension.     It  is  involved  in  the   fact  that  Extension  nee es- 
some  parts  are  outside  other  parts ;   &  this  of  necessity  must  be  the  case,  if  several  points  repuLve1Sforces.° 
cannot  at  the  same  time  occupy  the  same  point  of  space.     Indeed,  even  if  we  knew  nothing 

from  any  other  source  about  the  distribution  of  the  points  of  matter,  it  would  be  manifest 
from  the  rules  of  probability  that  they  were  dispersed  through  a  space  extended  in  length, 
breadth  &  depth  ;  &  it  would  be  so  clear,  that  there  could  not  be  the  slightest  doubt  about 
it ;  &  thus  we  should  obtain  extension  in  length,  breadth  &  depth  as  a  consequence  of 
my  Theory  alone.  For,  in  any  plane,  for  any  straight  line  in  it,  there  are  an  infinite  number 
of  kinds  of  curves,  which  starting  in  the  same  direction  from  a  given  point  extend  in  length 
&  breadth  with  respect  to  this  same  straight  line  ;  &  for  any  one  of  these  curves  there  are 
an  infinite  number  of  curves  that,  starting  from  that  point,  have  also  a  third  dimension 
through  distance  from  the  point.  Hence,  there  are  infinitely  more  cases  of  positions  with 
three  dimensions  than  with  two  alone  or  only  one ;  &  thus  there  is  infinitely  greater  probability 
in  favour  of  one  of  the  former  than  for  one  of  the  latter  ;  &  as  the  probability  is  absolutely 
infinite,  it  removes  any  doubt  about  a  case  which  is  infinitely  improbable,  though  absolutely 
possible.  Indeed,  if  the  matter  is  carefully  considered,  &  the  number  of  cases  compared 
with  one  another,  we  shall  find  that  it  is  infinitely  improbable  that  more  than  two  points 
will  anywhere  lie  accurately  in  the  same  straight  line,  or  more  than  three  in  the  same 
plane. 

372.  This  extension  is  not  mathematically,  but  only  physically,  continuous ;   &  on  the  s"ch  extension  is 
matter  of  the  prejudgment,  from  which  we  have  formed  for  ourselves  the  idea  of  absolutely  mtthematically, 
continuous  extension  from  infancy,  enough  has  been  said  in  the  First  Part,  starting  with  continuous;  it   is 
Art.  158.     There,  too,  we  saw  that  there  could  not  be  brought  forward  against  my  Theory  consists! 

the  arguments  which  of  old  were  brought  against  the  followers  of  Zeno,  &  which  now  are 
urged  against  the  disciples  of  Leibniz,  by  which  it  is  proved  that  extension  cannot  be 
produced  from  non-extension.  For  these  disputants  assume  that  their  non-extended  points 
are  placed  in  contact  with  one  another,  so  as  to  form  a  mathematical  continuum  ;  &  this 
cannot  happen,  since  things  that  are  contiguous  as  well  as  non-extended  must  compenetrate  ; 
but  I  assume  non-extended  points  that  are  separated  from  one  another.  Nor  indeed  have 
the  arguments,  which  some  others  use,  any  validity  in  opposition  to  my  Theory  ;  when  they 
say  that  there  is  no  such  extension,  since  it  is  founded  on  non-extended  points  &  empty 
space,  which  is  absolute  nothing.  According  to  my  Theory,  it  is  founded,  not  on  points 
simply,  but  on  points  having  distance  relations  with  one  another  ;  these  relations,  in  my 
Theory,  are  not  founded  upon  an  empty  intermediate  space  ;  for  this  space  has  no  actual 
existence.  It  is  only  something  that  is  possible,  indefinitely  imagined  by  us ;  that  is  to 
say,  it  is  the  possibility  of  real  local  modes  of  existence,  pictured  by  us  after  we  have 
mentally  excluded  every  gap,  as  I  explained  in  the  First  Part  in  Art.  142,  &  more  fully 
in  the  dissertation  on  Space  &  Time,  which  I  give  at  the  end  of  this  work.  The  relations 
are  founded  on  real  modes  of  existence  ;  &  these  in  every  case  yield  a  real  relation  which 
is  in  reality,  &  not  merely  in  supposition,  different  for  different  distances.  Further,  if 
anyone  should  argue  that  these  non-extended  points,  or  non-extended  modes  of  existence, 
cannot  constitute  anything  extended,  the  reply  is  easy.  I  say  that  they  cannot  constitute 
a  mathematically  extended  continuum,  but  they  can  a  physically  extended  continuum. 
The  latter  only  I  admit,  &  I  prove  its  existence  by  positive  arguments ;  none  of  these 
arguments  being  favourable  to  the  other  continuum,  namely  one  mathematically  extended. 
This  latter,  even  apart  from  any  arguments  of  mine,  has  very  many  difficulties.  The 
extension,  which  I  admit,  is  of  such  a  nature  that  it  has  some  points  of  matter  that  lie  outside 
of  others,  &  the  points  have  some  distance  between  them,  nor  do  they  all  lie  on  the  same 
straight  line,  nor  all  of  them  in  the  same  plane  ;  but  many  of  them  are  so  close  to  one  another 
that  the  intervals  between  them  are  quite  beyond  the  scope  of  the  senses.  In  that  is  involved 
the  extension  which  I  admit ;  &  it  is  something  real,  not  imaginary,  &  it  will  be  physically 
continuous. 


274  PHILOSOPHIC  NATURALIS  THEORIA 

Quomodo  existat  373.  At  erit  fortasse,  qui  dicet,  sublata  extensione  absolute  mathematica  tolli  omnem 

con^uo'Ltu^fsi?  Geometriam.     Respondeo,  Geometriam  non  tolli,  quae  considerat  relationes  inter  distantias, 
ente.  &  inter  intervalla  distantiis  intercepta,  quae  mente  concipimus,  &  per  quam  ex  hypothesibus 

quibusdam  conclusiones  cum  iis  connexas  ex  primis  quibusdam  principiis  deducimus. 
Tollitur  Geometria  actu  existens,  quatenus  nulla  linea,  nulla  superficies  mathematice 
continua,  nullum  solidum  mathematice  continuum  ego  admitto  inter  ea,  quae  existunt ; 
an  autem  inter  ea,  quse  possunt  existere,  habeantur,  omnino  ignoro.  Sed  aliquid  ejusmodi 
in  communi  etiam  sententia  accidit.  Nulla  existit  revera  in  Natura  recta  linea,  nullus 
circulus,  nulla  ellipsis,  nee  in  ejusmodi  lineis  accurate  talibus  fit  motus  ullus,  cum  omnium 
Planetarum,  &  Terrae  in  communi  sententia  motus  habeantur  in  curvis  admodum  compli- 
catis,  atque  altissimis,  &,  ut  est  admodum  probabile,  transcendentibus.  Nee  vero  in 
magnis  corporibus  ullam  habemus  superficiem  accurate  planam,  &  continuam,  aut  sphsericam, 
aut  cujusvis  e  curvis,  quas  Geometrae  contemplantur,  &  plerique  ex  iis  ipsis,  qui  solida 
volunt  elementa,  simplices  ejusmodi  figuras  ne  in  ipsis  quidem  elementis  admittent. 

Quid  in  ea  imagi-  374.  Quamobrem  Geometria  tota  imaginaria  est,  &  idealis,  sed  propositiones  hypo- 

narium.  quid  reale  :      i       •  •     j       j    j  r          i  a      •        •  j-   •  i     -11 

eiegans  anaiogia  theticse,  quae  inde  deducuntur,  [171]  sunt  verse,  &  si  existant  conditiones  ab  ilia  assumptae, 
loci  cum  tempore  existent  utique  &  conditionata  inde  eruta,  ac  relationes  inter  distantias  punctorum 
Hatis  '"mensural "  imaginarias  ope  Geometriae  ex  certis  conditionibus  deductae,  semper  erunt  reales,  &  tales, 
quales  eas  invenit  Geometria,  ubi  illae  ipsae  conditiones  in  realibus  punctorum  distantiis 
existant.  Ceterum  ubi  de  realibus  distantiis  agitur,  nee  illud  in  sensu  physico  est  verum, 
ubi  punctum  interiacet  aliis  binis  in  eadem  recta  positis,  a  quibus  aeque  distet,  binas  illas 
distantias  fore  partes  distantiae  punctorum  extremorum  juxta  ea  quae  diximus  num.  67. 
Physice  distantia  puncti  primi  a  secundo  constituitur  per  puncta  ipsa,  &  binos  reales  ipsorum 
existendi  modos,  ita  &  distantia  secundi  a  tertio  ;  quorum  summa  continet  omnia  tria 
puncta  cum  tribus  existendi  modis,  dum  distantia  primi  a  tertio  constituitur  per  sola  duo 
puncta  extrema,  &  duos  ipsorum  existendi  modos,  quae  ablato  intermedio  reali  puncto 
manet  prorsus  eadem.  Illas  duae  sunt  partes  illius  tertiae  tantummodo  in  imaginario,  & 
geometrico  statu,  qui  concipit  indefinite  omnes  possibiles  intermedios  existendi  modos 
locales,  &  per  earn  cognitionem  abstractam  concipit  continua  intervalla,  ac  eorum  partes 
assignat,  &  ope  ejusmodi  conceptuum  ratiocinationes  instituit  ab  assumptis  conditionibus 
petitas,  quae,  ubi  demum  ad  aliquod  reale  deducunt,  non  nisi  ad  verum  possint  deducere, 
sed  quod  verum  sit  tantummodo,  si  rite  intelligantur  termini,  &  explicentur.  Sic  quod 
aliqua  distantia  duorum  punctorum  sit  aequalis  distantiae  aliorum  duorum,  situm  est  in 
ipsa  natura  illorum  modorum,  quibus  existunt,  non  in  eo,  quod  illi  modi,  qui  earn  individuam 
dlstantiam  constituunt,  transferri  possint,  ut  congruant.  Eodem  pacto  relatio  duplae, 
vel  triplae  distantiae  habetur  immediate  in  ipsa  essentia,  &  natura  illorum  modorum.  Vel 
si  potius  velimus  illam  referre  ad  distantiam  aequalem  ;  dici  poterit,  earn  esse  duplam 
alterius,  quae  talis  sit,  ut  si  alteri  ex  alterius  punctis  ponatur  tertium  novum  ad  aequalem 
distantiam  ex  parte  altera  ;  distantia  nova  hujus  tertii  a  primo  sit  aequalis  illi,  quae  duplae 
nomen  habet,  &  sic  de  reliquis,  ubi  ad  realem  statum  transitur.  Neque  enim  in  statu 
reali  haberi  potest  usquam  congruentia  duarum  magnitudinum  in  extensione,  ut  haberi 
nee  in  tempore  potest  unquam  ;  adeoque  nee  aequalitas  per  congruentiam  in  statu  reali 
haberi  potest,  nee  ratio  dupla  per  partium  asqualitatem.  Ubi  decempeda  transfertur 
ex  uno  loco  in  alium,  succedunt  alii,  atque  alii  punctorum  extremorum  existendi  modi, 
qui  relationes  inducunt  distantiarum  ad  sensum  aequalium  :  ea  aequalitas  a  nobis  supponitur 
ex  causis,  nimirum  ex  mutuo  nexu  per  vires  mutuas,  uti  hora  hodierna  ope  egregii  horologii 
comparatur  cum  hesterna,  itidem  aequalitate  supposita  ex  causis,  sed  loco  suo  divelli,  & 
ex  uno  die  in  alterum  hora  eadem  traduci  nequaquam  potest.  Verum  haec  omnia  ad 
Metaphysicam  potius  pertinent,  &  ea  fusius  cum  omnibus  [172]  loci,  ac  temporis  relationibus 
persecutus  sum  in  memoratis  dissertationibus,  quas  hie  in  fine  subjicio. 


FigurabiHtas    orta  375.  Ex    extensione    oritur    figurabilitas,    cum    qua    connectitur    moles,    &  densitas 

quid6  "siV  figure" C&  supposita  massa.  Quoniam  puncta  disperguntur  per  spatium  extensum  in  longum,  latum, 
quam  vaga,  '  &  &  profundum  ;  spatium,  per  quod  extenduntur,  habet  suos  terminos,  a  quibus  figura 
etiam^n*  communi  pendet.  Porro  figuram  determinatam  ab  ipsa  natura,  &  existentem  in  re,  possunt  agnoscere 
sententia.  tantummodo  in  elementis  ii,  qui  admittunt  elementa  ipsa  solida,  atque  compacta,  &  continua, 


A  THEORY  OF  NATURAL  PHILOSOPHY  275 

373.  But  perhaps  some  one  will  say  that,  if  absolutely  continuous  extension  is  barred,  How  Geometry  can 
then  the  whole  of  Geometry  is  demolished.     I  reply,  that  Geometry  is  not  demolished,  existing  continuum 
since  it  deals  with  relations  between  distances,  &  between  intervals  intercepted  in  these  is  excluded, 
distances ;    that  these  we  mentally  conceive,  &  by  them  we  derive  from  certain  hypotheses 
conclusions  connected  with  them,  by  means  of  certain  fundamental  principles.     Geometry, 

as  actually  existent,  is  demolished  ;  in  so  far  as  there  is  no  line,  no  surface,  &  no  solid  that 
is  mathematically  continuous,  which  I  admit  as  being  among  things  actually  existing  ; 
whether  they  are  to  be  numbered  amongst  things  that  might  possibly  exist,  I  do  not  know. 
But  something  of  the  sort  does  take  place,  according  to  the  usual  idea  of  things.  As  a 
matter  of  fact,  there  is  in  Nature  no  such  thing  as  a  straight  line,  or  a  circle,  or  an  ellipse  ; 
nor  is  there  motion  in  lines  that  are  accurately  such  as  these  ;  for  in  the  opinion  of  everybody, 
the  motions  of  all  the  planets  &  the  Earth  take  place  in  curves  that  are  very  complicated, 
having  equations  of  a  very  high  degree,  or,  as  is  quite  possible,  transcendent.  Nor  in  large 
bodies  do  we  have  any  surfaces  that  are  quite  plane,  &  continuous,  or  spherical,  or  shaped 
according  to  any  of  the  curves  which  geometers  investigate  ;  &  very  many  of  these  men, 
who  accept  solid  elements,  will  not  admit  simple  figures  even  in  the  very  elements. 

374.  Hence  the  whole  of  geometry  is  imaginary  ;    but  the  hypothetical  propositions  The   imaginary  & 
that  are  deduced  from  it  are  true,  if  the  conditions  assumed  by  it  exist,  &  also  the  conditional  Geometry  ^arfeie- 
things  deduced  from  them,  in  every  case  ;  &  the  relations  between  the  imaginary  distances  gant  analogy    be- 
of  points,  derived  by  the  help  of  geometry  from  certain  conditions,  will  always  be  real,  *ween  ^Iace  &  time 

11  el  11  1  1  T      •  •  e  IT 

&  such  as  they  are  found  to  be  by  geometry,  when  those  conditions  exist  for  real  distances  of  equality, 
of  points.  Besides,  when  we  are  dealing  with  real  distances,  it  is  not  true  in  a  physical 
sense,  when  a  point  lies  between  two  others  in  the  same  straight  line,  equally  distant  from 
either,  to  say  that  the  two  distances  are  parts  of  the  distance  between  the  two  outside  points, 
according  to  what  we  have  said  in  Art.  67.  Physically  speaking,  the  distance  of  the  first 
point  from  the  second  is  fixed  by  the  two  points  &  their  two  real  modes  of  existence,  &  so 
also  for  the  distance  between  the  second  &  the  third.  The  sum  of  these  contains  all  three 
points  &  their  three  modes  of  existence  ;  whilst  the  distance  of  the  first  from  the  third 
is  fixed  by  the  two  end  points  only,  together  with  their  two  modes  of  existence  ;  &  this 
remains  unaltered  if  the  intermediate  real  point  is  taken  away.  The  two  distances  are 
parts  of  the  third  only  in  imagination,  &  in  the  geometrical  condition,  which  in  an  indefinite 
manner  conceives  all  the  possible  intermediate  local  modes  of  existence ;  &  from  that 
abstract  conception  forms  a  picture  of  continuous  intervals,  &  assigns  parts  to  them  ;  then, 
by  the  aid  of  such  imagery  institutes  chains  of  reasoning  founded  on  assumed  conditions ; 
&  these,  when  at  last  they  lead  to  something  real,  will  only  do  so,  if  it  is  possible  for  them 
to  lead  to  something  that  is  true,  &  something  that  is  only  true  if  the  terms  are  correctly 
understood  &  explained.  Thus,  the  fact,  that  the  distance  between  two  points  is  equal 
to  the  distance  between  two  other  points,  rests  upon  the  nature  of  their  modes  of  existence, 
&  not  upon  the  idea  that  the  modes,  which  constitute  the  individual  distances,  can  be 
transferred,  so  as  to  agree  with  one  another.  In  the  same  way,  the  idea  of  twice,  or  three 
times  a  distance,  is  obtained  directly  from  the  essential  nature  of  those  modes  of  existence. 
Or,  if  we  prefer  to  refer  it  to  the  idea  of  equal  distances,  we  can  say  that  one  distance  is 
twice  another  when  it  is  such  that,  if  beyond  the  second  point  of  the  latter  we  place  a  new 
third  point  at  a  distance  equal  to  that  of  the  first  point  from  the  second,  then  the  distance 
of  this  new  third  point  from  the  first  point  will  be  equal  to  that  to  which  the  name  double 
distance  is  given  ;  &  so  on  for  other  multiples,  when  the  matter  is  reduced  to  a  consideration 
of  real  state.  For,  in  the  real  state,  there  never  can  be  a  congruence  of  two  magnitudes 
in  extension,  just  as  there  never  can  be  such  a  congruence  in  time  ;  &  therefore  there  never 
can  be  an  equality  depending  on  congruence  in  the  real  state,  nor  a  double  ratio  through 
equality  of  parts.  When  a  length  of  ten  feet  is  transferred  from  one  place  to  another,  there 
follow,  one  after  the  other,  different  modes  of  existence  of  the  end  points ;  &  these  modes 
introduce  relations  of  practically  equal  distances.  This  equality  is  supposed  by  us  to  be 
due  to  causes ;  for  instance,  to  the  mutual  connection  in  consequence  of  mutual  forces  ; 
just  as  an  hour  of  to-day  may  be  compared  with  one  of  yesterday  by  the  help  of  an  accu- 
rate clock  ;  but  the  same  hour  cannot  be  disjointed  from  its  own  position  &  transferred 
from  one  day  to  another  in  any  way.  But  really,  such  matters  have  more  to  do  with 
Metaphysics  ;  &  I  have  investigated  them  more  fully,  together  with  all  the  relations  of 
space  &  time,  in  the  dissertations  I  have  mentioned,  which  I  add  at  the  end  of  this  work. 

375.  From  extension  arises  the  idea  of  figurability  ;   with  this  is  connected  volume  &,  Figurabiiity    arises 

,      J'J     ,  ,     ,       .,  -  '  £.  .  i     from     extension; 

when  we  have  conceived  the  idea  of  mass,  density.      Since  points  are  scattered  through  the  nature  of  shape, 
extended  space  in  length,  breadth  &  depth,  the  space  through  which  they  are  extended  &  how  vague  the 

i_       •      1  j      •  ,  i      •        i  T-*        i          •     •     •        i_        i  idea  of  it  is,  even  m 

has  its  boundaries  ;   &  upon  these  boundaries  depends  shape,     rurther,  it  is  in  the  elements  the  opinion  usually 
alone  that  a  shape,  determinate  by  its  very  nature,  &  existing  of  itself,  can  be  acknowledged  held. 
by  those  who  suppose  the  elements  to  be  solid,  compact  &  continuous ;    &  by  those  who 


276  PHILOSOPHIC  NATURALIS  THEORIA 

&  qui  ab  inextensis  extensum  continuum  componi  posse  arbitrantur,  ubi  nimirum  tota 
ilia  materia  superficie  continua  quadam  terminetur.  Ceterum  in  corporibus  hisce,  quae 
nobis  sub  sensum  cadunt,  idea  figurae,  quae  videtur  maxime  distincta,  est  admodum  vaga, 
&  indefinita,  quod  quidem  diligenter  exposui  agens  superiore  anno  de  figura  Telluris  in 
dissertatione  inserta  postremo  Bononiensium  Actorum  tomo,  in  qua  continetur  Synopsis 
mei  operis  de  Expeditione  Litteraria  -per  Pontificiam  ditionem,  ubi  sic  habeo  ;  Inprimis  hoc 
ipsum  nomen  figure  terrestris,  quod  certam  quandam,  ac  determinatam  significationem  videtur 
habere,  habet  illam  quidem  admodum  incertam,  y  vagam.  Superficies  ilia,  quce  maria,  y 
lacus,  y  fluvios,  ac  mantes,  y  campos,  vallesque  terminat,  est  ilia  quidem  admodum,  nobis 
saltern,  irregularis,  y  vero  etiam  instabilis :  mutatur  enim  quovis  utcunque  minima  undarum, 
y  glebarum  motu,  nee  de  hac  Telluris  figura  agunt,  qui  in  figuram  Telluris  inquirunt :  aliam 
ipsi  substituunt,  quce  regularis  quodammodo  sit,  sit  autem  illi  priori  proxima,  quce  nimirum 
abrasis  haberetur  montibus,  collibusque,  vallibus  vero  oppletis.  At  hcec  iterum  terrestris  figures 
notio  vaga  admodum  est,  &  incerta.  Uti  enim  infinita  sunt  curvarum  regMlarium  genera, 
quce  per  datum  datorum  punctorum  numerum  transire  possint,  ita  infinita  sunt  genera  curvarum 
superficierum,  quce  Tellurem  ita  ambire  possint,  atque  concludere,  ut  vel  omnes,  vel  datos 
contingant  in  datis  punctis  mantes,  collesque,  vel  si  per  medios  transire  colles,  ac  monies  debeat 
superficies  qucedam  ita,  ut  regularis  sit,  y  tantundem  materics  concludat  extra,  quantum 
vacui  aeris  infra  sese  concludat  usque  ad  veram  hanc  nobis  irregularem  Telluris  superficiem, 
quam  intuemur  :  infinites  itidem,  &  a  se  invicem  diverse?  admodum  superficies  haberi  possunt, 
quce  problemati  satisfaciant,  atque  ece  ejusmodi  etiam,  ut  nullam,  quce  sensu  percipi  possif, 
prce  se  ferant  gibbositatem,  quce  ipsa  vox  non  ita  determinatam  continet  ideam. 

Quanto    magis   in  376.  Haec  ego  ibi  de  Telluris  figura,  quae  omnino  pertinent  ad  figuram   corporis 

cujuscunque  in  communi  etiam  sententia  de  continua  extensione  materiae  :  nam  omnium 
fere  corporum  superficies  hie  apud  nos  utique  multo  magis  scabrae  sunt  pro  ratione  suae 
magnitudinis,  quam  Terra  pro  ratione  magnitudinis  suae,  &  vacuitates  interrias  habent 
quamplurimas.  Ve-[i73]-rum  in  mea  Theoria  res  adhuc  magis  indefinita,  &  incerta  est. 
Nam  infinite  sunt  etiam  superficies  curvae  continues,  in  quibus  tamen  omnia  jacent  puncta 
massae  cujusvis  :  quin  immo  infinitae  numero  curvae  sunt  lineae,  quae  per  omnia  ejusmodi 
puncta  transeant.  Quamobrem  mente  tantummodo  confingenda  est  qusedam  superficies, 
quae  omnia  puncta  includat,  vel  quae  pauciora,  &  a  reliquorum  coacervatione  remotiora 
excludat,  quod  aestimatione  quadam  morali  fiet,  non  accurata  geometrica  determinatione. 
Ea  superficies  figuram  exhibebit  corporis ;  atque  hie  jam,  quae  ad  diversa  figurarum  genera 
pertinent ;  id  omne  mini  commune  est  cum  communi  Theoria  de  continua  extensione 
materiae. 

Moles  a  figura  377.  A  figura  pendet  moles,  quae  nihil  est  aliud,  nisi  totum  spatium  extensum  in 

e]usenideain&e7n  longum,    latum,    &   profundum    externa    superficie   conclusum.     Porro    nisi   concipiamus 

sententia  communi,  superficiem  illam,   quam  innui,  quae  figuram  determinet ;    nulla  certa  habebitur  molis 

haclxhteoriaf8      "  i^ea  :  <lum  immo  si  superficiem  concipiamus  tortuosam  illam,  in  qua  jaceant  puncta  omnia  ; 

jam  moles  triplici  dimensione  praedita  erit  nulla  ;   si  lineam  curvam  concipimus  per  omnia 

transeuntem  :    nee  duarum  dimensionum  habebitur  ulla  moles.     Sed  in  eo  itidem  incerta 

asstimatione  indiget  sententia  communis  ob  interstitia  ilia  vacua,  quae  habentur  in  omnibus 

corporibus,  &  scabritiem,  juxta  ea,  quae  diximus,  de  indeterminatione  figurae.     Hie  autem 

itidem  concepta  superficie  extima  terminante  figuram  ipsam,  quae  deinde  de  mole  relata 

ad  superficiem  tradi  solent,  mihi  communia  sunt  cum  aliis  omnibus,  ut  illud  :  posse  eandem 

magnitudine  molem  terminari  superficiebus  admodum  diversis,  &  forma,  &  magnitudine, 

ac  omnium  minimam  esse  sphaericae  figurae   superficiem  respectu  molis  :   in  figuris   autem 

similibus  molem    esse    in    ratione    triplicata  laterum     homologorum,  &    superficiem    in 

duplicata,  ex  quibus  pendent  phaenomena  sane   multa,   atque  ea  inprimis,  quae  pertinent 

ad  resistentiam  tarn  fluidorum,  quam  solidorum. 

Massa  :  quid  inejus  578.  Massa  corporis  est  tota  quantitas  materiae  pertinentis  ad  id  corpus,  quae  quidem 

idea   mcertum     ob        •,•'••  •  •  j    -11     j  A       !_•       • 

matenam  exteram  mmi  erit  i?56  numerus    punctorum  pertmentium  ad  illud  corpus.     At  hie    jam  ontur 

immixtam.  Omnia  indeterminatio  quaedam,  vel  saltern  summa  difficultas  determinandi  massae  ideam,  nee  id 

partibus1   "diversaj  tantum  in  mea,  verum  etiam  in  communi  sententia,  ob  illud  additum  punctorum  pertinentium 

naturae.  ad  muj,  corpus,  quod  heterogeneas  substantias  excludit.     Ea  de  re  sic  ego  quidem  in  Stayanis 

Supplementis  §  10  Lib.  i  :    Nam  admodum  difficile  est  determinare,  quce  sint  illce  substan- 

tice  beterogenees,  quce  non  pertinent  ad  corporis  constitutionem.     Si  materiam  spectemus  ;  ea 

y  mihi,  y  aliis  plurimis  homogenea  est,  y  solis  ejus  diversis  combinationibus  diverse  oriuntur 


A  THEORY  OF  NATURAL  PHILOSOPHY  277 

think  that  an  extended  continuum  can  be  formed  out  of  non-extended  points,  when  indeed 
the  whole  of  the  matter  is  bounded  by  a  continuous  surface.  Besides,  in  those  bodies  that 
fall  within  the  scope  of  our  senses,  the  idea  of  figure,  which  seems  to  be  very  distinct,  is 
however  quite  vague  &  indefinite  ;  &  I  pointed  this  out  fairly  carefully,  when  dealing  some 
time  ago  with  the  figure  of  the  Earth,  in  a  dissertation  inserted  in  the  last  volume  of  the 
Acta  Bononiensia  ;  this  contains  the  synopsis  of  my  work,  Expeditio  Litteraria  -per  Pontificiam 
ditionem,  &  there  the  following  words  occur.  Now,  in  the  first  place,  this  term,  "  the  figure 
of  the  Earth  "  which  seems  to  have  a  certain  definite  &  determinate  meaning,  is  really  very 
vague  y  indefinite.  The  surface  which  bounds  the  seas,  the  lakes,  the  rivers,  the  mountains, 
the  plains  &  the  valleys,  is  really  something  quite  irregular,  at  least  to  us ;  y  moreover  it  is 
also  unstable  ;  for  it  changes  with  the  slightest  motion  of  the  waves  y  the  soil.  But  those  who 
investigate  "  the  figure  of  the  Earth,"  do  not  deal  with  this  figure  of  the  Earth  ;  they  substitute 
for  it  another  figure  which,  although  to  some  extent  regular,  yet  approximates  closely  to  the 
former  true  figure  ;  that  is  to  say,  it  has  the  mountains  y  the  hills  levelled  off,  whilst  the  valleys 
are  fitted  up.  Now  once  more  the  idea  of  this  figure  of  the  Earth  is  vague  y  uncertain.  For, 
just  as  there  are  infinite  classes  of  regular  curves  that  can  be  made  to  pass  through  a  given  number 
of  given  points  ;  so  also  there  are  infinite  classes  of  curved  surfaces  that  can  be  made  to  go  round 
the  Earth  y  circumscribe  it  in  such  a  manner  that  they  touch  all  the  mountains  y  hills,  or  at 
least  certain  given  ones  ;  or,  if  you  like,  some  surface  is  bound  to  pass  through  the  middle  of  the 
hills  y  mountains  in  such  a  way  that  it  cuts  off  as  much  matter  outside  itself,  as  it  encloses 
empty  air-spaces  within  it  y  our  true  surface  of  the  Earth,  to  our  eyes,  so  irregular.  Also, 
there  can  be  an  infinite  number  of  surfaces,  y  these  too  quite  different  frqm  one  another,  which 
satisfy  the  problem  ;  y  all  of  them,  too,  of  such  a  kind  that  they  have  no  manifest  humps,  as 
far  as  can  be  detected  ;  y  this  term  even  contains  no  true  definiteness. 

376.  These  are  my  words  in  that  dissertation  with  regard  to  the  figure  of  the  Earth  ;  The    vagueness   is 
&  they  apply  in  general  to  the  figure  of  any  body  also,  if  considered  according  to  the  usual  jheogreater  m  th'S 
way  with  regard  to  the  continual  extension  of  matter.     For,  the  surfaces  of  nearly  all  bodies 

here  around  us  are  in  every  case  much  rougher  in  comparison  with  their  size  than  is  the 
Earth  in  comparison  with  its  magnitude  ;  &  they  have  many  internal  empty  spaces.  But, 
in  my  Theory,  the  matter  is  much  more  indefinite  &  uncertain  still.  For  there  are  an 
infinite  number  of  continuous  curved  surfaces,  in  which  nevertheless  all  the  points  of  any 
mass  lie  ;  nay,  further,  there  are  an  infinite  number  of  curved  lines  passing  through  all  the 
points.  Therefore  we  can  only  mentally  conceive  a  certain  surface  which  shall  include  all 
the  points  or  exclude  a  few  of  them  which  are  more  remote  by  gathering  the  rest  together  ; 
this  can  be  done  by  a  kind  of  moral  assessment,  but  not  by  an  accurate  geometrical  construc- 
tion. This  surface  gives  the  shape  of  the  body  ;  &  with  that  idea,  all  that  relates  to  the 
different  kinds  of  shapes  of  bodies  is  in  agreement  in  my  Theory  with  the  usual  theory  of  the 
continual  extension  of  matter. 

377.  Volume  depends  upon  shape  ;  &  volume  is  nothing  else  but  the  whole  of  the  space,  Volume  depends  on 
extended  in  length,  breadth  &  depth,  which  is  included  by  the  external  surface.     Further,  thfsPtoo  ^vague  h! 
unless  we  picture  that  surface  which  I  mentioned  as  determining  the  shape,  there  can  be  the  usual  theory,  & 
no  definite  idea  of  volume.     Nay  indeed,  if  we  think  of  the  tortuous  surface  in  which  all  ™"^eh  mor' 

the  points  lie,  we  shall  never  have  a  volume  possessed  of  a  third  dimension  ;  whilst  if  we  think 
of  a  curved  line  passing  through  all  the  points,  no  volume  will  be  obtained  that  has  even 
two  dimensions.  But  in  that  the  usual  idea  is  also  wanting,  as  regards  indefinite  assessment, 
owing  to  those  empty  interstices  that  are  present  in  all  bodies,  &  the  roughness,  as  we  have 
said,  which  arises  from  the  indeterminateness  of  figure.  Here  again,  if  an  outside  surface  is 
conceived  as  bounding  the  figure,  all  those  things  that  are  usually  enunciated  about  volume 
in  relation  to  figure  agree  in  my  theory  with  those  of  all  others ;  for  instance,  that  the 
same  volume  as  regards  magnitude  can  be  bounded  by  surfaces  that  are  quite  different, 
both  in  shape  &  size,  &  that  the  least  surface  of  all  having  the  same  volume  is  that  of  a 
sphere.  Also  that,  in  similar  figures,  the  volumes  are  in  the  triplicate  ratio  of  homologous 
sides,  the  surfaces  in  the  duplicate  ratio  ;  &  upon  these  depend  a  truly  great  number  of 
phenomena,  &  especially  those  which  are  connected  with  the  resistance  both  of  fluids  & 
of  solids. 

378.  The  mass  of  a  body  is  the  total  quantity  of  matter  pertaining  to  that  body  ;   &  Mass  ;  what  there 

J'n-ii  i  •     •  -11  ,  •  i  <•  r  r.       is  m  the  idea  of  it 

in  my  1  heory  this  is  precisely  the  same  thing  as  the  number  of  points  that  go  to  form  the  that    is    indefinite 
body.     Here  now  we  have  a  certain  indefiniteness,  or  at   least  the  greatest  difficulty,  in  owin_8    to   <?utside 

e       ' .  ,,,..,  ,  ..'  .          fl  .     ,       J  matter    mingling 

forming  a  definite  idea  of  mass ;  &  that,  not  only  in  my  theory,  but  in  the  usual  theory  as  with  it.    All  bodies 
well,  on  account  of  the  addition  of  the  words  points  that  go  to  form  the  body  ;   this  excludes  are   composed    of 

,  i  •  •         .     i  T  i        i        <•   11       •  i      •        i.      parts    of    different 

heterogeneous  substances.     On  this  point  indeed,  I  made  the  following  remarks  m  the  natures. 
Supplements  to  Stay's  Philosophy  : — For  it   is  very  difficult  to  define  what  those  heterogeneous 
substances  may  be,  if  they  do  not  pertain  to  the  constitution  of  a  body.     If  we  consider  matter, 
it  is  in  my  opinion,  y  in  that  of  very  many  others,  homogeneous  ;    y  the  different  species  of 


2  78  PHILOSOPHISE  NATURALIS  THEORIA 

corporum  species.  Quare  ab  ipsa  materia  non  potest  desumi  discrimen  illud  inter  substantias 
pertinentes,  y  non  pertinentes.  Si  autem  y  diver sam  [174]  illam  combinationem  spectemus, 
corpora  omnia,  quce  observamus,  mixta  sunt  ex  substantiis  adm.od.um  dissimilibus,  quce  tamen 
omnes  ad  ejus  corporis  constitutionem  pertinent.  Id  in  animalium  corporibus,  in  plant-is,  in 
marmoribus  plerisque,  oculis  etiam  patet,  in  omnibus  autem  corporibus  Cbemia  docet,  quce 
mixtionem  illam  dissohit. 

piures    substantial  370,.  Ex  alia  parte  tenuissima  cetherea  materia,  quce  omnino  est  aliqua  nostro  aere  varior, 

substantiam"  "cor-  a^  constitutionem  masses  nequaquam  pertinere  censetur,  ut  nee  pro  corporibus  plerisque  aer, 

pom.  qui  meatibus  internis  interjacet.     Sic  aer  inclusus  spongice  meatibus,  ad  ipsius  constitutionem 

nequaquam  censetur  pertinere.     Idem  autem  ad  multorum  corporum  constitutionem  pertinet : 

saltern  ad  fixam  naturam  redactus,  ut   Halesius  demonstravit,  piures  y  animalis  regni,   & 

vegetabilis  substantias  magna  sui  parte  constare  aere  fixitatem  adepto.     Rursus  substanties 

volatiles,  aere  ipso  tenuiores  multo,  quce  in  corporum  dissolutions  chemica  in  balitus,  y  fumos 

abeunt,  y  piures  fortasse,  quas  nos  nullo  sensu  percipimus,  ad  ipsa  corpora  pertinebant. 

Nee  exciudi  omnia  380.  Nee  illud  assumi  potest,  quidquid  solidum,  y  fixum  est,  id  tantummodo  pertinere 

fnci<u^inposse°Iqu1^  a^  corp°r^s  massam  ;    quis  enim  a  corporis  humani  massa  sanguinem  omnem,  y  tot  lymphas 

translate     corpore  excludat,  a   lignis  resectis  succos  nondum  concretos  ?     Prceterquam  quod  masses  idea   non  ad 

untur!PS°  transfe  "  solida  solum  corpora  pertinet,  sed  etiam  ad  fluida,  in  quibus  ipsis  alia  tenuiora  aliorum  densiorum 

meatibus  interjacent.     Nee  vero  did  potest,  pertinere  ad  corporis  constitutionem,  quidquid 

materice  translato  corpore,  simul  cum  ipso  transfertur  ;  nam  aer,  qui  intra  spongiam  est,  partim 

mutatur  in  ea  translatione,  is  nimirum,  qui  orificio  est  propior,   partim  manet,  qui  nimirum 

intimior,  y  qui  aliquandiu  manet,  mutatur  deinde. 

Hinc    indistinctam  381.  Hcec,  &  alia  mihi  diligentius  perpendenti,  illud  videtur  demum,  ideam  masses  non 

Quid  'dwisitaseai&  esse  Accurate  determinatam,  y  distinctam,  sed  admodum  vagam,  arbitrariam,  y  confusam. 
raritas  ;  utranque  Erit  massa  materia  omnis  ad  corporis  constitutionem  pertinens ;  sed  a  crassa  quadam,  y 
poss"'inhact!ieoria  arbitraria  cestimatione  pendebit  illud,  quod  est  pertinere  ad  ipsam  ejus  constitutionem.  Hsec 
in  quacunque  ego  ibi  :  turn  ad  molem  transeo,  de  cujus  indeterminatione  jam  hie  superius  egimus,  ac 
deinde  ad  densitatem,  quae  est  relatio  massae,  ad  molem,  eo  major,  quo  pari  mole  est  major 
massa,  vel  quo  pari  massa  est  minor  moles.  Hinc  mensura  densitatis  est  massa  divisa  per 
molem  ;  &  qusecunque  vulgo  proferuntur  de  comparationibus  inter  massam,  molem,  & 
densitatem,  haec  omnia  &  mihi  communia  sunt.  Massa  est  ut  factum  ex  mole  &  densitate  ; 
moles  ut  massa  divisa  per  densitatem.  Raritas  autem  etiam  mihi,  ut  &  aliis,  est  densitatis 
inversa,  ut  nimirum  idem  sit  dicere,  corpus  aliquod  esse  decuplo  minus  densum  alio  aliquo 
corpore,  ac  dicere,  esse  decuplo  magis  rarum.  Verum  quod  ad  densitatem  &  raritatem 
pertinet,  in  eo  ego  quidem  a  communi  sententia  discrepo,  uti  exposui  num.  89,  quod  [175] 
ego  nullum  habeo  limitem  densitatis  &  raritatis,  nee  maximum,  nee  minimum  ;  dum  illi 
minimam  debent  aliquam  raritatem  agnoscere,  &  maximam  densitatem  possibilem,  utut 
finitam,  quse  illis  idcirco  per  saltum  quendam  necessario  abrumpitur  ;  licet  nullam  agnoscant 
raritatem  maximam,  &  minimam  densitatem.  Mihi  enim  materiae  puncta  possunt  & 
augere  distantias  a  se  invicem,  &  imminuere  in  quacunque  ratione  ;  cum  data  linea  quavis, 
possit  ex  ipsis  Euclideis  elementis  inveniri  semper  alia,  quae  ad  ipsam  habeat  rationem 
quancunque  utcunque  magnam,  vel  parvam  ;  adeoque  potest,  stante  eadem  massa,  augeri 
moles,  &  minui  in  quacunque  ratione  data  ;  at  illis  potest  quidem  quaevis  massa  dividi  in 
quenvis  numerum  particularum,  quae  dispersae  per  molem  utcunque  magnam  augeant 
raritatem,  &  minuant  densitatem  in  immensum  ;  sed  ubi  massa  omnis  ita  ad  contactus 
immediatos  devenit,  ut  nihil  jam  supersit  vacui  spatii ;  turn  vero  densitas  est  maxima,  & 
raritas  minima  omnium,  quae  haberi  possint,  &  tamen  finita  est,  cum  mensura  prioris 
habeatur,  massa  finita  per  finitam  molem  divisa,  &  mensura  posterioris,  divisa  mole  per 
massam. 


inertia    massarum  382.  Inertia  corporum  oritur  ab  inertia  punctorum,  &  a  viribus  mutuis ;    nam  illud 

punctorum'^ipsl  demonstravimus  num.  260,  si  puncta  quaecunque  vel  quiescant,  vel  moveantur  directionibus, 
respondens  conser.  &  celeritatibus  quibuscunque,  sed  singula  aequabili  motu  ;  centrum  commune  gravitas 
gra^tafe^&^dla  ve^  quiesccre,  vel  moveri  uniformiter  in  directum,  ac  vires  mutuas  quascunque  inter  eadem 
massae  unitae  in  puncta  nihil  turbare  statum  centri  communis  gravitatis  sive  quiescendi,  sive  movendi 
Ipsa  uniformiter  in  directum.  Porro  vis  inertise  in  eo  ipso  est  sita  :  nam  vis  inertiae  est 


A  THEORY  OF  NATURAL  PHILOSOPHY  279 

bodies  arise  solely  from  different  combinations  of  it.  Hence  it  is  impossible  to  take  away  from 
matter  the  distinction  between  substances  that  pertain  to  a  body  W  those  that  do  not.  Again 
if  we  consider  the  difference  of  combination,  all  bodies  that  come  under  our  observation  are  mixtures 
of  substances  that  are  perfectly  unlike  one  another  ;  &  yet  all  of  them  are  necessary  to  the 
constitution  of  the  body.  We  have  ocular  evidence  of  this  in  the  bodies  of  animals,  in  plants, 
in  most  of  the  marbles  ;  moreover,  in  all  bodies,  chemistry  teaches  us  how  to  separate  that  mixture. 

379.  In  another  respect,  that  very  tenuous  ethereal  matter,  which  is  something  indeed  A  !arge  number  of 
much  less  dense  than  our  air,  can  in  no  sense  be  considered  to  be  a  constituent  part  of  a  body  ;  pertafn'To  the  sub- 
nor  indeed,  in  the  case  of  most  bodies  can  the  air  which  is  contained  in  its  internal  parts.     Thus  stance  of  a  body. 
the  air  that  is  included  in  the  passages  of  a  sponge  can  in  no  sense  be  considered  as  being  necessary 

to  the  constitution  of  the  sponge.  But  the  same  thing  pertains  to  the  constitution  of  many  bodies  ; 
at  least,  when  reduced  to  a  fixed  nature.  For  Hales  has  proved  that  many  substances  of  the 
animal  &  vegetable  kingdoms  in  a  great  part  consist  of  air  that  has  attained  fixity.  Again, 
volatile  substances,  more  tenuous  than  air  itself,  which  go  off  in  vapours  W  fumes  from  bodies 
chemically  decomposed,  y  perchance  many  which  are  not  perceived  by  any  of  our  senses,  all 
pertained  to  these  bodies. 

380.  Nor  can  it  be  assumed  that  only  something  solid  &  fixed  can  pertain  to  the  mass  of  a  Nor  can  ail  fluids  be 
body.     For  who  would  exclude  from  the  mass  of  the  human  body  the  whole  of  the  blood,  y  the  ^those  'things0]*; 
large  number  of  watery  fluids,  or  from  chips  of  wood  the  juices  that  are  not  yet  congealed?  included,  which 
Especially  as  the  idea  of  mass  pertains  not  only  to  solids  alone  but  also  to  fluids  ;    fc?  in  these  moved^r 

some  of  the  more  tenuous  parts  lie  in  the  interstices  of  the  more  dense.     On  the  other  hand,  it  with  the  body. 

cannot  be  said  that  any  kind  of  matter,  which  when  the  body  is  moved  is  carried  with  it,  pertains 

of  necessity  to  the  constitution  of  the  body.     For  the  air  which  is  within  a  sponge  is  partly 

moved  by  that  translation,  that  is  to  say  that  part  which  is  near  an  orifice  ;   whilst  it  partly 

remains,  that  is  to  say  that  part  which  is  more  internal,  W  remains  for  some  length  of  time, 

£5"  then  is  moved. 

381.  After  carefully  considering  these  &  other  matters,  I  have  come  to  the  conclusion  Hence  also  the  idea 
that  the  idea  of  mass  is  not  strictly  definite  &  distinct,  but  that  it  is  quite  vague,  arbitrary  nitemasxhe5  nature 

confused.     Mass  will  be  the  whole  of  the  matter  pertaining  to  the  constitution  of  a  body  ;  of  density,  &  rarity; 


but  what  part  of  it  actually  does  pertain  to  its  constitution,  will  depend  upon  a  non-scientific  thi^Theorv116™  be 
y  arbitrary  assessment.  These  are  my  words  ;  &  after  that  I  pass  on  to  volume,  the  increased  or  dimin- 
indefiniteness  of  which  I  have  already  dealt  with  above,  &  after  that  to  density,  which  is  lshed  to  anyextent- 
the  relation  of  mass  to  volume  ;  being  so  much  the  greater  as  in  equal  volume  there  is  so 
much  the  greater  mass,  or  according  as  for  equal  mass  there  is  so  much  the  less  volume. 
Hence  the  measure  of  density  is  mass  divided  by  volume  ;  &  whatever  is  usually  said  about 
comparisons  between  mass,  volume  &  density,  everything  is  in  agreement  with  what  I  say. 
Mass  is,  so  to  speak,  the  product  of  volume  &  density  ;  &  volume  is  mass  divided  by  density. 
Rarity,  with  me,  as  well  as  with  others,  is  the  inverse  of  density  ;  thus  it  is  the  same  thing 
to  say  that  one  body  is  ten  times  less  dense  than  another  body  as  to  say  that  it  is  ten  times 
more  rare.  But  as  regards  the  properties  of  rarity  &  density,  here  I  indeed  differ  from  the 
usual  opinion.  For,  as  I  showed  in  Art.  89,  I  have  no  limiting  value  for  either  density 
or  rarity,  no  maximum,  no  minimum  ;  whereas  others  must  admit  a  minimum  rarity,  or 
a  maximum  density,  as  being  possible  ;  &,  since  this  must  be  something  finite,  it  must 
of  necessity  involve  a  sudden  break  in  continuity  ;  although  they  may  not  admit  any  maximum 
rarity  or  minimum  density.  For  with  me  the  points  of  matter  can  both  increase  &  diminish 
their  distances  from  one  another  in  any  ratio  whatever  ;  since,  given  any  line,  it  is  possible, 
by  the  elementary  principles  of  Euclid,  to  find  another  in  every  case,  which  shall  bear  to 
the  given  line  any  ratio  however  great  or  small.  Thus,  it  is  possible  that,  whilst  the  mass 
remains  the  same,  the  volume  should  be  increased  or  diminished  in  any  ratio  whatever. 
But,  in  the  case  of  other  theories,  it  is  indeed  possible  that  a  mass  can  be  divided  into  any 
number  of  particles,  which  when  dispersed  throughout  a  volume  of  any  size  however  great 
will  increase  the  rarity  or  diminish  the  density  to  an  indefinitely  great  extent  ;  but  when 
the  whole  mass  has  been  brought  into  a  state  of  immediate  contact  of  its  particles  in  such 
a  manner  that  there  no  longer  exist  any  empty  spaces  between  these  particles,  then  indeed 
there  is  a  maximum  density  or  a  minimum  rarity  obtainable,  although  this  is  finite  ;  for, 
a  measure  of  the  first  may  be  obtained  by  dividing  a  finite  mass  by  a  finite  volume,  or  of 
the  second  by  dividing  volume  by  mass.  The  iner^^  oj  a 

382.  The  inertia  of  bodies  arises  from  the  inertia  of  their  points  &  their  mutual  forces,  mass  arises  from 
For,  in  Art.  260,  it  was  proved  that,  if  any  points  are  either  at  rest,  or  moving  in  any  ^ntsTThe  corre* 
directions  with  any  velocities,  so  long  as  each  of  the  motions  is  uniform,  then  the  centre  spending  conserva- 
of  gravity  of  the  set  will  either  be  at  rest  or  move  uniformly  in  a  straight  line  ;  &  that,  thTcentre^Tgrav- 
whatever  mutual  forces  there  may  be  between  the  points,  these  will  in  no  way  affect  the  tty  ;  the  idea  of 
state  of  the  common  centre  of  gravity,  whether  it  is  at  rest  or  whether  it  is  moving  uniformly 
in  a  straight  line.  Further  the  force  of  inertia  is  involved  in  this  ;  for  the  force  of  inertia  of  gravity. 


280  PHILOSOPHIC  NATURALIS  THEORIA 

determinatio  perseverandi  in  eodem  statu  quiescendi,  vel  movendi  uniformiter  in  directum  : 
nisi  externa  vis  cogat  statum  suum  mutare  :  &  cum  ex  mea  Theoria  demonstretur,  earn 
proprietatem  debere  habere  centrum  gravitatis  massae  cujuscunque  compositae  punctis 
quotcunque,  &  utcunque  dispositis  ;  patet,  earn  deduci  pro  corporibus  omnibus  :  &  hie 
illud  etiam  intelligitur,  cur  concipiantur  corpora  tanquam  collecta,  &  compenetrata  in 
ipso  gravitatis  centre. 

Mobjiitas  :  quies  383.  Mobilitas  recenseri  solet  inter  generales  corporum  proprietates,  quse  quidem 

haberi.tolcxclnBa  sPonte  consequitur  vel  ex  ipsa  curva  virium  :  cum  enim  ipsa  exprimat  suarum  ordinatarum 
prorsus  quiete  a  ope  determinationcs  ad  accessum,  vel  recessum,  requirit  necessario  mobilitatem,  sive 
possibilitatem  motuum,  sine  quibus  accessus,  &  recessus  ipsi  haberi  utique  non  possunt. 
Aliqui  &  quiescibilitatem  adscribunt  corporibus  :  at  ego  quidem  corporum  quietem  saltern 
in  Natura,  uti  constituta  est,  haberi  non  posse  arbitror,  uti  exposui  num.  86.  Earn  excludi 
oportere  censeo  etiam  infinitae  improbabilitatis  argumento,  quo  sum  usus  in  ea  dissertatione 
De  Spatio,  y  Tempore,  quam  toties  jam  nominavi,  &  in  Supplementis  hie  proferam  §  I, 
ubi  [176]  evinco,  casum,  quo  punctum  aliquod  materiae  occupet  quovis  momento  temporis 
punctum  spatii,  quod  alio  quopiam  quocunque  occuparit  vel  ipsum,  vel  aliud  punctum 
quodcunque,  esse  infinities  improbabilem,  considerate  nimirum  numero  punctorum  material 
finite,  numero  momentorum  possibilium  infinite  ejus  generis,  cujus  sunt  infinita  puncta 
in  una  recta,  qui  numerus  momentorum  bis  sumitur,  semel  cum  consideratur  puncti  dati 
materise  cujuscunque  momentum  quodvis,  &  iterum  cum  consideratur  momentum  quodvis, 
quo  aliud  quodpiam  materiae  punctum  alicubi  fuerit,  ac  iis  collatis  cum  numero  punctorum 
spatii  habentis  extensionem  in  longum,  latum,  &  profundum,  qui  idcirco  debet  esse  infinitus 
ordinis  tertii  respectu  superiorum.  Deinde  ab  omnium  corporum  motu  circa  centrum 
commune  gravitatis,  vel  quiescens,  vel  uniformiter  progrediens  in  recta  linea,  quies  actualis 
itidem  a  Natura  excluditur. 


tate  3^4-  Verum    ipsam    quietem  excludit  alia  mihi  proprietas,  quam    omnibus    itidem 

omnium  motuum  :  materiae  punctis,  &  omnium  corporum  centris  gravitatis  communem  censeo,   nimirum 
probiema   generate  continuitas  motuum,  de  qua  egi  num.  883,  &  alibi.     Quodvis  materiae  punctum  seclusis 

eo   pertmens.  .,          V1  .  '      .     ^          °     .  .      ,.,  .   .  ,r      .. 

motibus  libens,  qui  onuntur  ab  imperio  liberorum  spmtuum,  debet  describere  curvam 
quandam  lineam  continuam,  cujus  determinatio  reducitur  ad  hujusmodi  probiema  generale  : 
Dato  numero  punctorum  materiae,  ac  pro  singulis  dato  puncto  loci,  quod  occupent  dato 
quopiam  momento  temporis,  ac  data  directione,  &  velocitate  motus  initialis,  si  turn  primo 
projiciuntur,  vel  tangentialis,  si  jam  ante  fuerunt  in  motu,  ac  data  lege  virium  expressa 
per  curvam  aliquam  continuam,  cujusmodi  est  curva  figurae  I,  quas  meam  hanc  Theoriam 
continet,  invenire  singulorum  punctorum  trajectorias,  lineas  nimirum,  per  quas  ea  moventur 
singula.  Id  probiema  mechanicum  quam  sublime  sit,  quam  omnem  humanae  mentis 
excedat  vim,  ille  satis  intelliget,  qui  in  Mechanica  versatus  non  nihil  noverit,  trium  etiam 
corporum  motus,  admodum  simplici  etiam  vi  praeditorum,  nondum  esse  generaliter  definitos, 
uti  monui  num.  204,  &  consideret  immensum  punctorum  numerum,  ac  altissimam  curvae 
virium  tantis  flexibus  circa  axem  circumvolutae  elevationem. 

Quid     curvae    de-  385.  Sed  licet  ejusmodi  probiema  vires  omnes  humanae  mentis  excedat  ;  adhuc  tamen 

no^iSbean^Pro!  unusquisque   Geometra   videbit   facile,   probiema   esse  prorsus   determinatum,   &   curvas 

biema    inversum  ejusmodi  fore  omnes  continuas  sine  ullo  saltu,  si  in  lege  virium  nullus  sit  saltus.      Quin 

s^ptis1tenvpusScufo  immo  &  ^u^  arbitror,  in  ejusmodi  curvis  nee  ullas  usquam  cuspides  occurrere  ;  nam  nodos 

utcunque  parvo.      nullos  esse  consequitur  ex  eo,  quod  nullum  materiae  punctum  redeat  ad  idem  punctum 

spatii,  in  quo  ipsum  aliquando  fuerit,  adeoque  nullus  habeatur  regressus,  qui  tamen  ad 

nodum  est  necessarius.     Hujusmodi  curvae  necessariae  essent  omnes,  &  mens,  [177]  <!U3e 

tantum  haberet  vim,  quanta  requiritur  ad  ejusmodi  problemata  rite  tranctanda,  &  intimius 

perspiciendas  solutiones  (quas  quidem  mens  posset  etiam  finita  esse,  si  finitus  sit  punctorum 

numerus,  &  per  finitam  expressionem  sit  data  notio  curvse  exprimentis  legem  virium)  posset 

ex  arcu  continue  descripto  tempore  etiam  utcunque  exiguo  a  punctis  materiae  omnibus 

derivare  ipsam  virium  legem,  cum  quidam  finiti  tantummodo  positionum  numeri  fmitos 

determinare  possint  numeros  punctorum  curvae  virium,  &  arcus  continuus  legem  ipsam 

continuam  :   &  fortasse  solae  etiam  positiones  omnium  punctorum  cum  dato  arcu  continuo 

percurso  ab  unico  etiam  puncto  motu  continuo,  exiguo  etiam  aliquo  tempusculo  ad  rem 

prsestandam  satis  essent.     Cognita  autem  lege  virium  &  positione,  ac  velocitate,  &  directione 

punctorum  omnium  dato  tempore,  posset  ejusmodi  mens  praevidere  omnes  futures  neces- 

saries motus,  ac  status,  &  omnia  Naturae  phenomena  necessaria,  ab  iis  utique  pendentia, 

atque  prsedicere  :   &  ex  unico  arcu  descripto  a  quovis  puncto,  tempore  continuo  utcunque 


A  THEORY  OF  NATURAL  PHILOSOPHY  281 

consists  in  a  propensity  for  staying  in  a  state  of  rest  or  of  maintaining  a  uniform  state  of  motion 
in  a  straight  line,  unless  some  external  force  compels  a  change  of  this  state.  Now,  since  by 
my  Theory  it  is  proved  that  the  centre  of  gravity  of  any  mass,  composed  of  any  number  of 
points  disposed  in  any  manner  whatever,  is  bound  to  have  this  property,  it  is  clear  that 
the  same  property  can  be  deduced  for  all  bodies  ;  £  by  this  it  can  also  be  understood  why 
bodies  can  be  conceived  to  be  collected  &  condensed  at  their  centres  of  gravity. 

383.  Mobility  is  usually  considered  as  one.  of  the  general  properties  of  bodies;    &  Mobility;    quiesci- 
indeed  it  follows  immediately  from  the  curve  of  forces.     For,  since  this  curve,  by  means  of  ^ty  ^^P0^^ 
its  ordinates,  represents  the  propensity  to  approach  or  recede,  it  necessarily  requires  mobility,  such   thing    in 
or  the  possibility  of  motion,  without  which  approach  or  recession  can  certainly  not  be  ^ure  as  absolute 
obtained.     Now  there  are  some,  who  ascribe  quiescibility  to  bodies  ;    but  I  consider  that 

absolute  rest,  at  any  rate  in  Nature  as  it  is  at  present  constituted,  is  impossible,  as  I  explained 
in  Art.  86.  I  think  also  that  it  must  be  excluded  by  the  argument  of  infinite  improbability, 
which  I  used  in  the  dissertation  De  Spatio,  &  Tempore,  which  I  have  mentioned  so  many 
times  already,  &  which  I  quote  in  this  work  as  Supplement,  §  I  ;  in  it  I  prove  that  the 
case  in  which  any  point  of  matter  occupies  at  any  instant  of  time  a  point  of  space,  which 
at  any  other  instant  whatever  either  it  or  any  other  point  whatever  would  occupy,  is  infinitely 
improbable ;  this,  by  considering  the  finite  number  of  points  of  matter,  &  the  infinite 
number  of  instants  of  time  possible,  of  that  class  for  which  there  are  an  infinite  number  of 
points  in  the  same  straight  line  ;  this  number  of  instants  is  considered  twice,  once  when 
any  instant  for  any  given  point  of  matter  is  considered,  &  again  when  any  instant  is  considered 
in  which  any  other  point  of  matter  was  somewhere  else  ;  when  these  are  compared  with 
the  number  of  points  of  a  space  which  has  extension  in  length,  breadth  &.  depth,  the  latter 
must  be  infinite  of  the  third  order  with  respect  to  those  mentioned  above.  Finally,  by 
the  motion  of  all  bodies  about  a  common  centre  of  gravity,  whether  this  is  at  rest  or  travelling 
uniformly  in  a  straight  line,  absolute  rest  is  excluded  from  Nature. 

384.  In  my  opinion  also,  there  is  another  property  that  excludes  absolute  rest,  one  Absolute   rest  is 
which  I  consider  is  common  also  to  all  points  of  matter  &  to  the  centres  of  gravity  of  all  the^continuity  'of 
bodies ;    namely,  continuity  of  motion,  with  which  I  dealt  in  Art.  88  &  elsewhere.     Any  all  motions ;  gene- 
point  of  matter,  setting  aside  free  motions  that  arise  from  the  action  of  arbitrary  will,  respect°to^t1  Wth 
must  describe  some  continuous  curved  line,  the  determination  of  which  can  be  reduced  to 

the  following  general  problem.  Given  a  number  of  points  of  matter,  &  given,  for  each  of 
them,  the  point  of  space  that  it  occupies  at  any  given  instant  of  time  ;  also  given  the  direction 
&  velocity  of  the  initial  motion  if  they  were  projected,  or  the  tangential  velocity  if  they 
are  already  in  motion  ;  &  given  the  law  of  forces  expressed  by  some  continuous  curve, 
such  as  that  of  Fig.  i,  which  contains  this  Theory  of  mine ;  it  is  required  to  find  the  path 
of  each  of  the  points,  that  is  to  say,  the  line  along  which  each  of  them  moves.  How  difficult 
this  mechanical  problem  may  become,  how  it  may  surpass  all  powers  of  the  human  mind, 
can  be  easily  enough  understood  by  anyone  who  is  versed  in  Mechanics  &  is  not  quite  unaware 
that  the  motions  of  even  three  bodies  only,  &  these  possessed  of  a  perfectly  simple  law  of 
force,  have  not  yet  been  completely  determined  in  general,  &  then  will  consider  an  immense 
number  of  points,  &  the  extremely  high  degree  of  a  curve  of  forces  twisting  round  the  axis 
with  so  many  sinuosities. 

385.  Now,  although  a  problem  of  such  a  kind  surpasses  all  the  powers  of  the  human  What  does   not 
intellect,  yet  any  geometer  can  easily  see  thus  far,  that  the  problem  is  determinate,  &  that  described  byThe 
such  curves  will  all  be  continuous  without  any  break  in  them,  so  long  as  there  is  no  discon-  points.  The  inverse 
tinuity  in  the  law  of  forces,     Indeed,  I  think  that,  in  such  curves,  there  never  occur  curves"1'  Described 
any  cusps ;   for,  it  follows  that  there  are  no  nodes,  from  the  fact  that  no  point  of  matter  in  any  interval  of 
returns  to  the  same  point  of  space  that  it  occupied  at  any  time  ;   &  thus  there  is  none  of 

that  regression  which  is  necessary  for  a  node.  All  the  curves  must  be  of  this  kind  ;  &  a  mind 
which  had  the  powers  requisite  to  deal  with  such  a  problem  in  a  proper  manner  &  was 
brilliant  enough  to  perceive  the  solutions  of  it  (&  such  a  mind  might  even  be  finite,  provided 
the  number  of  points  were  finite,  &  the  notion  of  the  curve  representing  the  law  of  forces 
were  given  by  a  finite  representation),  such  a  mind,  I  say,  could,  from  a  continuous  arc 
described  in  an  interval  of  time,  no  matter  how  small,  by  all  points  of  matter,  derive  the  law 
of  forces  itself ;  for,  any  merely  finite  number  of  positions  can  determine  a  finite  number 
of  points  on  the  curve  of  forces,  &  a  continuous  arc  the  continuous  law.  Perhaps  even 
the  positions  of  all  the  points,  together  with  a  given  continuous  arc  traversed  with  continuous 
motion  by  but  a  single  one  of  them,  &  that  too  in  an  interval  of  time  no  matter  how  small, 
would  be  sufficient  to  obtain  a  solution  of  the  problem.  Now,  if  the  law  of  forces  were 
known,  &  the  position,  velocity  &  direction  of  all  the  points  at  any  given  instant,  it  would 
be  possible  for  a  mind  of  this  type  to  foresee  all  the  necessary  subsequent  motions  &  states, 
&  to  predict  all  the  phenomena  that  necessarily  followed  from  them.  It  would  be  possible 
from  a  single  arc  described  by  any  point  in  an  interval  of  continuous  time,  no  matter  how 


282 


PHILOSOPHIC  NATURALIS  THEORIA 


parvo,  quern  aliqua  mens  satis  comprehenderet,  eadem  determinare  posset  reliquum  omnem 
ejusdem  continuae  curvae  tractum  utraque  e  parte  in  infinitum  productum. 
Cur    ab  humana  386.  Nos  eo  aspirare  non  possumus,  turn  ob  nostrae  mentis  imbecillatatem,  turn  quia 

mente     solvi     non    •  .  .  r  •         i  / 

possit.     Quid  offi-  ignoramus  numerum,  &  positionem,  ac  motum  punctorum  smgulorum  (nam  nee  motus 
cjat  ei  determina-  absolutos    intuemur,   sed   respectivos   tantummodo-  respectu  Telluris,   vel  ad    summum 

tioni  hbertas  :  Har-  •         i  .  .r         .  •      r  •        \ 

monia;  pra:stabiiit<e  respectu  systematis  planetani,  vel  systematis  rixarum  omnium)   turn  etiam,  quia  curvas 
impugnatio.  illas  turbant  liberi  motus,  quos  producunt  spirituales  substantiae.     Harmonia  praestabilita 

Leibnitianorum  ejusmodi  perturbationem  tollit  omnem,  saltern  respectu  animae  nostrae, 
cum  omne  immediatum  commercium  demat  inter  corpus,  &  animam  ;  &  id,  quod  tantopere 
improbatum  est  in  Theoria  Cartesiana,  quse  bruta  redegerat  ad  automata,  ad  homines 
etiam  ipsos  transfer!,  quorum  motus  a  machina  provenire  omnes,  &  necessaries  esse  in  ea 
Theoria,  facile  constat  :  &  quidem  idcirco  etiam  mihi  Theoria  displicet  plurimum,  quam 
praeterea  si  admitterem,  nullam  sane  viderem,  ne  tenuissimam  quidem  rationem,  quae 
mihi  suadere  posset,  praeter  animam  meam,  cujus  ideae  per  se,  &  sine  ullo  immediate  nexu 
cum  corpore  evolvantur,  me  habere  aliquod  corpus,  quod  motus  ullos  habeat,  &  multo 
minus,  ejusmodi  motus  esse  conformes  iis  ideis,  aut  ullos  alios  esse  homines,  ullam  naturam 
corpoream  extra  me  ;  ad  quae  omnia,  &  multo  adhuc  pejora,  mentem  suis  omnia  momentis 
librantem  deducat  omnino  oportet  ejusmodi  sententia,  quam  promoveri  passim,  &  vero 
etiam  recipi,  ac  usque  adeo  gliscere,  quin  &  omnino  tolerari,  semper  miratus  sum. 


saltu. 


Motus  Hberos  om-  387.  Censeo  ieitur,  &  id  intima  vi,  qua  anima  suarum  [178!  idearum  naturam,  & 

nino    ab   anima  •  •    •  i  •  vi  •       i 

progigni,   sed   non  proprietates  quasdam,  atque  ongmem  novit,  constare  arbitror,  motus  liberos  corpons  ab 
imprimi,  nisi  aequa-  anima  provenire  :   ac  quemadmodum  virium  lex  necessaria,  in  ipsa  fortasse  materiae  natura 

liter  in   partes*  •  i*  «  t  •  •  11  ^          •      •  i 

oppositas,  &  sine  slta>  ejusmodi  est ;  ut  juxta  earn  bma  materiae  puncta  debeant  ad  se  invicem  accedere, 
vel  a  se  invicem  recedere,  determinata  &  quantitate  motus,  &  directione  per  distantias ; 
ita  esse  alias  leges  virium  liberas  animae,  secundum  quas  debeant  quaedam  puncta  materias 
habentia  ejusmodi  dispositionem,  quae  ad  vivum,  &  sanum  corpus  organicum  requiritur, 
ad  ipsius  animae  nutum  moveri ;  sed  hujusmodi  leges  itidem  censeo  requirere  illud,  ut 
nulli  materice  puncto  imprimatur  motus  aliquis,  nisi  alicui  alteri  imprimatur  alius  contrarius, 
&  aequalis,  quod  constat  ex  ipso  nisu,  quern  semper  exercemus  in  partes  contrarias,  juxta  ea, 
quae  diximus  num.  74  :  ac  itidem  arbitror,  &  id  ipsum  diligent!  observatione,  &  reflexione 
facile  colligitur,  ejusmodi  quoque  motus  imprimi  non  posse,  nisi  servata  lege  continuitatis 
sine  ullo  saltu,  quod  si  ab  omnibus  spiritibus  observari  debeat ;  discedent  quidem  veri 
motus  a  curvis  illis  necessariis,  &  a  libera  voluntatis  determinatione  pendebunt  curvae 
descriptae  ;  sed  motuum  continuitas  nequaquam  turbabitur. 


Conclusion^  de-  388.  Porro  inde  constat,  cur  in  motibus  nullum  uspiam  deprehendamus  saltum,  cur 
exdu1ioPquieti™Um  nullum  materiae  punctum  ab  uno  loci  puncto  abeat  ad  aliud  punctum  loci  sine  transitu 
per  intermedia,  cur  nulla  densitas  mutetur  per  saltum,  cur  &  motus  reflexi,  £  refracti  fiant 
per  curvaturam  continuam,  ac  alia  ejusmodi,  quas  hue  pertinent.  Verum  simul  patebit 
&  illud,  in  cujus  gratiam  haec  congessimus,  nullam  fore  absolutam  quietam,  in  qua  nimirum 
continuatus  ille  curvae  descriptae  ductus  abrumpatur  ea  continuitate  laesa  nihilo  minus, 
quam  laederetur,  si  curva  continua  desineret  alicubi  in  rectam. 


Aequalitas  action- 
is,  &  reactionis,  & 
ejus  consectaria. 


389.  Jam  vero  ad  actionis,  &  reactionis  asqualitatem  gradu  facto,  earn  abunde  deduximus 
a  num.  265.  pro  binis  quibusque  corporibus  ex  actione,  &  reactione  aequalibus  in  punctis 
quibuscunque.  Cum  nimirum  mutuae  vires  nihil  turbent  statum  centri  gravitatis  com- 
munis,  &  centra  gravitatis  binarum  massarum  debeant  cum  ipso  communi  centre  jacere 
in  directum  ad  distantias  hinc,  &  inde  reciproce  proportionales  ipsis  massis,  ut  ibidem 
demonstravimus ;  consequitur  illud,  motus  quoscunque,  quos  ex  mutua  actione  habebunt 
binarum  massarum  centra  gravitatis,  debere  fieri  in  lineis  similibus,  &  proportionalibus 
distantiae  singularum  ab  ipso  gravitatis  centro  communi,  adeoque  reciproce  proportionalibus 
ipsis  massis ;  &  quod  inde  consequitur,  summam  motuum  computatorum  secundum 
directionem  quancunque,  quam  ex  mutuis  actionibus  acquiret  altera  massa,  fore  semper 
aequalem  summae  motuum  computatorum  secundum  oppositam,  quam  massa  altera  acquiret 
simul,  in  quo  ipso  sita  est  actionis  &  reactionis  aequalitas,  ex  qua  corporum  [179]  collisiones 
deduximus  in  secunda  parte,  &  ex  qua  multa  phsenomena  pendent,  in  Astronomia  inprimis. 


A  THEORY  OF  NATURAL  PHILOSOPHY  283 

small,  which  was  sufficient  for  a  mind  to  grasp,  to  determine  the  whole  of  the  remainder 
of  such  a  continuous  curve,  continued  to  infinity  on  either  side. 

386.  We  cannot  aspire  to  this,  not  only  because  our  human  intellect  is  not  equal  to  the  why  the  problem 
task,  but  also  because  we  do  not  know  the  number,  or  the  position  &  motion  of  each  of  these  cannot   be    solved 
points  (for  we  do  not  observe  absolute  motions,  but  merely  relative  motions  with  respect  intellect  ;U  what 
to  the  Earth,  or  at  most  those  with  respect  to  the  planetary  system  or  the  system  of  all  obstacle  to  its 

.1       t~      J  \        o     ^u  i_  i        i.          t.      £  •  j         j    determination    is 

the  fixed  stars)  ;    &  there  is  yet  another  reason,  namely  that  the  free  motions  produced  due   to    freedom ; 

by  spiritual  substances  affect  these  curves.     The  "  pre-established  harmony  "  of  the  followers  argument    against 

of  Leibniz  abrogates  all  such  disturbing  effect,  at  least  as  far  as  regards  our  will,  since  it  does  harmony!" 

not  admit  any  direct  intercourse  between  body  &  spirit.     What  was  so  strongly  condemned 

in  the  theory  of  Descartes,  which  reduced  animals  to  automata,  is  transferred  to  men  as  well ; 

&  it  is  easily  shown  that  all  their  motions  arise  from  a  mechanism,  &  that  these  are  necessary 

upon  that  theory.     For  this  reason,  indeed,  I  am  very  much  against  the  Cartesian  theory ; 

for,  besides  other  things,  if  I  admitted  its  principles,  I  should  not  be  able  to  see  any  real 

reason,  nay,  not  of  the  slightest  kind,  which  would  lead  me  to  think  that,  in  addition  to 

my  mind,  ideas  about  which  are  evolved  of  itself  &  without  any  direct  connection  with 

the  body,  I  had  a  body  that  had  motions ;  much  less,  that  these  motions  conformed  to  those 

ideas,  or  that  there  were  any  other  men,  or  any  corporeal  nature  outside  myself.     Such  a 

philosophy  must  of  necessity  lead  a  mind  that  puts  everything  in  the  scales  of  its  own 

impulses  to  such  absurdities,  &  still  worse  ;    &  I  have  always  been   astonished  that  this 

philosophy  has  gained  ground  &  has  even  been  accepted  everywhere,  &  up  to  the  present 

has  been  growing ;    I  am  amazed  that  it  should  have  been  tolerated  at  all. 

387.  I  think,  therefore,  that  the  free  motions  of  bodies  arise  from  the  mind;   &  that  Free    motions  are 
this  is  due  to  an  inner  force,  by  which  the  mind  knows  the  nature,  certain  properties  &  the  byrtth"lymmddUbut 
origin  of  its  ideas,  I  think  can  be  easily  established.     Just  as  we  must  have  a  law  of  forces,  are  not  impressed 
perhaps  involved  in  the  very  nature  of  matter,  of  such  a  kind  that  according  to  it  two  points  opposite ^irec'aons1 
of  matter  must  approach  towards,  or  recede  from,  one  another  with  a  motion  determined  &  without  breach 
in  magnitude  &  direction  by  the  distance  between  the  points ;    so  there  must  be  other  of  contmulty 
free  laws  for  the  mind,  according  to  which  any  points  that  have  that  disposition  which  a 

living  &  healthy  body  requires,  must  obey  the  command  of  the  mind.  But  such  laws,  I 
also  think,  require  the  condition  that  a  motion  cannot  be  impressed  on  any  point  of  matter, 
unless  an  equal  &  opposite  motion  is  impressed  on  some  other  point  of  matter  ;  this  follows 
from  the  stress  that  we  always  exert  in  opposite  directions,  according  to  what  has  been 
said  in  Art.  74.  Lastly,  I  consider,  &  the  fact  can  be  derived  by  diligent  observation  & 
reflection,  that  such  motion  can  not  be  impressed,  unless  it  follows  a  law  of  continuity 
without  any  break  ;  &  if  this  law  is  bound  to  be  observed  by  all  object-souls,  the  real  motions 
will  truly  depart  from  the  necessary  curves,  &  the  curves  actually  described  will  depend 
on  a  free  determination  of  the  will ;  but  the  continuity  of  the  motions  will  not  thereby 
be  affected. 

388.  Further,  it  is  hence  evident  why  we  nowhere  get  any  discontinuity  in  motions,  Conclusions  de- 
why  no  point  of  matter  can  ever  pass  from  one  position  to  another  without  passing  through  £ h^eiciusion^o3! 
all  intermediate  positions,  why  density  can  in  no  case  be  suddenly  changed,  why  reflected  absolute  rest. 

&  refracted  motions  come  about  through  continuous  curvature,  &  other  things  of  the  sort 
relating  to  the  matter  in  hand.  But,  in  particular,  there  will  at  the  same  time  be  evident 
the  fact,  which  is  the  purpose  of  all  we  have  just  done,  namely,  that  there  is  no  such  thing 
as  absolute  rest ;  that  is  to  say,  such  a  thing  as  the  sudden  breaking  off  of  the  continuous 
drawing  of  the  curve  described,  the  continuity  being  destroyed  just  as  much  as  it 
would  be  if  a  continuous  curve  finally  became  a  straight  line  after  reaching  a  certain 
point. 

389.  Passing  on  to  the  equality  of  action  &  reaction,  we  have  already,  in  Art.  265,  Equality  of  action 
fully  proved  its  truth  for  any  two  bodies  from  the  equality  of  the  action  &  reaction  between  &    reaction  ;    its 

'   J  ,  _        .  '          .  «  »     *        i '  rr  i  f   consequences. 

any  two  points,  bor  instance,  since  the  mutual  forces  do  not  in  any  way  affect  the  state  of 
the  common  centre  of  gravity,  &  the  centres  of  gravity  of  two  masses  must  lie  in  a  straight  line 
with  the  common  centre  of  the  two,  at  distances  on  each  side  of  the  latter  that  are  inversely 
proportional  to  the  masses,  as  was  also  proved  in  the  same  article ;  it  must  follow  that  any 
motions,  which  owing  to  mutual  action  are  possessed  by  the  centres  of  gravity  of  the  two 
masses,  must  take  place  along  lines  that  are  similar  &  proportional  to  the  distances  of  each 
from  the  common  centre  of  gravity,  &  thus  inversely  proportional  to  the  masses.  Also  it 
then  follows  that  the  sum  of  the  motions,  reckoned  in  any  direction,  acquired  by  either  of 
the  masses  on  account  of  the  mutual  actions,  must  always  be  equal  to  the  sum  of  the  motions 
in  the  directly  opposite  direction,  acquired  simultaneously  by  the  other  mass ;  &  in  this 
is  involved  the  equality  of  action  &  reaction  ;  &  from  it  we  deduced  the  laws  of  the 
collisions  of  bodies  in  the  second  part ;  &  upon  it  depend  many  phenomena,  especially  in 
Astronomy. 


284 


PHILOSOPHISE   NATURALIS  THEORIA 


inde   an  motus 


an  ab  externis. 


390.  Illud  unum  hie  adnotandum  censeo,  per  hanc  ipsam  legem  comprobari  plurimum 
*Psas  v*res  mutuas  inter  materiae  particulas,  &  deveniri  ad  originem  motuum  plurimorum, 
'  quae  inde  pendet  ;  si  nimirum  particulae  massae  cujuslibet  ingentem  habeant  motum 
reciprocum  hac,  iliac,  &  interea  centrum  commune  gravitatis  iisdem  iis  motibus  careat  ; 
id  sane  indicio  est,  eos  motus  provenire  ab  internis  viribus  mutuis  inter  puncta  ejusdcm 
massae.  Id  vero  accidit  inprimis  in  fermentationibus,  quae  habentur  post  quarundam 
substantiarum  permixtionem,  quarum  particulse  non  omnes  simul  jam  in  unam  feruntur 
plagam,  jam  in  aliam,  sed  singillatim  motibus  diversissimis,  &  inter  se  etiam  contrariis, 
quos  idcirco  motus  omnes  illarum  centra  gravitatis  habere  non  possunt  ;  ii  motus  provenire 
omnino  debent  a  mutuis  viribus,  &  commune  gravitatis  centrum  interea  quiescet  respectu 
ejus  vasis,  in  quo  fermentatio  sit,  &  Terrae,  respectu  cujus  quiescit  vas. 


Divisibmtas  in  in-  391.  Quod  ad  divisibilitatem  pertinet,  earn  quidem  in  infinitum  progredientem  sine 

tinu!1"1  immaterial  u^°  ^mite  m  spatio  continue  ille  solus  non  agnoscet,  qui  Geometriae  etiam  elementaris 
itidem  si  sit  con-  vim  non  sentiat,  a  qua  pro  ejusmodi  divisibilitate  in  infinitum  tarn  multa,  &  simplicia,  & 
Perspicua  sane  argumenta  desumuntur.  Ubi  ad  materiam  sit  transitus  ;  si,  ubi  de  ea  agitur, 
quae  distinctas  occupant  loci  partes,  distincta  etiam  sunt  ;  ab  ilia  spatii  continui  divisibilitate 
in  infinitum,  materiae  quoque  divisibilitas  in  infinitum  consequitur  evidentissime,  & 
utcunque  prima  materiae  elementa  atomos,  sive  Naturae  vi  insectilia  censeant  multi,  ut 
&  Newtonus  ;  adhuc  tamen  absolutam  eorum  divisibilitatem  agnoscunt  passim  illi  ipsi. 


virtuaiem     exten-  392.  Materiae  elementa  extensa  per  spatium  divisibile,  sed  omnino  simplicia,  &  carentia 

swnem  non  haben.  partjijUS)  admiserunt  nonnulH  e  Peripateticis,  &  est  etiam  nunc,  qui  recentiorem  Philoso- 
phiam  professus  admittat  ;  at  earn  sententiam  non  ex  praejudicio  quodam,  quanquam  id 
etiam  est  ingens,  &  commune,  sed  ex  inductionis  principio,  &  analogia  impugnavi  in  prima 
parte  num.  83.  Quamobrem  arbitror,  si  quid  corporeum  extensionem  habeat  per  totum 
quodpiam  continuum  spatium,  id  ipsum  debere  absolute  habere  partes,  &  esse  divisibile 
in  infinitum  aeque,  ac  illud  ipsum  est  spatium. 


Puncta  esse  indi- 
visibilia  ;  massas 
divisibiles  usque  ad 
certum  limit  em 
singulas. 


Componibilitas     i  n 
infinitum. 


Ejus 


in  infinitum. 


393-  At  in  mea  Theoria,  in  qua  prima  elementa  materiae  mihi  sunt  simplicia,  ac  inex- 
tensa,  nullam,  eorum  divisibilitatem  haberi  constat.  Massae  autem,  queecunque  actu 
existant,  sunt  mihi  congeries  punctorum  ejusmodi  numero  finitae.  Hinc  eae  congeries 
dividi  utique  possunt  in  partes,  sed  non  plures,  quam  sit  ipse  punctorum  numerus  massam 
constituentium,  cum  nulla  pars  minus  continere  possit,  quam  unum  ex  iis  punctis.  Nee 
Geometrica  argumenta  quidquam  probant  in  mea  Theo-[i8o]ria  pro  divisibilitate  ultra 
eum  limitem ;  posteaquam  enim  deventum  fuerit  ad  intervalla  minora,  quam  sit  distantia 
duorum  punctorum,  sectiones  ulteriores  secabunt  intervalla  ipsa  vacua,  non  materiam. 

394.  Verum  licet  ego  non  habeam  divisibilitatem  in  infinitum,  habeo  tamen  componi- 
bilitatem,  ut  appellare  soleo,  in  infinitum.  In  quovis  dato  spatio  habebitur  quidem  semper 
certus  quidam  punctorum  numerus,  qui  idcirco  etiam  finitus  erit ;  neque  enim  ego  admitto 
infinitum  ullum  in  Natura,  aut  in  extensione,  neque  infinite  parvum  in  se  determinatum, 
quod  ego  positiva  demonstratione  exclusi  primum  in  mea  Dissertatione  de  Natura  Iff  usu 
infinitorum,  &  infinite  parvorum  ;  turn  &  aliis  in  locis  ;  quod  tamen  requireretur  ad  hoc, 
ut  intra  finitum  spatium  contineretur  punctorum  numerus  indefinitus  :  at  longe  aliter  se 
res  habet ;  si  consideremus,  qui  numerus  punctorum  in  dato  spatio  possit  existere  :  turn 
enim  nullus  est  numerus  finitus  ita  magnus,  ut  alius  adhuc  finitus  ipso  major  haberi  in  eo 
spatio  non  possit.  Nam  inter  duo  puncta  quaecunque  potest  in  medio  interseri  aliud, 
quod  quidem  neutrum  continget ;  aliter  enim  etiam  ea  duo  se  contingerent  mutuo,  & 
non  distarent,  sed  compenetrarentur.  Potest  autem  eadem  ratione  inter  hoc  noyum,  & 
priora  ilia  interseri  novum  utrinque,  &  ita  porro  sine  ullo  limite  :  adeoque  deveniri  potest 
ad  numerum  punctorum  quovis  determinato  utcunque  magno  majorem  in  unica  etiam 
recta,  &  proinde  multo  magis  in  spatio  extenso  in  longum,  latum,  &  profundum.  Hanc 
ego  voco  componibilitatem  in  infinitum.  Numerus,  qui  in  quavis  data  massa  existit, 
finitus  est ;  sed  dum  eum  Naturae  Conditor  determinare  voluit,  nullos  habuit  limites, 
quos  non  potuerit  praetergredi,  nullum  ultimum  habente  terminum  serie  ilia  possibilium 
finitorum  in  infinitum  crescentium. 

sequivaientia  ^95.  Haec  componibilitas  in  infinitum  aequivalet  divisibilitati  in  ordine  ad  explicanda 

Naturae    phaenomena.     Posita    divisibilitate    materiae    in   infinitum,    solvitur   facile    illud 


A  THEORY  OF  NATURAL  PHILOSOPHY  285 

390.  I  consider  that  in  this  connection  it  should  be  remarked  that  by  means  of  this  Hence,  the  point  as 
law  especially  the  existence  of  these  mutual  forces  between  particles  of  matter  is  established,  rn'otion^f"'!  mass 
&  that  in  it  we  attain  to  the  source  of  most  of  the  motions,  which  arises  from  it.     For  arises  from  internal 
instance,  considering  that  the  particles  of  a  mass  may  have  an  immense  reciprocal  motion,  or  external  forces, 
whilst  the  common  centre  of  gravity  is  without  any  such  motion,  surely  that  is  a  token 

that  these  motions  come  from  mutual  internal  forces  between  the  particles  of  the  mass. 
Now,  this  takes  place,  in  particular,  in  fermentations,  such  as  are  obtained  after  making 
a  mixture  of  certain  substances ;  here  the  particles  of  the  substances  are  not  all  at  the  same 
time  moving  first  in  one  direction,  then  in  another,  but  each  of  them  separately  in  the 
most  widely  diverging  directions,  &  even  in  opposite  directions,  to  one  another.  Hence, 
as  the  centres  of  gravity  cannot  have  all  these  motions,  the  motions  must  arise  from  mutual 
forces ;  &,  besides,  the  common  centre  of  gravity  is  at  rest  with  regard  to  the  vessel  in 
which  the  fermentation  takes  place,  &  also  with  regard  to  the  Earth,  with  respect  to  which 
the  vessel  is  at  rest. 

391.  Now,  as  concerning  divisibility,  that  this  can  be  carried  on  indefinitely  without  infinite  divisibility 
any  limit  in  continuous  space  will  be  denied  only  by  one  who  does  not  feel  the  force  of  o™^?1}.^  «"„?»"  f 

.  r  /      /  i        i     •       i  •        i      s>Pace  •  tne  same  ot 

the  most  elementary  principles  of  geometry;   for,  from  it  may  be  derived  so  many  simple  matter,  if  it  is 
&  perfectly  clear  arguments  in  favour  of  such  infinite  divisibility.     When  we  come  to  outtlvktuai&exten- 
consider  matter,  if  in  dealing  with  it,  we  take  it  that  what  occupies  a  distinct  part  of  space  sion. 
is  itself  distinct,   then,   from  the  infinite  divisibility  of  continuous   space,   the  infinite 
divisibility  of  matter  also  follows  very  clearly ;  &,  although  there  are  many  who  think  that 
the  primary  elements  of  matter  are  atoms,  that  is  to  say,  things  that  are  incapable  of  further 
division  by  any  Natural  force,  as  Newton  also  thought,  yet  even  they  must  still  in  all  cases 
admit  their  absolute  divisibility. 

392.  Some  of  the  Peripatetics  admitted  elements  of  matter  extended  through  divisible  Virtual  extension  is 
space,  but  quite  simple  &  without  parts ;    &  at  the  present  day  there  is  one  professing  a  E 

more  modern  philosophy  who  admits  such  elements.  This  idea,  in  Art.  83  of  the  first 
part  of  this  work,  I  contradicted,  not  by  the  employment  of  any  prejudgment,  although 
there  certainly  exists  one  that  is  very  forcible  &  generally  acknowledged,  but  by  the 
employment  of  the  principle  of  induction  &  analogy.  Hence,  I  think  that,  if  anything 
has  corporeal  extension  throughout  the  whole  of  any  continuous  space,  it  must  also  absolutely 
have  parts  &  must  be  infinitely  divisible,  in  exactly  the  same  manner  as  the  space  is  infinitely 
divisible. 

393.  Now,  in  my  Theory,  in  which  the  primary  elements  of  matter  are  simple  &  non-  Points  are    indivi- 
extended,  it  is  easily  seen  that  there  can  be  no  divisibility  of  the  elements.     Also  masses,  sibie,  whilst  every 

,  '       ,,  .  _'..-..  mass  is  divisible  up 

in  so  far  as  they  actually  exist,  are  to  me  merely  sets  of   such   points  finite   in   number,  to  a  certain  limit. 

Hence  these  sets  of  points  can  at  any  rate  be  divided  into  parts,  but  not  into  a  greater  number 

of  points  than  that  given  by  the  number  of  points  constituting  the  mass,  since  no  part  can 

contain  less  than  one  of  these  points.     Nor  do  geometrical  arguments  prove  anything, 

as  far  as  my  Theory  is  concerned,  in  favour  of  divisibility  beyond  this  limit ;    for,  as  soon 

as  we  reach  intervals  that  are  less  than  the  distance  between  two  points,  further  sections 

will  cut  these  empty  intervals  &  not  matter. 

394.  Now,  although  I  do  not  hold  with  infinite  divisibility,  yet  I  do  admit  infinite  infinite    componi- 
componibility,  as  it  is  usually  called.     In  any  given  space  we  can  always  have  a  certain 

number  of  points ;  &  hence  this  number  is  finite.  For,  I  do  not  admit  anything  infinite 
in  Nature,  or  in  extension,  or  a  self-determined  infinitely  small.  Such  a  thing  I  excluded 
by  direct  proof,  for  the  first  time  in  my  dissertation  De  Natura,  y  usu  infinitorum,  y  infinite 
parvorum  ;  &  later,  in  other  writings ;  this,  however,  is  required,  if  an  indefinite  number 
of  points  is  to  be  included  within  a  finite  space.  But  the  facts  of  the  matter  are  quite 
different,  if  we  consider  how  great  a  number  of  points  can  exist  within  a  given  space  ;  for, 
then  there  is  no  finite  number  so  great,  but  that  a  still  greater  finite  number  can  be  had 
within  the  space.  For,  between  any  two  points  it  is  possible  to  insert  another  midway, 
which  will  touch  neither  of  the  former  ;  if  this  is  not  the  case,  then  the  two  former  points 
must  touch  one  another,  &  not  be  at  a  distance  from  one  another,  but  compenetrated. 
Further,  in  the  same  manner,  between  the  new  point  &  the  first  two  points,  we  can  insert 
a  new  one  on  either  side  ;  &  so  on  without  any  limit.  Thus  we  could  arrive  at  a  number 
of  points  greater  than  any  given  number,  no  matter  how  large,  all  of  them  even  lying  in 
a  single  straight  line  ;  much  more  then  would  this  be  the  case  in  space  extended  in  length, 
breadth  &  depth.  This  I  call  infinite  componibility.  The  number  of  points  present  in 
any  given  mass  is  finite  ;  but  when  the  Creator  of  the  Universe  willed  what  that  number 
was  to  be,  he  had  no  limits ;  for  the  series  of  possible  finites  increasing  indefinitely  has  no 
last  term. 

395.  This  infinite  componibility  is  equivalent  to  divisibility  for  the  purpose  of  explaining  The  equivalence  of 

i        v  r  XT  rr  i         •    f    •        T    •  -1  •!•        f  componibility   to 

the  phenomena  of  Nature,     If  we  postulate  infinite  divisibility  for  matter,  we  have  an  easy  infinite  divisibility 


286  PHILOSOPHIC  NATURAL1S  THEORIA 

problema  :  Datam  massam  utcunque  parvam,  ita  distribuere  -per  datum  spatium  utcunque 
magnum,  ut  in  eo  nullum  sit  spatiolum  majus  dato  quocunque  utcunque  parvo  penitus  vacuum, 
y  sine  ulla  ejus  materice  particula.  Concipitur  enim  numerus,  quo  illud  magnum  spatium 
datum  continere  possit  hoc  spatiolum  exiguum,  qui  utique  finitus  est,  &  in  se  determinatus  : 
concipitur  in  totidem  particulas  divisa  massula,  &  singulse  particulse  destinantur  singulis 
spatiolis ;  qua  iterum  dividi  possunt,  quantum  libuerit,  ut  parietes  spatioli  sui  convestiant, 
qui  utique  ad  unam  ejus  transversam  sectionem  habent  finitam  rationem,  adeoque  continua 
sectione  planis  parallelis  facta  possunt  ipsi  parietes  convestiri  segmentis  suas  particulse, 
vel  possunt  ejus  particulas  segmenta  iterum  per  illud  spatiolum  utcunque  dispergi.  In 
[181]  mea  Theoria  substituitur  hujusmodi  aliud  problema  :  Intra  datum  spatiolum  collocare 
eum  punctorum  numerum,  qui  deinde  disiribui  possit  per  spatium  utcunque  magnum  ita,  ut 
in  eo  nullum  sit  spatiolum  cubicum  majus  dato  quocunque  utcunque  parvo  penitus  vacuum,  & 
quod  in  se  non  habeat  numerum  punctorum  utcunque  magnum. 


Demonstrate     ea  3^5    Quocl  jn  ordine  ad  explicanda  phaenomena  hoc  secundum  problema  sequivaleat 

illi  primo,  patet  utique  :  nam  solum  deest  convestitio  parietum  continua  mathematice  : 
sed  illi  succedit  continuatio  physica,  cum  in  singulis  parietibus  collocari  possit  ejus  ope 
quicunque  numerus  utcunque  magnus,  distantiis  idcirco  imminutis  utcunque.  Quod  in 
mea  Theoria  secundum  illud  problema  solvi  possit  ope  expositse  componibilitatis  in  infinitum, 
patet  :  quia  ut  inveniatur  numerus,  qui  ponendus  est  in  spatiolo  dato,  satis  est,  ut  numerus 
vicium,  quo  ingens  spatium  datum  continet  illud  spatiolum  posterius  multuplicetur  per 
numerum  punctorum,  quern  velimus  collocari  in  hoc  ipso  quovis  posteriore  spatiolo  post 
dispersionem,  &  auctor  Naturae  potuit  utique  intra  illud  spatiolum  primum  hunc  punctorum 
numerum  collocare. 

Divisibiiitas  in  397.  Jam  quod  pertinet  ad  divisibilitatem  immanem,  quam  nobis  ostendunt  Naturas 

Natura  immams;  phaenomena  in  coloratis  quibusdam  corporibus,  immanem  molem  aquae  inficientibus  eodem 
colore,  in  auro  usque  adeo  ductili,  in  odoribus,  &  ante  omnia  in  lumine,  omnia  mihi  cum 
aliis  communia  erunt ;  &  quoniam  nulla  ex  observationibus  nobis  potest  ostendere  divisi- 
bilitatem absolute  infinitam,  sed  ingentem  tantummodo  respectu  divisionum,  quibus 
plerumque  assuevimus ;  res  ex  meo  problemate  aeque  bene  explicabitur  per  componibilitatem 
ac  in  communi  Theoria  ex  illo  alio  per  divisibilitatem  materiae  in  infinitum. 

immutabiiitas  pri-  *gg_  prima  materiae  elementa  volunt  plerunque  immutabilia,  &   eiusmodi,  ut  atteri, 

morum         elemen-  J'         .  .       .  .  .  \     .        *      ,  ,J       „  XT 

torum materiae :  or-  atque  conirmgi  ommno  non  possint,  ne  mrmrum  pnaenomenorum  ordo,  &  tota  Naturae 
dines  diversi  parti-  facies  commutetur.     At  elementa  mea  sunt  sane  eiusmodi,  ut  nee  immutari  ipsa,  nee 

cularum    minus,  ac  ,  .   .  ,.  ,          .  .  .   J  .,  ,, 

minus  immuta-  legem  suam  vinum,  ac  agendi  modum  in  compositiombus  commutare  ullo  modo  possmt ; 
cum  nimirum  simplicia  sint,  indivisibilia,  &  inextensa.  Ex  iis  autem  juxta  ea,  quse  diximus 
num.  239  ad  distantias  perquam  exiguas  collocatis  in  limitibus  virium  admodum  validis 
oriri  possunt  primae  particulae  minus  jam  tenaces  suae  formae,  quam  simplicia  elementa, 
sed  ob  ingentem  illam  viciniam  adhuc  tenacissimae  idcirco,  quod  alia  particula  quasvis 
ejusdem  ordinis  in  omnia  simul  ejus  puncta  fere  aequaliter  agat,  &  vires  mutuae  majores 
sint,  quam  sit  discrimen  virium,  quibus  diversa  ejus  puncta  solicitantur  ab  ilia  particula. 
Ex  hisce  primi  ordinis  particulis  possunt  constare  particulae  ordinis  secundi ;  adhuc  minus 
tenaces,  &  ita  porro  ;  quo  enim  plures  compositiones  sunt,  &  majores  distantiae,  eo  facilius 
fieri  potest,  ut  inaequalitas  [182]  virium,  quas  sola  mutuam  positionem  turbat,  incipiat 
esse  major,  quam  sint  vires  mutuae,  quae  tendunt  ad  conservandam  mutuam  positionem, 
&  formam  particularum  ;  &  tune  jam  alterationes,  &  transformationes  habebuntur,  quas 
videmus  in  corporibus  hisce  nostris,  &  quae  habentur  etiam  in  pluribus  particulis  postremorum 
ordinum,  haec  ipsa  nova  corpora  componentibus.  Sed  prima  materiae  elementa  erunt 
omnino  immutabilia,  &  primorum  etiam  ordinum  particulae  formas  suas  contra  externas 
vires  validissime  tuebuntur. 

Gravitas      exhibita 
a     postremo     arcu 

curvffi  accedens  ad  399.  Gravitas  etiam  inter  generales  proprietates  a  Newtonianis  inprimis  numeratur, 

p^xfmT^oTs™  quibus  assentior ;  dummodo  ea  reipsa  non  habeat  rationem  reciprocam  duplicatam 
nostro  concipiendi  distantiarum  extensam  ad  omnes  distantias,  sed  tantum  ad  distantias  ejusmodi,  cujusmodi 
'  eae,  quse  interjacent  inter  distantiam  nostrorum  corporum  a  parte  multo  maxima 


A  THEORY  OF  NATURAL  PHILOSOPHY  287 

solution  of  the  following  problem.  Distribute  a  given  mass,  however  small,  within  a  given 
space,  however  large,  in  such  a  manner  that  there  shall  be  no  little  space  in  it  greater  than  any 
given  one,  no  matter  how  small,  that  shall  be  quite  empty,  W  without  any  particle  of  that  matter. 
For  we  assume  a  certain  number  to  represent  the  number  of  times  the  large  given  space 
can  contain  the  exceedingly  small  space,  this  number  being  in  every  case  finite  &  self- 
determined  ;  we  assume  the  mass  to  be  divided  into  the  same  number  of  particles,  &  one 
of  the  particles  to  be  placed  in  each  of  the  small  spaces.  The  former  can  again  be  divided, 
as  much  as  is  desired,  so  that  the  new  parts  of  each  particle  cover  the  boundary  walls  of 
the  corresponding  small  space  ;  &  these  in  every  case  bear  a  finite  ratio  to  one  transverse 
section  of  it,  so  that,  by  making  continuous  sections  with  parallel  planes,  these  boundary 
walls  can  be  covered  each  with  segments  of  the  particle  corresponding  to  it  ;  or  the  segments 
of  a  particle  can  be  scattered  in  any  manner  throughout  the  small  space,  repeating  the 
above  process.  In  my  Theory  another  problem  is  substituted,  such  as  the  following  :  — 
Place  within  a  given  small  space  such  a  number  of  points,  that  these  can  then  be  distributed 
throughout  any  space,  however  great,  in  such  a  manner  that  there  shall  be  no  little  cubical  space 
in  it  greater  than  any  given  one,  however  small,  that  shall  be  quite  empty,  &  which  does  not 
contain  in  itself  any  number  of  points  however  great. 

396.  It  is  quite  clear  that,  for  the  purpose  of  explaining  the  phenomena  of  Nature,  Demonstration. 
the  second  problem  is  equivalent  to  the  first  ;   for,  the  only  thing  that  is  wanting  in  it  is 

a  continuous  covering  of  the  boundary  walls,  in  a  strictly  mathematical  sense  ;  &  instead 
of  this  we  have  a  physical  continuity,  since  in  each  of  the  walls  there  can  be  placed  by  means 
of  it  any  number  of  particles,  however  great,  &  therefore  at  distances  from  one  another 
which  are  indefinitely  diminished.  It  is  also  clear  that,  in  my  Theory,  the  second  problem 
can  be  solved  by  the  employment  of  the  infinite  componibility  that  I  have  explained  ; 
for,  in  order  to  find  the  number  to  be  placed  in  a  given  small  space,  it  is  sufficient  that  the 
number  of  times  that  the  large  given  space  contains  the  latter  small  space  should  be  multiplied 
by  the  number  of  points  which  we  desire  to  be  placed  in  this  latter  small  space  after 
dispersion  ;  &  certainly  the  Author  of  Nature  was  able  to  place  this  number  of  points  within 
that  first  small  space. 

397.  Now,  as  regards  the  immense  divisibility,  which  the  phenomena  of  Nature  present  The  immense  divi- 
to  us  in  certain  coloured  bodies,  when  they  stain  an  immense  volume  of  water  with  the  same  Slbl          Nature. 
colour,  in  the  extremely  great  ductility  of  gold,  in  odours,  &  more  than  all  in  light,  everything 

will  be  in  agreement  in  my  Theory  with  the  theories  of  others.  Moreover,  since  no 
observations  can  show  us  any  divisibility  that  is  absolutely  infinite,  but  only  such  as  is 
immensely  great  when  compared  with  such  divisions  as  we  are  for  the  most  part  accustomed 
to  ;  it  follows  that  the  matter  can  be  explained  just  as  well  from  my  problem  by  means 
of  componibility,  as  in  the  usual  theory  it  can  be  from  the  other  problem  by  the  infinite 
divisibility  of  matter. 

398.  The  primary  elements  of  matter  are  considered  by  most  people  to  be  immutable,  immutability  of 
&  of  such  a  kind  that  it  is  quite  impossible  for  them  to  be  subject  to  attrition  or  fracture,  ments^o^matte'ri 
unless  indeed  the  order  of  phenomena  &  the  whole  face  of  Nature  were  changed.     Now,  different   kinds  of 
my  elements  are  really  such  that  neither  themselves,  nor  the  law  of  forces  can  be  changed  ;  ^immutable3    & 
&  the  mode  of  action  when  they  are  grouped  together  cannot  be  changed  in  any  way  ;  for 

they  are  simple,  indivisible  &  non-extended.     From  these,  by  what  I  have  said  in  Art. 

239,  when  collected  together  at  very  small  distances  apart,  in  sufficiently  strong  limit-points 

on  the  curve  of  forces,  there  can  be  produced  primary  particles,  less  tenacious  of  form 

than  the  simple  elements,  but  yet,  on  account  of  the  extreme  closeness  of  its  parts,  very 

tenacious  in  consequence  of  the  fact  that  any  other  particle  of  the  same  order  will  act 

simultaneously  on  all  the  points  forming  it  with  almost  the  same  strength,  &  because  the 

mutual  forces  are  greater  than  the  difference  between  the  forces  with  which  the  different 

points  forming  it  are  affected  by  the  other  particle.     From  such  particles  of  the  first  order 

there  can  be  formed  particles  of  a  second  order,  still  less  tenacious  of  form  ;  &  so  on.     For 

the  greater  the  composition,  &  the  larger  the  distances,  the  more  readily  can  it  come  about 

that  the  inequality  of  forces,  which  alone  will  disturb  the  mutual  position,  begins  to  be 

greater  than  the  mutual  forces  which  endeavour  to  maintain  that  mutual  position,  i.e.  the 

form  of  the  particles.     Then  indeed  we  shall  have  changes  &  transformations,  such  as  we 

see  in  these  bodies  of  ours,  &  which  are  also  obtained  in  most  of  the  particles  of  the  last 

orders,  which  compose  these  new  bodies.     But  the  primary  elements  of  matter  will  be  Gravity,  as  repre- 

quite  immutable,  &  particles  of  the  first  orders  will  preserve  their  forms  in  opposition  to  sente.d  ,,bv  the  last 

,    '          -  .   ,  arc  of  the  curve,  ap- 

even  very  strong  forces  from  without.  proximates  to  that 

399.  Gravity  also  is  counted  as  a  general  property,  especially  by  followers  of  Newton  ;  siv<rn  by  the  New- 

,     T    J''     ,      ,        '  ..  °       .      .    *  ''  i      }.         .  .'    toman  law;   possi- 

&  1  am  ot  the  same  opinion,  so  long  as  it  is  not  supposed  to  be  in  the  inverse  ratio  of  biiity  of  its  being 


the  squares  of  the  distances  for  all  distances,  but  merely  for  distances  such  as  those  that  lie  ^^nthet  Sam1e> 


n 
between  the  distance  of  our  bodies  from  the  far  greatest  part  of  the  mass  of  the  Earth,  hypothesis. 


288  PHILOSOPHIC  NATURALIS  THEORIA 

massae  terrestris,  &  distantias  a  Sole  apheliorum  pertinentium  ad  cometas  remotissimos, 
&  dummodo  in  hoc  ipso  tractu  sequatur  non  accuratissime,  sed,  quam  libuerit,  proxime, 
rationem  ipsam  reciprocam  duplicatam,  juxta  ea,  quae  diximus  num.  121.  Ejusmodi 
autem  gravitas  exhibetur  ab  arcu  illo  postremo  meae  curvae  figurae  i,  qui,  si  gravitas  exten- 
ditur  cum  eadem  ilia  lege  ad  sensum,  vel  cum  aliqua  simili,  in  infinitum,  erit  asymptoticus. 
Posset  quidem,  ut  monui  num.  119,  concipi  gravitas  etiam  accurate  talis,  quae  extendatur 
ad  quascunque  distantias  cum  eadem  lege,  &  praeterea  alia  quaedam  vis  exposita  per  aliam 
curvam,  in  quam  vim,  &  in  gravitatem  accurate  reciprocam  quadratis  distantiae  resolvatur 
lex  virium  figurae  I  ;  quae  quidem  vis  in  illis  distantiis,  in  quibus  gravitas  sequitur  quam 
proxime  ejusmodi  legem,  esset  insensibilis ;  in  aliis  autem  distantiis  plurimis  ingens  esset  : 
ac  ubi  figura  I  exhibet  repulsiones,  deberet  esse  vis  hujus  alterius  conceptae  legis  itidem 
repulsiva  tanto  major,  quam  vis  legis  primitivae  figurae  i,  quanta  esset  gravitas  ibi  concepta, 
quae  nimirum  ab  illo  additamento  vis  repulsivae  elidi  deberet.  Sed  haec  jam  a  nostro 
concipiendi  modo  penderent,  ac  in  ipsa  mea  lege  primitiva,  &  reali,  gravitas  utique  est 
generalis  materiae,  ac  legem  sequitur  rationis  reciprocae  duplicatas  distantiarum,  quanquam 
non  accurate,  sed  quamproxime,  nee  ad  omnes  extenditur  distantias  ;  sed  illas,  quas  exposui. 


i  irftoto  4°°'  Ceterum    gravitatem    generalem    haberi    in    toto    planetario    systemate,    ego 

soiari  systemate,  quidem  arbitror  omnino  evinci  iisdem  argumentis  ex  Astronomia  petitis,  quibus  utuntur 
pressi(^SSfluidrbm  Newtoniani,  quae  hie  non  repeto,  cum  ubique  prostent,  &  quae  turn  alibi  ego  quidem 
congessi  pluribus  in  locis,  turn  in  Adnotationibus  ad  poema  P.  Noceti  De  Aurora  Boreali. 
Illud  autem  arbitror  evidentissimum,  ilium  accessum  ad  Solum  cometarum,  &  planetarum 
primariorum,  ac  secundariorum  ad  primaries,  quem  videmus  in  descensu  a  recta  tangente 
ad  arcum  curvae,  &  multo  magis  alios  motus  a  mutua  gravitate  pendentes  haberi  omnino 
[183]  non  posse  per  ullius  fluidi  pressionem  ;  nam  ut  alia  praetermittam  sane  multa,  id 
fluidum,  quod  sola  sua  pressione  tantum  possit  in  ejusmodi  globos,  multo  plus  utique  posset 
occursu  suo  contra  illorum  tangentialem  velocitatem,  quae  omnino  deberet  imminui  per 
ejusmodi  resistentiam,  cum  ingenti  perturbatione  arearum,  &  totius  Astronomiae  Mechanicae 
perversione  ;  adeoque  id  fluidum  vel  resistentiam  ingentem  deberet  parere  planetae,  aut 
cometae  progredienti,  vel  ne  pressione  quidem  ullum  ipsi  sensibilem  imprimit  motum. 

Theoria  respondere          4ai-  Ejus  autem  praecipuae  leges  sunt,  ut  directe  respondeat  massae,  &  reciproce 
massae    directe,   &  quadratis  distantiarum  a  singulis  punctis  massae  ipsius.  quod  in  mea  Theoria  est  admodum 

quadrate  distantiae  •/•  •  i  •  j  MI 

reciproce.  mamfestum  ita  esse  debere  ;  nam  ubi  ventum  est  ad  arcum  mum  meae  curvae,  qui  gravitatem 

refert,  vires  omnes  jam  sunt  attractivae,  &  eandem  illam  ad  sensum  sequuntur  legem, 
adeoque  aliae  alias  non  elidunt  contrariis  directionibus,  sed  summa  earum  respondet  ad 
sensum  summae  punctorum  ;  nisi  quatenus  ob  inaequalem  punctorum  distantiam,  & 
positionem,  ad  habendam  accurate  ipsam  summam,  ubi  moles  sunt  aliquanto  majores, 
opus  erit  ilia  reductione,  qua  Mechanic!  utuntur  passim,  &  cujus  ope  inveniuntur  leges, 
secundum  quas  punctum  in  data  distantia,  &  positione  situm  respectu  massae  habentis 
datam  figuram,  ab  ipsa  attrahitur  ;  ubi,  quemadmodum  indicavimus  num.  347,  globus 
in  globum  ita  gravitat,  ut  gravitaret,  si  totae  eorum  massae  essent  compenetratae  in  eorum 
centris  :  at  in  aliis  figuris  longe  aliae  leges  obveniunt. 


Th,"  datio  402.  Verum  hie  illud  maxime  Theoriam  commendat  meam,  quod  num.  212  notandum 

i  neon**;     ex     con-  f  »  •  ...«.  »»  1» 

formitate   omnium  dixi,  quod  videamus  tantam  hanc  conformitatem  in  vi  gravitatis  in  omnibus  massis  ;   licet 
S?^mur.  rn6?,^  eaedem  in  ordine  ad  alia  phaenomena,  quae  a  minoribus  distantiis  pendent,  tantum  discrimen 

A  *i**l  1  TVT 

aliis.  habeant,  quantum  habent  diversa  corpora  in  duntie,  colore,  sapore,  odore,  sono.     JNam 

diversa  combinatio  punctorum  materiae  inducit  summas  virium  admodum  diversas  pro 
iis  distantiis,  in  quibus  adhuc  curva  virium  contorquetur  circa  axem  ;  proinde  exigua 
mutatio  distantiae  vires  attractivas  mutat  in  repulsivas,  ac  vice  versa  summis  differentias 
substituit  ;  dum  in  distantiis  illis,  in  quibus  gravitas  servat  quamproxime  leges,  quas  diximus, 
curva  ordinatas  omnes  ejusdem  directionis  habet,  &  vero  etiam  distantia  parum  mutata,- 
fere  easdem  ;  quod  necessario  inducit  tanta  priorum  casuum  discrimina,  &  tantam  in 
hoc  postremo  conformitatem. 

Omnia  fere  a  gravi- 

tate pendentia  sunt  .  , 

communia  huic  403.  Distinctio  gravitatis  (quae  est  ut  massa,  in  quam  tenditur,  directe,  &  quadratum 

Theory  cum  ^com-  j;,,^^  reciproce)  a  pondere  (quod  est  praeterae  ut  massa,  quae  gravitat)  est  mihi  eadem, 
rum  in  ea  faciiior  ac  Newtonianis,  &  omnibus  Mechanicis  ;    £  ilia  vim    acceleratricem  exhibet,  hoc  vim 

i 


deductio. 


A  THEORY  OF  NATURAL  PHILOSOPHY  289 

&  the  distances  from  the  Sun  of  the  aphelia  of  the  most  remote  comets ;  &  so  long  as  in 
this  region  it  is  not  assumed  to  follow  the  law  of  the  inverse  squares  exactly,  but  only  very 
approximately  to  any  desired  degree  of  closeness,  as  I  said  in  Art.  121.  Now  gravity  of 
this  kind  is  represented  by  the  last  arc  of  my  curve  in  Fig.  I  ;  &  this,  if  gravity  goes  on 
indefinitely  according  to  this  same  or  any  similar  law,  will  be  an  asymptotic  branch.  Indeed, 
it  may  be,  as  I  remarked  in  Art.  119,  assumed  that  gravity  is  even  accurately  as  the  inverse 
square,  &  that  it  extends  to  all  distances  according  to  the  same  law,  but  that  in  addition 
there  is  some  other  force  represented  by  another  curve  ;  then  the  law  of  forces  of  Fig.  I 
is  to  be  resolved  into  this  force  &  into  gravity  reckoned  as  being  exactly  as  the  inverse  square 
of  the  distance.  This  force,  then,  at  those  distances,  for  which  gravity  follows  very 
approximately  such  a  law,  will  be  an  insensible  force  ;  but  at  most  other  distances  it  would 
be  very  great.  Where  Fig.  i  gives  repulsions,  the  force  that  is  assumed  to  follow  this  other 
law  would  also  have  to  be  repulsive,  &  greater  than  the  force,  given  by  the  law  of  the  primitive 
curve  of  Fig.  i,  by  an  amount  equal  to  the  supposed  value  of  gravity  at  that  place  ;  &  this 
must  be  cancelled  by  the  addition  of  this  repulsive  force.  However,  this  would  depend 
upon  our  manner  of  assumption  ;  &  in  this  my  own  primitive  &  actual  law,  I  consider  that 
gravity  is  indeed  universal  &  follows  the  law  of  the  inverse  squares  of  the  distances,  although 
not  exactly,  but  very  closely ;  I  consider  that  it  does  not  extend  to  all  distances,  but  only 
to  those  I  have  set  forth. 

400.  For  the  rest,  that  gravity  exists  universally  throughout  the  whole  planetary  Gravity  exists 
system,  I  think  is  thoroughly  demonstrated  by  those  arguments  derived  from  Astronomy  whcTeUsoW°  system6 
which  are  used  by  the  disciples  of  Newton  ;   these  I  do  not  repeat  here,  since  they  are  set  &  it  cannot  possibly 
forth  everywhere  ;  I  too  have  discussed  them  in  several  places,  besides  including  them  in 
Adnotationes  ad  poema  P.  Noceti  De  Aurora  Boreali.     But  I  consider  that  it  is  most  evident 

that  the  approach  to  the  Sun  of  the  comets  &  primary  planets,  &  that  of  the  secondaries 
to  the  primaries,  such  as  we  see  in  the  descent  from  the  rectilinear  tangent  to  the  arc  of 
the  curve,  &  to  a  far  greater  degree  other  motions  depending  on  mutual  gravitation 
cannot  possibly  be  due  to  fluid  pressure.  For,  to  omit  other  reasons  truly  numerous, 
the  fluid  that  can  avail  so  much  in  its  action  on  spheres  of  this  kind  merely  by  its  pressure, 
would  certainly  have  a  much  greater  effect  upon  their  tangential  velocities,  by  its  opposi- 
tion ;  these  would  in  every  case  be  bound  to  be  diminished  by  such  resistance,  with  a  huge 
perturbation  of  areas,-  &  the  perversion  of  the  whole  of  astronomical  mechanics.  Thus 
the  fluid  would  either  be  bound  to  set  up  a  huge  resistance  to 'the  progress  of  a  planet  or 
a  comet,  or  else  it  does  not  even  by  its  pressure  impress  any  sensible  motion  upon  it. 

401.  Now,  the  principal  laws  of  gravitation  are  that  it  varies  directly  as  the  mass  &  Gravitation,  ac 
inversely  as  the  square  of  the  distances  from  each  of  the  points  of  that  mass ;    &  in  my  xheo'ry    varies"^! 
Theory  it  is  quite  clear  that  this  must  be  the  case.     For,  as  soon  as  we  reach  the  arc  of  the   mass  directly 
my  curve  that  represents  gravitation,  all  the  forces  are  attractive,  &  to  all  intents  obey  *f  " 

the  same  law ;    &  so  some  of  them  do  not  cancel  others  in  opposite  directions,  but  their  inversely. 

sum  approximately  corresponds  to  the  number  of  points.     Except  in  so  far  as,  on  account 

of  the  inequality  between  the  distances  of  the  points,  &  their  relative  positions,  there  will 

be  need,  in  order  to  obtain  the  sum  of  the  forces  accurately  when  the  volumes  are  somewhat 

large,  to  make  use  of  the  reduction  usually  employed  by  mechanicians ;  by  the  aid  of  which 

are  found  the  laws  according  to  which  a  point  situated  at  a  given  distance  &  in  a  given  position 

from  a  mass  of  given  shape  is  attracted  by  that  mass.     Here,  as  I  indicated  in  Art.  347, 

one  sphere  gravitates  towards  another  sphere  in  the  manner  that  it  would  if  the  whole 

of  their  masses  were  for  each  condensed  at  their  respective  centres ;  whilst  for  other  figures 

we  meet  with  altogether  different  laws. 

402.  But  the  greatest  support  for  my  Theory   lies  in  a  statement  in  Art.  212,  which  I  j"^0/^  ^  me 
said  ought  to  be  noticed  ;  namely,  in  the  fact  that  we  see  so  much  uniformity  in  all  masses  Theory    from    the 
with  regard  to  the  force  of  gravity ;    in  spite  of  the  fact  that  these  same  masses,  for  the  conformity   of   ail 

,•       i  i  6    i       '     v          r       i  11         T  i  vrr  bodies     in     having 

purpose  of  other  phenomena  depending  on  the  smaller  distances  apart,  have  differences  gravitation,  whilst 
so  great  as  those  possessed  by  different  bodies  as  regards  hardness,  colour,  taste,  smell  &  there  are  so  many 

„  ,.£  /.         .  .    .  .  .     .  11       vrr  differences  in  other 

sound.     For,  a  different  combination  of  the  points  of  matter  induces  totally  different  sums  properties. 

for  those  distances  up  to  which  the  curve  of  forces  still  twists  about  the  axis  ;  where  a  very 

slight  change  in  the  distances  changes  attractive  forces  into  repulsive,  &  substitutes,  vice  versa, 

differences  for  sums.     Whereas,  at  those  distances  for  which  gravity  obeys  the  laws  we 

have  stated  very  approximately,  the  curve  has  its  ordinates  all  in  the  same  direction  &, 

even  if  the  distance  is  slightly  altered,  practically  unaltered  in  length.     This  of  necessity  Neariv  everything 

produces  a  huge  difference  in  the  former  case,  &  a  very  great  uniformity  in  the  latter.       depending  on  gray- 

403.  The  distinction  between  gravitation  (which  is  proportional  to  the  mass  on  which  |nyaOT^ement°with 
it  acts,  directly,  &  as  the  square  of  the  distance,  inversely)  &  weight  (which  is,  in  addition,  the  usual  theory : 
proportional  to  the  mass  causing  the  gravitation)  is  just  the  same  in  my  Theory  as  in  that  Oj 

of  Newton  &  all  mechanicians.     The  former  gives  the  accelerating  force,  the  latter  the  motive  easier  in  mine. 

u 


290 


PHILOSOPHIC  NATURALIS   THEORIA 


motricem,  cum  ilia  determinet  vim  puncti  gravitantis  cujusvis,  a  qua  pendet  celeritas 
massae  ;  [184]  hoc  summam  virium  ad  omnia  ejusmodi  puncta  pertinentium.  Pariter 
communia  mihi  sunt,  quaecunque  pertinet  ad  gravium  motus  a  Galilaeo,  &  Hugenio  definitos, 
nisi  quod  gravitatis  resolutionem  in  descensu  per  plana  inclinata,  &  in  gravibus  sustentatis 
per  bina  obliqua  plana,  vel  obliqua  fila,  reducam  ad  compositionem  juxta  num.  284,  &  286, 
&  centrum  oscillationis,  una  cum  centro  Eequilibrii,  &  vecte,  &  libra,  &  machinarum  principiis 
deducam  e  consideratione  systematis  trium  massarum  in  se  mutuo  agentium,  ac  potissimum 
a  simplici  theoremate  ad  id  pertinente,  quae  fuse  persecutus  sum  a  num.  307.  Communia 
pariter  mihi  sunt,  quaecunque  habentur  in  caelesti  Newtoniana  Mechanica  jam  ubique 
recepta  de  planetarum,  &  cometarum  motibus,  de  perturbationibus  motuum  potissimum 
Jovis,  &  Saturni  in  distantiis  minoribus  a  se  invicem,  de  aberrationibus  Lunae,  de  maris 
aestu,  de  figura  Telluris,  de  praecessione  aequinoctiorum,  &  nutatione  axis  ;  quin  immo 
ad  hasc  postrema  problemata  rite  solvenda,  multo  tutior,  &  expeditior  mihi  panditur  via, 
quae  me  eo  deducet  post  considerationem  systematis  massarum  quatuor  jacentium  etiam 
non  in  eodem  piano  communi,  &  connexarum  invicem  per  vires  mutuas,  uti  ad  centrum 
oscillationis  etiam  in  latus  in  eodem  piano,  &  ad  centrum  percussionis  in  eadem  recta  tarn 
facile  me  deduxit  consideratio  systematis  massarum  trium. 


cetur. 


immobiiitas  fix  a-  404.  Illud  mihi  prseterea  non  est  commune,  quod  pertinet  ad  immobilitatem  stellarum 
iia  fixarum,  quam  contra  generalem  Newtoni  gravitatem  vulgo  solent  objicere,  quae  nimirum 
debeant  ea  attractione  mutua  ad  se  invicem  accedere,  &  in  unicam  demum  coire  massam. 
Respondent  alii,  Mundum  in  infmitum  protendi,  &  proinde  quamvis  fixam  aeque  in  omnes 
partes  trahi.  Sed  in  actu  existentibus  infmitum  absolutum,  ego  quidem  censeo,  haberi 
omnino  non  posse.  Recurrent  alii  ad  immensam  distantiam,  quae  non  sinat  motum  in 
fixis  oriundum  a  vi  gravitatis,  n-e  post  immanem  quidem  saeculorum  seriem  sensu  percipi. 
li  in  eo  verum  omnino  affirmant  ;  si  enim  concipiamus  fixas  Soli  nostro  aequales  &  similes, 
vel  saltern  rationem  luminum,  quae  emittunt,  non  multum  discedere  a  ratione  massarum  ; 
quoniam  &  vis  ipsis  massis  proportionalis  est,  ac  praeterea  tarn  vis,  quam  lumen  decrescit 
in  ratione  reciproca  duplicata  distantiarum  ;  erit  vis  gravitatis  nostri  Solaris  systematis 
in  omnes  Stellas,  ad  vim  gravitatis  nostrae  in  Solem,  quae  multis  vicibus  est  minor,  quam 
vis  gravitatis  nostrorum  gravium  in  Terram,  ut  est  tota  lux,  quae  provenit  a  fixis  omnibus, 
ad  lucem,  quae  provenit  a  Sole,  quae  ratio  est  eadem,  ac  ratio  noctis  ad  diem  in  genere  lucis. 
Quam  exiguus  motus  inde  consequi  possit  eo  tempore,  cujus  temporis  ad  nos  devenire 
potuit  notitia,  nemo  non  videt.  Si  fixae  omnes  ad  eandem  etiam  jaceant  plagam,  is  motus 
omnino  haberi  posset  pro  nullo. 


Difficuitas  residua  405.  Adhuc  tamen,  quoniam  nostra  vita,  &  memoria  respectu  immensi  fortasse  subse- 
subiata  ab  hac  cuturi  sevi  est  itidem  fere  nihil  ;  [185]  si  gravitas  generalis  in  infmitum  protendatur  cum 
eadem  ilia  lege,  &  eodem  asymptotico  crure,  utique  non  solum  hoc  systema  nostrum  solare, 
sed  universa  corporea  natura  ita,  paullatim  utique,  sed  tamen  perpetuo  ab  eo  statu  recederet, 
in  quo  est  condita,  &  universa  ad  interitum  necessario  rueret,  ac  omnis  materia  deberet 
demum  in  unicam  informem  massam  conglobari,  cum  fixarum  gravitas  in  se  invicem,  nullo 
obliquo,  &  curvilineo  motu  elidatur.  Id  quidem  haud  ita  se  habere,  demonstrari  omnino 
non  potest  ;  adhuc  tamen  Divinae  Providentise  videtur  melius  consulere  Theoria,  quae 
ejus  etiam  ruinse  universalis  evitandae  viam  aperiat,  ut  aperit  sane  mea.  Fieri  enim  potest, 
uti  notavimus  n.  170,  ut  postremus  ille  curvae  meae  arcus,  qui  exhibet  gravitatem,  posteaquam 
recesserit  ad  distantias  majores,  quam  sint  cometarum  omnium  ad  nostrum  solare  systema 
pertinentium  distantiae  maximae  a  Sole,  incipiat  recedere  plurimum  ab  hyperbola  habente 
ordinatas  reciprocas  quadratorum  distantiae,  ac  iterum  axem  secet,  &  contorqueatur.  Eo 
pacto  posset  totum  aggregatum  fixarum  cum  Sole  esse  unica  particula  ordinis  superioris 
ad  eas,  quae  hoc  ipsum  systema  componunt,  &  pertinere  ad  systema  adhuc  in  immensum 
majus  &  fieri  posset  ut  plurimi  sint  ejus  generis  ordines  particularum  ejusmodi  etiam, 
ut  ejusdem  ordinis  particulae  sint  penitus  a  se  invicem  segregatae  sine  ullo  possibili 
commeatu  ex  una  in  aliam  per  asymptoticos  arcus  plures  meae  curvae  juxta  ea,  quae 
exposui  a  num.  171. 


Cohiesio  :  expiicatio  .Qg    pjoc   pacto   difficultas    qu33   a    necessario  fixarum   accessu  repetebatur   contra 

per   quietem,  vel    _T       T      ,  Jf          .  .  T.  .  .  .  .  . 

motus  conspirantes.  Newtomanam  T  heonam,  in  mea  penitus  evanescit  ac  simul  a  gravitate  jam  gradum  fecimus 
ad  cohaesionem,  quam  ex  generalibus   materiae   proprietatibus  posueram  postremo  loco. 


A  THEORY  OF  NATURAL  PHILOSOPHY  291 

force  ;  since  the  former  gives  the  force  of  any  gravitating  point,  upon  which  depends 
the  velocity  of  the  mass,  &  the  latter  the  sum  of  all  the  forces  pertaining  to  all  such  points. 
Similarly,  the  agreement  is  the  same  in  my  Theory  with  regard  to  anything  relating  to 
the  motions  of  heavy  bodies  stated  by  Galileo  &  Huygens ;  except  that,  in  descent  along 
inclined  planes,  or  bodies  supported  by  two  inclined  planes  or  inclined  strings,  I  substitute 
for  their  resolution  of  gravity  the  principle  of  composition,  as  in  Art.  284,  286  ;  &  I  deduce 
the  centre  of  oscillation,  as  well  as  the  centre  of  equilibrium,  the  lever,  the  balance  &  the 
principles  of  machines  from  a  consideration  of  three  masses  acting  mutually  upon  one  another ; 
&  this  more  especially  by  means  of  a  simple  theorem  depending  on  that  consideration, 
which  I  investigated  fully  in  Art.  307.  The  agreement  is  just  as  close  in  my  Theory  with 
regard  to  anything  occurring  in  the  celestial  mechanics  of  Newton,  now  universally  accepted, 
with  regard  to  the  motions  of  planets  &  comets,  particularly  the  perturbations  of  the  motions 
of  Jupiter  &  Saturn  when  at  less  than  the  average  distances  from  one  another,  the  aberrations 
of  the  Moon,  the  flow  of  the  tides,  the  figure  of  the  Earth,  the  precession  of  the 
equinoxes,  &  the  nutation  of  the  axis.  Finally,  for  the  correct  solution  of  these  latter 
problems,  a  much  safer  &  more  expeditious  path  is  opened  to  me,  such  as  will  lead  me 
to  it  after  an  investigation  of  the  system  of  four  masses,  not  even  lying  in  the  same  common 
plane,  connected  together  by  mutual  forces ;  just  as  the  consideration  of  a  system  of  three 
masses  led  me  with  such  ease  to  the  centre  of  oscillation  even  to  one  side  in  the  same 
plane,  &  to  the  centre  of  percussion  in  the  same  straight  line. 

404.  In  addition  to  these,  there  is  one  thing  in  which  I  do  not  agree,  namely,  in  that  The  manner  in 
which  relates  to  the  immobility  of  the  fixed  stars ;   it  is  usually  objected  to  the  universal  which  the   immo- 
gravitation  of  Newton,  that  in  accordance  with  it  the  fixed  stars  should  by  their  mutual  starJwas  explained 
attraction  approach  one  another,  &  in  time  all  cohere  into  one  mass.     Others  reply  to  this,  by  Newton, 
that  the  universe  is  indefinitely  extended,  &  therefore  that  any  one  fixed  star  is  equally 

drawn  in  all  directions.  But  in  things  that  actually  exist,  I  consider  that  it  is  totally  impossible 
that  there  can  be  any  absolute  infinity.  Others  fall  back  on  the  immense  distance,  which 
they  say  will  not  permit  the  motion  arising  in  the  fixed  stars  from  the  force  of  gravity  to 
be  perceived  by  the  senses,  even  after  an  immense  number  of  ages.  In  this  they  assert 
nothing  but  the  truth  ;  for  if  we  consider  the  fixed  stars  equal  &  similar  to  our  sun,  or 
at  any  rate  the  amounts  of  light  that  they  emit,  as  not  being  far  different  from  the  ratio 
of  their  masses ;  then  since  also  the  force  is  proportional  to  the  masses,  &  in  addition  both 
force  &  light  decrease  in  the  inverse  ratio  of  the  squares  of  the  distances,  it  must  be  that 
the  force  of  gravity  of  our  solar  system  on  all  the  stars  is  to  the  force  of  our  gravity  on  the 
Sun,  which  latter  is  many  times  less  than  the  force  of  gravity  of  our  heavy  bodies  on  the 
Earth,  as  the  total  light  which  comes  from  all  the  stars  is  to  the  light  which  comes  from 
the  Sun  ;  &  this  ratio  is  the  same  as  the  ratio  of  night  to  day  in  respect  of  light.  How 
slight  is  the  motion  that  can  arise  from  this  in  the  time  (the  comparatively  short  time 
available  for  observation)  nobody  can  fail  to  see.  Even  if  all  the  fixed  stars  were 
situated  in  the  same  direction,  the  motion  could  be  considered  as  absolutely  nothing. 

405.  However,  since  our  period  of  life  &  memory,  in  comparison  with  the  immense  The  remaining  diffi- 
number  of    ages  perchance  to    follow,  is  almost    as    nothing,  if    universal    gravitation  th'is^heory"™7 " 
extends  indefinitely  with  the  same  law,  &  the  same  asymptotic  branch,  not  only  this  solar 

system  of  ours  indeed,  but  the  universe  of  corporeal  nature,  would,  little  by  little  in  truth, 
but  still  continuously,  recede  from  the  state  in  which  it  was  established,  &  the  universe 
would  necessarily  fall  to  destruction  ;  all  matter  would  in  time  be  conglomerated  into  one 
shapeless  mass,  since  the  gravity  of  the  fixed  stars  on  one  another  will  not  be  cancelled  by 
any  oblique  or  curvilinear  motion.  That  this  is  not  the  case  cannot  be  absolutely  proved  ; 
&  yet  a  Theory  which  opens  up  a  possible  way  to  avoid  this  universal  ruin,  in  the  way  that 
my  Theory  does,  would  seem  to  be  more  in  agreement  with  the  idea  of  Divine  Providence. 
For  it  may  be  that,  as  I  remarked  in  Art.  170,  the  last  arc  of  my  curve,  which  represents 
gravity,  after  it  has  reached  distances  greater  than  the  greatest  distances  from  the  Sun  of 
all  the  comets  that  belong  to  our  solar  system,  will  depart  very  considerably  from  the  hyperbola 
having  its  ordinates  the  reciprocals  of  the  squares  of  the  distances,  &  once  more  will  cut 
the  axis  &  be  twined  about  it.  In  this  way,  it  may  be  that  the  whole  aggregate  of  the 
fixed  stars,  together  with  the  Sun,  is  a  single  particle  of  an  order  higher  than  those  of 
which  the  system  is  composed  ;  &  that  it  belongs  to  a  system  immensely  greater  still.  It 
may  even  be  the  case  that  there  are  very  many  such  orders  of  particles,  of  such  a  kind  that 
particles  of  the  same  class  are  completely  separated  from  one  another  without  any 
possible  means  of  getting  from  one  to  the  other,  owing  to  several  asymptotic  arcs  to  my 
curve,  as  I  explained  in  Art.  171. 

406.  In  this  way,  the  difficulty, which  has  been  repeatedlybrought  against  the  Newtonian  Cohesion ;  expiana- 
theory  on  account  of  this  necessary  mutual  approach  of  the  fixed  stars,  disappears  altogether  ^"orof  motions  ?n 
in  my  Theory.     At  the  same  time,  we  have  now  passed  on  from  gravity  to  cohesion,  which  the  same  direction. 


292  PHILOSOPHISE  NATURALIS  THEORIA 

Cohsesionem  explicuerunt  aliqui  per  puram  quietam  ut  Cartesian!  alii  per  motus  conspir- 
antes,  ut  Joannes  Bernoullius,  ac  Leibnitius,  quam  explicationem  illustrarunt  exemplo 
illius  veli  aquse,  quod  in  fontibus  quibusdam  cernimus,  quod  velum  sit  tantummodo  ex 
conspirante  motu  guttularum  tenuissimarum,  &  tamen  si  quis  digito  velit  perrumpere,  eo 
majorem  resistentiam  sentit,  quo  velocitas  aquae  effluentis  est  major,  ut  idcirco  multo 
adhuc  major  conspirantis  motus  velocitas  videatur  nostrorum  cohsesionem  corporum 
exhibere,  quae  non  nisi  immani  vi  confringimus,  ac  in  partes  dividimus.  Utraque  explicandi 
ratio  eodem  redit,  si  quietis  nomine  intelligatur  non  utique  absoluta  quies,  quse  translata 
Tellure  a  Cartesianis  nequaquam  admittebatur,  sed  respectiva  :  nam  etiam  conspirantes 
motus  nihil  sunt  aliud,  nisi  quies  respectiva  illarum  partium,  quse  conspirant  in  motibus. 


nias  exponere  407.  At  neutra  earn  explicat,  quam  cohsesionem  reipsa  dicimus,  sed  cohsesionis  quendam 

velut  effectum.  Ea,  quse  cohserent,  utique  respective  quiescunt,  sive  motus  conspirantes 
habent,  &  id  quidem  ipsum  in  hac  mea  Theoria  accidit  [186]  itidem,  in  qua  cum  singula 
puncta  materiae  suam  pergant  semper  eandem  continuam  curvam  describere,  ea,  quse 
cohserent  inter  se,  toto  eo  tempore,  quo  cohserent,  arcus  habent  curvarum  suarum  inter 
se  proximos,  &  in  arcubus  ipsis  conspirantes  motus.  Sed  in  iis,  quse  cohaerent,  id  ipsum, 
quod  motus  ibi  sint  conspirantes,  non  est  sine  causa  pendente  a  mutuis  eorum  viribus,  quse 
causa  impediat  separationem  alterius  ab  altero,  ac  in  ea  ipsa  causa  stat  discrimen  cohaeren- 
tium  a  contiguis.  Si  duo  lapides  in  piano  horizontali  jaceant,  utique  habent  motum 
conspirantem,  quern  circa  Solem  habet  Tellus ;  sed  si  tertius  lapis  in  alterutrum  incurrit, 
vel  ego  ipsum  submoveo  manu,  statim  sine  ulla  vi  mutua,  quae  separationem  impediat, 
dividuntur,  &  motus  desinit  esse  conspirans.  Hanc  ipsam  quasrimus  causam,  dum  in 
cohaesionem  inquirimus.  Nee  velocitas  motus,  &  exemplum  veli  aquse  rem  conficit. 
Motus  conspirans  duorum  lapidum  contiguorum  cum  tota  Tellure  est  utique  multo  velocior, 
quam  motus  particularum  aquse  proveniens  a  gravitate  in  illo  velo,  &  tamen  sine  ullo, 
difficultate  separantur.  In  aqua  experimur  difficultatem  perrumpendi  velum,  quia  ilia 
motus  conspirans  non  est  communis  etiam  nobis  &  Telluri,  ut  est  motus  illorum  lapidum  ; 
unde  fit,  ut  vis,  quam  pro  separatione  applicamus  singulis  particulis,  perquam  exiguo 
tempore  possit  agere,  &  ejus  effectus  citissime  cesset,  iis  decidentibus,  &  supervenientibus 
semper  novis  particulis,  quse  cum  tota  sua  ingenti  respectiva  velocitate  incurrunt  in  digitum. 
At  in  corporibus,  in  quibus  partes  cohaerentes  cernimus,  eae  partes  nullam  habent  veloci- 
tatem  respectivam  respectu  nostri,  nee  alise  succedunt  aliis  fugientibus.  Quamobrem 
longe  aliter  in  iis  se  res  habet,  &  oportet  invenire  causam  longe  aliam,  prseter  ipsum  solum 
conspirantem  motum,  ut  explicetur  difficultas,  quam  experimur  in  iis  separandis,  &  in 
inducendo  motu  non  conspirante. 


Expiicatio     petita  408.  Sunt,  qui  adducant  pressionem  fluidi  cujuspiam    tenuissimi,  uti  pressio  atmo- 

cur^aThtoeri^non  sphseras  extracto  acre  ex    hemisphaeriis   etiam  vacuis  ipsorum   separationem  impedit  vi 
possit.  respondente  ponderi  ipsius  atmosphaerse,  quse  vis  cum  in  vulgaribus  cohassionibus,  &  vero 

etiam  in  hemisphaeriis  bene  ad  se  invicem  adductis,  sit  multis  vicibus  major,  quam  pondus 
atmosphserse  ipsius,  quod  se  prodit  in  suspensione  mercurii  in  barometris ;  aliud  auxilio 
advocant  tenuius  fluidum.  At  inprimis  ejus  fluidi  hypothesis  precaria  est ;  deinde  hue 
illud  redit,  quod  supra  etiam  monui,  ubi  de  gravitatis  causa  egimus,  quod  nimirum  meo 
quidem  judicio  explicari  nullo  modo  possit,  cur  illud  fluidum,  quod  sola  pressione  tantum 
potest,  nihil  omnino  ad  sensum  possit  incursu  suo  contra  celerrimos  planetarum,  & 
cometarum  motus.  Accedit  etiam,  quod  distractio  &  compressio  fibrarum,  quae  habetur 
ante  fractionem  solidorum  corporum,  ubi  franguntur  appenso  inferne,  vel  superne  imposito 
[187]  pondere  ingenti,  non  ita  bene  cum  ea  sententia  conciliari  posse  videatur. 


ExpUcatio    New-  409.  Newtonus  adhibuit  ad  earn  rem  attractionem  diversam  ab  attractione  gravitatis, 

iiTmtnimU^istM6  quanquam  is  quidem  videtur  earn  repetere  itidem  a  tenuissimo  aliquo  fluido  comprimente  ; 

tils :    cur  admitti  repetit  certe  sub  finem  Opticse  a  spiritu  quodam  intimas  corporum  substantias  penetrante, 

cujus  spiritus  nomine  quid  intellexerit,  ego  quidem  nunquam  satis  assequi  potui ;    cujus 


A  THEORY  OF  NATURAL  PHILOSOPHY  293 

I  had  put  in  the  last  place  amongst  the  general  properties  of  matter.  Some  have  explained 
cohesion  from  the  idea  of  absolute  rest,  for  instance,  the  Cartesians ;  others,  like  Johann 
Bernoulli,  &  Leibniz,  by  means  of  equal  motions  in  the  same  direction.  They  illustrate 
the  explanation  by  means  of  the  film  of  water,  which  we  see  in  certain  fountains ;  this 
film  is  formed  merely  from  the  equal  motions  in  the  same  direction  of  the  tiniest  little 
drops,  &  yet,  if  anyone  tries  to  break  it  with  his  finger,  he  feels  a  resistance  that  is  the 
greater,  the  greater  the  velocity  of  the  effluent  water  ;  so  that  from  this  illustration  it 
would  seem  that  a  far  greater  velocity  of  equal  motion  in  the  same  direction  would  account  for 
the  cohesion  of  the  bodies  around  us,  which  we  cannot  fracture  &  divide  up  into  parts 
unless  we  use  a  huge  force.  Either  of  these  methods  of  explaining  the  matter  reduces  to 
the  same  thing,  if  by  the  term  '  rest '  we  understand  not  only  absolute  rest  which,  since 
the  Earth  is  in  motion,  has  in  no  sense  been  admitted  by  the  Cartesians,  but  also  relative 
rest.  For,  equal  motions  in  the  same  direction  are  nothing  else  but  the  relative  rest  of  the 
parts  that  have  equal  motions  in  the  same  direction. 

407.  Neither  of  these  ideas  explains  that  which  we  call  cohesion  in  a  real  sense,  but  ^ut  these  methods 
only  an  effect  of  cohesion.     Things  which  cohere  are  certainly  relatively  at  rest ;  or  they  effect  6&P  an"t  the 
have  equal  motions  in  the  same  direction.     This  is  exactly  what  happens  in  my  Theory  also ;  cause  of  cohesion, 
for,  in  it,  since  each  point  of  matter  always  keeps  on  describing  the  same  continuous  curve 

which  is  peculiar  to  itself,  those  points  that  cohere  to  one  another,  during  the  whole  of  the 
time  in  which  they  cohere,  have  the  arcs  of  their  respective  curves  very  near  to  one  another, 
&  the  motions  in  those  arcs  equal  &  in  the  same  direction.  But  in  points  that  cohere, 
the  fact  that  their  motions  are  then  equal  &  in  the  same  direction  is  not  without  a  cause ; 
&  this  depends  on  their  mutual  forces,  which  prevent  separation  of  one  point  from  another  ; 
&  in  this  cause  is  involved  the  difference  between  cohering  &  contiguous  points.  If  two 
stones  lie  in  the  same  horizontal  plane,  they  will  in  all  cases  have  equal  motions  in  the  same 
direction  as  the  Earth  has  round  the  Sun  ;  but  if  a  third  stone  strikes  against  either  of  them,  or 
if  I  move  this  third  stone  up  to  the  others  with  my  hand,  immediately,  without  any  mutual 
force  preventing  separation,  the  two  are  divided,  &  the  equal  motion  in  the  same  direction 
comes  to  an  end.  This  cause  of  cohesion  is  just  what  we  want  to  find,  when  we  seek  to  investi- 
gate cohesion  ;  &  velocity  of  motion,  or  the  example  of  the  film  of  water  will  not  effect  the 
solution.  The  equal  motions  in  the  same  direction  as  the  whole  Earth,  possessed  by  the  two 
contiguous  stones,  is  certainly  much  greater  than  the  motions  of  the  particles  of  water 
produced  by  gravity  in  the  film  ;  &  yet  the  two  stones  can  be  separated  without  any  difficulty. 
In  the  water  we  encounter  a  difficulty  in  breaking  the  film,  because  the  equal  motion  in  the 
one  direction  is  not  common  to  us  &  the  Earth,^as  the  motion  of  the  stones  is.  Hence  it 
comes  about  that  the  force,  which  we  apply  to  separate  the  several  particles,  can  only  act  for  an 
exceedingly  small  interval  of  time  ;  &  the  effect  of  this  force  ceases  very  quickly,  as  those 
particles  continually  fall  away  &  fresh  ones  come  up  ;  &  these  strike  the  finger  with  the  whole 
of  their  relatively  huge  velocity.  But,  in  bodies  in  which  we  perceive  coherent  parts,  those 
parts  have  no  relative  velocity  with  regard  to  ourselves,  nor  as  one  part  flies  off  does 
another  take  its  place.  Therefore  the  matter  has  to  be  explained  in  a  totally  different 
manner ;  &  we  must  find  a  totally  different  cause  to  the  idea  of  mere  equality  of  motion  in 
the  same  direction,  in  order  to  solve  the  difficulty  that  is  experienced  in  separating  the 
parts  &  inducing  in  them  motions  that  are  not  equal  &  in  the  same  direction. 

408.  There  are  some  who  bring  forward  the  pressure  of  some  fluid  of  very  small  density  Explanation  sought 

i          .  i  ,  ,.,  11  -II  j    fr°m  flul<i  pressure ; 

as  an  explanation.     Just  as  the  pressure  of  the  atmosphere,  when  the  air  has  been  abstracted  why  it  is  impossible 
from  a  pair  of  hollow  hemispheres,  prevents  them  from  being  separated  with  a  force  that  this  should  be 

r  , .  •    i         f    i  •  i  •      <•  •  i  •  i-  the  case. 

corresponding  to  the  weight  of  the  atmosphere  ;  &,  since  this  force  in  ordinary  cohesions, 
&  indeed  also  in  the  case  of  two  hemispheres  that  fit  one  another  very  well,  becomes  many 
times  greater  than  the  weight  of  the  atmosphere,  as  shown  in  the  suspension  of  mercury 
in  the  barometer,  they  invoke  the  aid  of  another  fluid  of  less  density.  But,  first  of  all, 
the  hypothesis  of  such  a  fluid  is  uncertain  ;  next,  there  here  arises  the  same  objection  that 
I  remarked  upon  above,  when  discussing  the  cause  of  gravity.  Namely,  that,  in  my  opinion 
no  manner  of  reason  could  be  given  as  to  why  this  fluid,  which  by  its  mere  pressure  could 
produce  so  great  an  effect,  had  as  far  as  observation  could  discern  absolutely  no  effect  on 
the  swiftest  motions  of  planets  &  comets,  owing  to  impact  with  them.  Also  there  is  this 
point  in  addition,  that  the  extension  &  compression  of  fibres,  which  takes  place  before 
fracture  in  solid  bodies,  when  they  are  broken  by  hanging  a  weight  beneath  or  by  setting 
a  weight  on  top  of  them,  does  not  seem  to  be  in  much  conformity  with  this  idea. 

409.  Newton  derived  an  explanation  of  the  matter  from  an  attraction  of  a  different  The  reason  why  it 
kind  to  gravitation  ;  although  he  indeed  seems  to  seek  to  obtain  this  attraction  from  some  j^m}™]i£|s^pia*a° 
compressing  fluid  of  very  small  density.     In  fact,  he  seeks  to  obtain  it,  at  the  end  of  his  tion  from  attraction 
Optics,  from  a  '  spirit '  permeating  the  inmost  substances  of  bodies ;   but  I  never  was  able  tances^a 

to  grasp  clearly  what  he  intended  by  the  term  '  spirit ' ;    &  even  he  confessed  that  the  Newton. 


294  PHILOSOPHIC  NATURALIS  THEORIA 

quidem  agendi  modum  &  sibi  incognitum  esse  profitetur.  Is  posuit  ejusmodi  attractionem 
imminutis  distantiis  crescentem  ita,  ut  in  contactu  sit  admodum  ingens,  &  ubi  primigeniae 
particulae  se  in  planis  continuis,  adeoque  in  punctis  numero  infinitis  contingant,  sit  infinities 
major,  quam  ubi  particulae  primigeniae  particulas  primigenias  in  certis  punctis  numero 
finitis  contingant,  ac  eo  minor  sit,  quo  pauciores  contactus  sunt  respectu  numeri  particu- 
larum  primigeniarum,  quibus  constant  particulae  majores,  quae  se  contingunt,  quorum 
contactuum  numerus  cum  eo  sit  minor,  quo  altius  ascenditur  in  ordine  particularum  a 
minoribus  particulis  compositarum,  donee  deveniatur  ad  hsec  nostra  corpora  ;  inde  ipse 
deducit,  particulas  ordinum  altiorum  minus  itidem  tenaces  esse,  &  minime  omnium  haec 
ipsa  corpora,  quae  malleis,  &  cuneis  dividimus.  At  mihi  positiva  argumenta  sunt  contra 
vires  attractivas  crescentes  in  infinitum,  ubi  in  infinitum  decrescant  distantise,  de  quibus 
mentionem  feci  num.  126;  &  ipsa  meae  Theoriae  probatio  evincit,  in  minimis  distantiis 
vires  repulsivas  esse,  non  attractivas,  ac  omnem  immediatum  contactum  excludit  :  quam- 
obrem  alibi  ego  quidem  cohaesionis  rationem  invenio,  quam  mea  mihi  Theoria  sponte 
propemodum  subministrat. 

Cohaesionem   repe-  410.  Cohaesio  mihi  est  igitur  iuxta  num.  165  in  iis  virium  limitibus,  in  quibus  transit ur 

tendam  a  limitibus  i  •         •  •         •  «          j-  ••         j  -L  o     i_  -j 

virium.  a  vl  repulsiva  in  minoribus  distantiis,  ad  attractivam  in  majonbus  ;    &  haec  quidem  est 

cohaesio  inter  duo  puncta,  qua  fit,  ut  repulsio  diminutionem  distantise  impediat,  attractio 
incrementum,  &  puncta  ipsa  distantiam,  quam  habent,  tueantur.  At  pro  punctis  pluribus 
cohassio  haberi  potest,  turn  ubi  singula  binaria  punctorum  sunt  inter  se  in  distantiis  limitum 
cohaesionum,  turn  ubi  vires  oppositae  eliduntur,  cujusmodi  exemplum  dedi  num.  223. 

Cohassio  duorum  411.  Porro  quod  ad  ejusmodi  cohaesionem  pertinet,   multa  ibi  sunt  notatu  digna. 

fes^o^aesionTs  Inprimis  ubi  agitur  de  binis  punctis,  tot  diversae  haberi  possunt  distantiae  cum  cohaesione, 
posse  esse  quot-  quot  exprimit  numerus  intersectionum  curvae  virium  cum  axe  unitate  auctus,  si  forte  sit 
fnrtp*6'  mm™™!!!!  impar,  ac  divisus  per  duo.  Nam  primus  quidem  limes,  in  quo  curva  ab  arcu  asymptotico 

ceb,      quo    inque    ...  *  .  .         r  ,   .        ..          r.  *     ....  i  -i  -i  •  i  • 

ordine  positos.  mo  pnmo,  sive  a  repulsionibus  impenetrabilitatem  exmbentibus  transit  ad  primum 
attractivum  arcum,  est  limes  cohaesionis,  &  deinde  alterni  intersectionum  limites  sunt  non 
cohaesionis,  &  coha2-[i88]sionis,  juxta  num.  179;  unde  fit,  ut  si  intersectionum  se  conse- 
quentium  assumatur  numerus  par  ;  dimidium  sit  limitum  cohaesionis.  Hinc  quoniam 
in  solutione  problematis  expositi  num.  117  ostensum  est,  curvam  simplicem  illam  meam 
habere  posse  quemcunque  demum  intersectionum  numerum  ;  poterit  utique  etiam  pro 
duobus  tantummodo  punctis  haberi  quicunque  numerus  distantiarum  differentium  a  se 
invicem  cum  cohaesione.  Poterunt  autem  ejusmodi  cohaesiones  ipsae  esse  diversissimae 
a  se  invicem  soliditatis,  ac  nexus,  limitibus  vel  validissimis,  vel  languidissimis  utcunque, 
prout  nimirum  ibi  curva  secuerit  axem  fere  ad  perpendiculum,  &  longissime  abierit,  vel 
potius  ad  ilium  inclinetur  plurimum,  &  parum  admodum  discedat ;  nam  in  priore  eorum 
casuum  vires  repulsivae  imminutis,  &  attractivae  auctis  utcunque  parum  distantiis,  ingentes 
erunt ;  in  posteriore  plurimum  immutatis,  perquam  exiguae.  Poterunt  autem  etiam  e 
remotioribus  limitibus  aliqui  esse  multo  languidiores,  &  alii  multo  validiores  aliquibus  e 
propioribus ;  ut  idcirco  cohaesionis  vis  nihil  omnino  pendeat  a  densitate,  sed  cohaesio 
possit  in  densioribus  corporibus  esse  vel  multo  magis,  vel  multo  minus  valida,  quam  in 
rarioribus,  &  id  in  ratione  quacunque. 


in  massis  numerus          412.  Quae  de  binis  punctis  sunt  dicta,  multo  magis  de  massis  continentibus  plurima, 

ma^oVtU%robiema  puncta,  dicenda  sunt.     In  iis  numerus  limitum  est  adhuc  major  in  immensum,  &  discrimen 

pro  Us  inveniendis  utique  majus.     Inventio  omnium  positionum  pro  dato  punctorum  numero,  in  quibus  tota 

quomodo  solve  n-  massa   haberet  limitem   quendam  virium,   esset   problema   molestum,   &  calculus  ad  id 

solvendum   necessarius    in    immensum    excresceret,    existente   aliquo    majore    punctorum 

numero.     Sed  tamen  data  virium  lege  solvi  utique  posset.     Satis  esset  assumere  positiones 

omnium  punctorum  respectu  cujusdam  puncti  in  quadam  arbitraria  recta  ad  arbitrium 

collocati,  &  substitutis  singulorum  binariorum  distantiis  a  se  invicem  in  aequatione  curvae 

primae  pro  abscissa,  ac  valoribus  itidem  assumptis  pro  viribus  singulorum  punctorum  pro 

ordinatis,  eruere  totidem  aequationes,  turn  reducere  vires  singulas  singulorum  punctorum 

ad    tres    datas    directiones,    &    summam    omnium    eandem    directionem    habentium    in 

quovis  puncto  ponere  =  o  :     orirentur  aequationes,   quae    paullatim   eliminatis   valoribus 

incognitis  assumptis,  demum  ad  aequationes  perducerent  definientes  punctorum  distantias 

necessarias  ad  aequilibrium,  &  respectivam  quietem,  quae  altissimae  essent,  &  plurimas 


A  THEORY  OF  NATURAL  PHILOSOPHY  295 

mode  of  action  was  unknown  to  him.  He  supposed  that  there  was  such  an  attraction, 
which,  as  the  distances  were  diminished,  increased  in  such  a  manner  that  at  contact  it 
was  exceedingly  great ;  &  when  the  primary  particles  touched  one  another  along  continuous 
planes,  &  thus  in  an  infinite  number  of  points,  this  attraction  became  infinitely  greater 
than  when  primary  particles  touched  primary  particles  in  a  definite  finite  number  of  points  ; 
&  the  less  the  number  of  contacts  compared  with  the  number  of  primary  particles  forming 
the  larger  particles  which  touch  one  another,  the  less  the  attraction  becomes ;  &  since 
the  number  of  these  contacts  becomes  smaller  the  higher  we  go  in  the  orders  of  particles 
formed  from  smaller  particles,  he  deduces  from  this  that  particles  of  higher  orders  are 
also  of  less  tenacity,  &  the  least  tenacious  of  all  are  those  bodies  that  we  can  divide  with 
mallet  &  wedge.  But  in  my  opinion  there  are  positive  arguments  against  attractive  forces 
increasing  indefinitely,  when  the  distances  decrease  indefinitely,  as  I  remarked  in  Art.  126 ; 
the  very  demonstration  of  my  Theory  gives  convincing  proof  that  the  forces  at  very  small 
distances  are  repulsive,  not  attractive,  &  ezcludes  all  immediate  contact.  So  that  I  find 
the  cause  of  cohesion  from  other  sources ;  &  my  Theory  supplies  me  with  this  cause  almost 
spontaneously. 

410.  Cohesion,  then,  in  rny  opinion  is,  as  I  have  said   in  Art.  165,  to  be  ascribed  to  Cohesion  is  to  be 
the  limit-points  on  the  curve  of  forces,  where  there  is  a  passage  from  a  repulsive  force  at  umit^oints^oiAhe 
a  smaller  distance  to  an  attractive  force  at  a  greater  distance  ;    that  is  to  say,  this  is  the  curve  of  forces, 
cause  of  cohesion  between  two  points,  for  here  a  repulsion  prevents  decrease,  &  attraction 

increase,  of  distance  ;  &  so  the  points  preserve  the  distance  at  which  they  are.  Cohesion 
for  more  than  two  points  can  be  obtained,  both  when  each  of  the  pairs  of  points  is  at  a 
distance  corresponding  to  a  limit-point  of  cohesion,  &  also  when  the  opposite  forces 
cancel  one  another,  an  example  of  which  I  gave  in  Art.  223. 

411.  Further,  with  regard  to  such  cohesion,  there  are  many  points  that  are  worthy  Cohesion   of   two 
of  remark.     First  of  all,  in  connection  with  two  points,  we  can  have  as  many  different  ^ints^f'  cohesion 
distances  corresponding  with  cohesion  as  is  represented  by  the  number  of  intersections  can   be    anything 
of  the  curve  of  forces  with  the  axis  (increased  by  one  if  perchance  the  number  is  odd)  divided  numte?*steJnjthd& 
by  two.     For  the  first  limit-point,  at  which  the  curve  passes  from  the  first  asymptotic  arc,  orderof  occurrence, 
i.e.,  from  repulsions  that  represent  impenetrability,  to  the  first  attractive  arc,  is  a  limit- 
point  of  cohesion  ;    &  after  that  the  points  of  intersection  are  alternately  limit-points  of 
non-cohesion  &  cohesion,  as  was  shown  in  Art.  179.     Hence  it  comes  about  that,  if  the 

number  of  intersections  following  one  after  the  other  are  assumed  to  be  even,  half  of  them  are 
limit-points  of  cohesion.  Hence,  since,  in  the  solution  of  the  problem  given  in  Art.  117, 
it  was  shown  that  that  simple  curve  of  mine  could  have  any  number  of  intersections,  it 
will  be  possible  for  two  points  only  to  have  any  number  of  different  distances  from  one 
another  that  would  correspond  to  limit-points  of  cohesion.  Moreover  these  cohesions 
could  be  of  very  different  kinds,  as  regards  solidity  &  connection,  the  limit-points  being 
either  very  strong  or  very  weak  ;  that  is  to  say,  according  as  the  curve  at  these  points  was 
nearly  perpendicular  to  the  axis  &  departed  far  from  it,  or  on  the  other  hand  was  much 
inclined  to  the  perpendicular  &  only  went  away  from  the  axis  by  a  very  small  amount. 
For,  in  the  first  case,  the  repulsive  forces  on  diminishing  the  distances,  or  the  attractive 
forces  on  increasing  the  distances,  ever  so  slightly,  will  be  very  great  ;  in  the  second 
case,  even  when  the  distances  are  altered  a  good  deal,  the  forces  are  very  slight.  Again 
also,  it  is  possible  that  some  of  the  more  remote  limit-points  would  be  much  weaker,  & 
others  much  stronger,  than  some  of  the  nearer  limit-points.  Thus,  with  me,  the  force  of 
cohesion  is  altogether  independent  of  density  ;  the  strength  of  cohesion,  in  denser  bodies, 
can  be  either  much  greater  or  much  less  than  that  in  less  dense  bodies,  &  the  ratio  can  be 
anything  whatever. 

412.  What  has  been  said  concerning  two  points  applies  also  &  in  a  far  greater  degree  in  masses  the 
to  masses  made  up  of  a  large  number  of  points.     In  masses,  the  number  of  limit-points  poTn/s  is  m'vufh 
is  immensely  greater  still,  &  the  difference  between  them  is  greater  in  every  case.     The  greater;  how  the 
finding  of  all  the  positions  for  a  given  number  of  points,  at  which  the  whole  mass  has  a  t^enTis 
limit-point  of  forces,  would  be  a  troublesome  undertaking  ;  &  the  calculation  necessary  for  its  solved, 
solution  would  increase  immensely  in  proportion  to  the  greater  number  of  points  taken. 

However,  it  can  certainly  be  solved,  if  the  law  of  forces  is  given.  It  would  be  sufficient  to 
assume  the  positions  of  all  the  points  with  respect  to  any  one  point  in  any  arbitrary  straight 
line  in  any  arbitrary  way,  &  having  substituted  the  distances  for  each  pair  from  one  another 
for  the  abscissa  in  the  equation  of  the  primary  curve,  &  taking  the  values  of  the  forces  for 
each  of  the  points  as  ordinates,  to  make  out  as  many  equations ;  then  to  resolve  each  of 
the  forces  into  three  chosen  directions,  &  to  put  the  sum  of  all  those  in  the  same  direction 
for  any  point  equal  to  zero.  We  shall  thus  obtain  equations  which,  as  the  unknown  assumed 
values  are  one  by  one  eliminated,  will  finally  lead  to  equations  determining  the  distances  of 
the  points  necessary  for  equilibrium,  &  relative  rest ;  but  these  would  be  of  very  high 


296  PHILOSOPHISE  NATURALIS   THEORIA 

haberent  radices  ;  nam  sequationes,  quo  altiores  sunt,  eo  plures  radices  habere  possunt,  ac 
singulis  radicibus  singuli  limites  exhiberentur,  vel  singulae  positiones  exhibentes  vim  nullam. 
Inter  ejusmodi  positiones  illse,  in  quibus  repulsioni  in  minoribus  distantiis  habitae,  succe- 
derent  attractiones  in  majoribus,  exhiberent  limites  cohaesionis,  qui  adhuc  essent  quam 
plurimi,  &  inter  se  magis  diversi,  quam  limites  ad  duo  tantummodo  pun-[i89]cta  pertin- 
entes  ;  cum  in  compositione  plurium  semper  utique  crescat  multitude,  &  diversitas  casuum. 
Sed  haec  innuisse  sit  satis. 


Cur    partes    solid!  AI*.  Ubi  confringitur  massa  aliqua,  &  dividitur  in  duas  partes,  quae  prius  tenacissime 

fracti  ad  se  invicem    •  11  •   •  -n  11  i  •  i        •  • 

appressae  non  acqui-  inter  se  conserebant,  si  iterum  mee  partes  adducantur  ad  se  invicem  ;    conaesio  prior  non 
rant     cohaeskmem  redit,  utcunque  apprimantur.     Eius  rei  ratio  apud  Newtonianos  est,  quod  in  ilia  divisione 

priorem,    ratio     in  j-      11  •        i  i  -j  •        •  j 

Theoria  Newton-  non  seque  divellantur  simul  omnes  particulse,  ut  textus  remaneat  idem,  qui  prius  :  sed 
prominentibus  jam  multis,  harum  in  restitutione  contactus  impediat,  ne  ad  contactum 
deveniant  tarn  multae  particular,  quam  multae  prius  se  mutuo  contingebant,  &  quam  multis 
opus  esset  ad  hoc,  ut  cohaesio  fieret  iterum  satis  firma  :  at  ubi  satis  lse.viga.tze.  binae  superficies 
ad  se  invicem  apprimantur,  sentiri  primo  resistentiam  ingentem  dicunt,  donee  apprimuntur  ; 
sed  ubi  semel  satis  appressae  sint,  cohasrere  multis  vicibus  majore  vi,  quam  sit  pondus  aeris 
comprimentis  ;  quia  antequam  deveniatur  ad  eos  contactus,  haberi  debet  repulsiva  vis 
ingens,  quam  in  majoribus  distantiis,  sed  adhuc  exiguis,  agnovit  Newtonus  ipse,  cui  cum 
deinde  succedat  in  minoribus  vis  attractiva,  quas  in  contactu  evadat  maxima,  &  in  laevigato 
marmore  satis  multi  contactus  obtineantur  simul  ;  idcirco  deinde  satis  validam  cohcesionem 
consequi.  . 


Ejusdem   ratio   in  AIA.  Quidquid    ipsi  de  contactibus  dicunt,  id  in  mea  Theoria  dicitur  aeque  de  satis 

mea  Theoria.  ,.  ,7  i        •       •       T      •   -i  r    .  .  .  .       , 

vaiidis  cohaesionis  limitibus.  In  scabra  superncie  satis  multae  prommentes  particulae 
progressae  ultra  limites,  in  quibus  ante  sibi  cohaerebant,  repulsionem  habent  ejusmodi, 
quae  impediat  accessum  reliquarum  ad  limites  illos  ipsos,  in  quibus  fuerant  ante  divulsionem. 
Inde  fit,  ut  ibi  nimis  paucae  simul  reduci  possint  ad  cohaesionem  particulse,  dum  in 
laevigatis  corporibus  adducuntur  simul  satis  multae.  Ubi  autem  duo  marmora,  vel  duo 
quaecunque  satis  solida  corpora,  bene  complanata,  &  laevigata  sola  appressione  cohaeserunt 
invicem,  ilia  quidem  admodum  facile  divelluntur  ;  si  una  superficies  per  alteram  excurrat 
motu  ipsis  superficiebus  parallelo  ;  licet  motu  ad  ipsas  superficies  perpendiculari  usque 
adeo  difficulter  distrahi  possint  :  quia  particulae  eo  motu  parallelo  delatae,  quae  adhuc 
sunt  procul  a  marginibus  partium  congruentium,  vires  sentiunt  hinc,  &  inde  a  particulis 
lateralibus,  a  quibus  fere  aequidistant,  fere  aequales,  adeoque  sentitur  resistentia  earum 
attractionum  tantummodo,  quas  in  se  invicem  exercent  marginales  particulae,  dum  augent 
distantias  limitum  :  nam  mihi  citra  limitem  quenvis  cohaesionis  est  repulsio,  ultra  vero 
attractio  ;  licet  ipsi  deinde  adhuc  aliae  &  attractiones,  &  repulsiones  possint  succedere. 
Ubi  autem  'perpendiculariter  distrahuntur,  debet  omnium  simul  limitum  resistentia 
vinci. 


Discrimen  mass*  415.  Nee  vero  idem  accidit,  ubi  marmora  integra,  &  nunquam  adhuc  divisa,  inter  se 
fruSiretS'nf  tevi-  cohaerent ;  turn  enim  fibrae  possunt  esse  multae,  quarum  particulae  adhuc  in  minori-[i9o]bus 
gatisadse  invicem  distantiis,  &  multo  validioribus  limitibus  inter  se  cohaereant,  ad  quos  sensim  devenerint 
alise  post  alias  iis  viribus,  quibus  marmor  induruit,  ad  quos  nunc  iterum  reduci  nequeant 
omnes  simul,  dum  marmora  apprimuntur,  quae  ulteriorum  limitum  minus  adhuc  validorum, 
sed  validorum  satis  repulsivas  vires  simul  sentiunt,  ob  quas  non  possunt  denticuli,  qui 
adhuc  supersunt  perquam  exigui  post  quamvis  laevigationem,  in  foveolas  se  immittere,  & 
ad  ulteriores  limites  validiores  devenire  ;  praeterquam  quod  attritione,  &  Isevigatione  ilia 
plurimarum  particularum  ordinis  proximi  massis  nobis  sensibilibus  inducitur  discrimen 
satis  amplum  inter  massam  solidam  primigeniam,  &  binas  massas  complanatas,  laevigatasque 
ad  se  invicem  appressas. 

Distractio,  &  com-  4J6-  Inde  autem  in  mea  Theoria  satis  commode  explicatur  &  distractio,  &  compressio 

pressio  nbrarum  fibrarum  ante  fractionem  ;    cum  nimirum  nihil  apud  me  pendeat  ab  immediato  contactu, 
10  sed  a  limitibus,  quorum  distantia  mutatur  vi  utcunque  exigua  :   sed  si  satis  validi  sint,  ad 


A  THEORY  OF  NATURAL  PHILOSOPHY  297 

degree  &  would  have  very  many  roots.  For,  the  higher  the  degree,  the  more  the  roots 
given  by  the  equations ;  &  for  each  of  the  roots  there  would  be  a  corresponding  limit-point, 
or  a  position  representing  zero  force.  Amongst  such  positions,  those,  in  which  we  have 
repulsion  at  a  less  distance  followed  by  attraction  at  a  greater  distance,  would  yield  limit-points 
of  cohesion  ;  &  these  would  be  as  great  in  number  &  as  different  from  one  another  as  were 
the  limit-points  pertaining  to  two  points  only  ;  for  in  a  composition  of  several  things  there 
certainly  is  always  an  increasing  multitude  &  diversity  of  cases.  But  let  it  suffice  that  I 
have  called  attention  to  these  matters. 

413.  When  a  mass  is  broken,  &  divided  into  two  parts,  which  originally  cohered  most  xhe   reason  given 
tenaciously,  if  the  parts  are  again  brought  into  contact  with  one  another,  the  previous  in  the  Newtonian 

< '  f     ,  j  ,  rri,  j-     i  •       theory   to   account 

cohesion  does  not  return,  however  much  they  are  pressed  together.      Ine  reason  of  this,  for  the  fact  that  the 
according  to  the  followers  of  Newton,  is  that  in  the  division  all  the  particles  are  not  equally  Parts  of  a  broken 

,  ...  ,  i_    r  i  e     t  solid,  when  brought 

torn  apart  simultaneously,  leaving  the  texture  the  same  as  before  ;    but  as  many  of  them  closely  together,  do 
now  iut  out  bevond  the  rest,  the  contact  between  these  in  restitution  prevents  as  many  not    attain    their 

. J,  J  .  ,  i  .  i_  •    •      if         i  •  i_  .    J    former  cohesion. 

particles  coming  into  contact  as  there  were  touching  one  another  originally,  which  number 
is  necessary  for  the  purpose  of  again  establishing  a  sufficiently  strong  cohesion.  But  when 
two  surfaces  that  are  sufficiently  well  polished  are  brought  closely  together,  they  say  that  at 
first  there  is  felt  a  resistance  of  very  great  amount,  until  they  are  pressed  into  contact ; 
but  when  once  the  surfaces  are  pressed  together  sufficiently  closely,  they  cohere  with  a 
force  that  is  many  times  greater  than  that  due  to  the  weight  of  the  air  pressing  upon  them. 
The  reason  they  give  is  that,  before  actual  contact  is  reached,  there  must  be  obtained  a 
very  great  repulsive  force,  such  as  Newton  himself  recognized  as  existing  at  comparatively 
large,  but  actually  very  small,  distances ;  &  after  that,  there  followed  an  attractive  force 
at  still  smaller  distances,  which  became  exceedingly  great  when  contact  was  reached.  Thus, 
in  polished  marble,  a  sufficiently  great  number  of  contacts  was  obtained  simultaneously ; 
&  in  consequence  a  comparatively  great  cohesion  was  obtained. 

414.  All  that  the  Newtonians  say  with  regard  to  contacts  applies  in  my  Theory  equally  The  reason  for  the 
well  with  regard  to  sufficiently  strong  limit-points  of  cohesion.     In  a  rough  surface,  a  ?amf  thing  accord- 

,.  ....  -111  TIT  T  lng  to  my  Theory. 

sufficient  number  of  jutting  particles,  pushed  out  beyond  the  distances  corresponding  to 
those  of  the  limit-points,  at  which  they  previously  cohered,  give  rise  to  a  repulsion  of  such 
sort  as  prevents  the  other  particles  from  approaching  to  the  distances  of  the  limit-points, 
at  which  they  were  before  being  torn  apart.  Thus  it  comes  about  that  in  this  case  too 
few  of  the  particles  can  be  brought  into  a  state  of  cohesion ;  whilst  in  the  case  of  polished 
bodies  we  have  a  sufficient  number  of  particles  brought  together  simultaneously.  Moreover, 
when  two  pieces  of  marble,  or  any  two  bodies  of  comparatively  great  solidity,  after  being 
well  smoothed  &  polished,  cohere  when  they  are  merely  pressed  together,  they  can  be  forced 
apart  perfectly  easily.  If,  for  instance,  one  surface  traverses  the  other  with  a  motion  parallel 
to  the  surfaces ;  although  they  can  with  difficulty  be  torn  apart  with  a  motion  perpendicular 
to  the  surfaces.  For,  particles  carried  along  by  this  parallel  motion,  such  as  are  still  far 
from  the  marginal  surfaces  of  the  parts  in  contact,  feel  the  effects  of  forces  on  one  side  & 
on  the  other,  due  to  laterally  situated  particles  from  which  they  are  nearly  equidistant, 
that  are  nearly  equal  to  one  another ;  &  thus  resistance  is  only  experienced  from  the 
attractions  which  the  particles  in  the  marginal  surfaces  exert  upon  one  another,  whilst 
they  increase  the  distances  of  the  limit-points.  The  reason  is  that  with  me  there  is  repulsion 
on  the  near  side  of  any  limit-point  of  cohesion,  &  attraction  on  the  far  side ;  although 
thereafter  still  other  attractions  &  repulsions  may  follow.  But  when  the  bodies  are 
drawn  apart  perpendicularly,  the  resistance  due  to  every  limit-point  must  be  overcome 
simultaneously. 

415.  The  same  arguments  do  not  apply  to  the  case  of  whole  pieces  of  marble  that  Distinction  be- 
have not  as  yet  been  broken  at  any  time,  when  they  cohere.     For,  in  that  case,  there  may  *we^itive  mass  °& 
be  many  filaments,  the  particles  of  which  hitherto  have  been  cohering  at  less  distances  &  two    pieces    that 
in  much  stronger  limit-points ;    these  limit-points  they  would  gradually  reach  one  after  h*ve  be<;",,b5oke£ 

•   i      i       c  11  •  i  i     •      i         i  11  i          ,    o«.       polished      & 

the  other  with  the  forces  that  have  given  the  marble  its  hardness ;  but  they  cannot  be  reduced  pressed  together. 

to  them  once  more  all  at  once,  whilst  the  pieces  of  marble  are  being  pressed  together.     At 

the  same  time  they  feel  the  effect  of  the  repulsive  forces  due  to  further  limit-points  still 

less  strong,  but  yet  fairly  powerful ;   &  on  account  of  these,  the  little  teeth  which  still  are 

left,  though  very  small,  after  any  polishing,  cannot  insert  themselves  into  the  little  hollows, 

&  so  reach  the  strong  limit-points  beyond.     Besides,  by  this  attrition  &  polishing  of  the 

greater  number  of  the  particles  of  an  order  next  to  such  masses  as  are  sensible  to  us  there 

is  induced  a  sufficiently  wide  distinction  between  a  primitive  solid  mass  &  two  masses  that 

have  been  smoothed  &  polished  &  then  pressed  together.  Distension  &  com- 

416.  Hence  also,  in  my  Theory,  we  can  give  a  fairly  satisfactory  explanation  of  the  pression    of  fibres 
distension  &  compression  of  fibres  that  precedes  fracture  ;  for,  with  me,  everything  depends  ^gooVwroianatton 
not  on  immediate  contact,  but  on  the  limit-points,  the  distance  of  which  is  changed  by  from  my  Theory. 


298  PHILOSOPHIC  NATURALIS  THEORIA 

vincendam  satis  magno  accessu  omnem  repulsionem,  vel  recessu  attractionem,  requiritur 
satis  magna  vis  :  quae  quidem  repulsio,  &  attractio  in  aliis  limitibus  longe  mihi  alia  est,  tarn 
si  vis  ipsa  consideretur  quam  si  consideretur  spatii,  per  quod  ea  agit,  magnitude,  quae  omnia 
pendent  a  forma,  &  amplitudine  arcuum,  quibus  hinc,  &  inde  circa  axem  contorquetur 
mea  virium  curva.  Hinc  in  aliis  corporibus  ante  fractionem  compressiones,  &  distractiones 
esse  possunt  longe  majores,  vel  minores,  &  longe  major,  vel  minor  vis  requiri  potest  ad 
fractionem  ipsam,  quae  vis,  ubi  distantiis  immutatis,  superaverit  maximam  arcus  ulterioris 
repulsivam  vim  in  recessu,  superatis  multo  magis  reliquis  omnibus  posterioribus  viribus 
repulsivis  ope  celeritatis  quoque  jam  acquisitae  per  ipsam  vincentem  vim,  &  per  attractivas 
intermixtas  vires,  quae  ipsam  juvant,  defert  particulas  massam  cqnstituentes  ad  illas  distantias, 
in  quibus  jam  nulla  vis  habetur  sensibilis,  sed  ad  tenuissimum  gravitatis  arcum  acceditur. 


Hinc  cur  sol  id  a  417.  Hinc  autem  etiam  illud  in  mea  Theoria  commodius  accidit,  quam  in  communi, 

der^pressaconfrin"  quod  in  mea  statim  apparet,  cur  pila  quaecunque  utcunque  solid!  corporis  post  certa  imposita 

gantur.  pondera  confringatur,  &  confringatur  etiam  solidus  globus  utriaque  compressus  ;    cum 

multo  magis  appareat,  quo  pacto  textus,  &  dispositio  particularum  necessaria  ad  summam 

virium  satis  validam  mutari  possit,  ubi  omnia  puncta  a  se  irwicem  distant  in  vacuo  libero, 

quam  ubi  continuae  compactae  partes  se  contingant,  nee  ulla  mihi  est  possibilis  solida  pila, 

quae  Mundum  totum,  si  vi  gravitatis  in  certain  plagam  feratur  totus,  sustineat,  ut  in  sententia 

de  continua  extensione  materiae  pila  perfecte  solida  utcunque  tenuis  ad  earn  rem  abunde 

sufficeret. 

Communia  esse          418.  Hisce  omnibus  jam  accurate  expositis,  communia   mihi  sunt  ea    omnia,  qua; 
' 


it  ™  pertinent  ad  methodos  explorandi  per  [191]  experimenta  diversam  diversorum  corporum 
qua?  pertinent   ad  cohaesionis  vim,   quod   argumentum  diligenter,   ut  solet,   excoluit  Musschenbroekius,  & 


:^  comparand!  resistentiam  ad  fractionem,  ubi  divisio  fieri  debeat  divulsione  perpendicular! 
resist  entiam  ad  ad  superficies  divellendas,  ut  ubi  trabi  vertical!  ingens  pondus  appenditur  inferne,  cum 
v^rsU°positionibds"  resistentia,  quas  habetur,  ubi  circa  latus  suum  aliquod  gyrare  debeat  superficies,  quae 
divellitur,  quod  accidit,  ubi  extremae  parti  trabis  horizontalis  pondus  appenditur  ;  quam 
perquisitionem  a  Galileo  inchoatam,  sed  sine  ulla  consideratione  flexionis  &  compressionis 
fibrarum,  quae  habetur  in  ima  parte,  alii  plures  excoluerunt  post  ipsum  ;  &  in  quibus  omnibus 
discrimina  inveniuntur  quamplurima.  Illud  unum  hie  addam  :  posse  cohaesionem 
ingentem  acquiri  ab  iis,  quae  per  se  nullum  haberent,  nova  materia  interposita,  ut  ubi 
cineres,  qui  oleis  actione  ignis  avolantibus  inter  se  inertes  remanserunt,  oleis  novis  in  massam 
cohaerentem  rediguntur  iterum,  ac  in  aliis  ejusmodi  casibus ;  sed  id  jam  pendet  a  discrimine 
inter  diversas  particulas,  &  massas,  ac  pertinet  ad  soliditatem  explicandam  inprimis,  non 
generaliter  ad  cohaesionem,  de  quibus  jam  agam  gradu  facto  a  generalibus  corporum 
proprietatibus  ad  multiplicem  varietatem  Naturae,  &  proprietates  corporum  particulars. 


Discrimen    inter          419.  Et  primo  quidem  se  hie  mihi  offert  ingens  illud  plurium  generum  discrimen, 

particulas  diversas,  i    1*1       •          .      .    •    .          t*  .  __•__  '     ^*^ ^    j: 


dpuncto-  <luod  haberi  potest  inter  diversas  punctorum  congeries,  quae  constituunt  diversa  genera 
rum,  a  mole,  a  particularum  corpora  constituentium.  Primum  discrimen,  quod  se  objicit,  repeti  potest 
quaeSltapotettfigesse  ^  '1PSO  numero  punctorum  constituentium  particular^,  qui  potest  esse  sub  eadem  etiam 


cum  quavis  mole  admodum  diversus.  Deinde  moles  ipsa  diversa  itidem  esse  potest,  ac  diversa  densitas, 
damd  eam  ret  ut  nimirum  duae  particular  i>ec  massam  habeant,  nee  molem,  n.ec  densitatem  aequalem. 
Deinde  data  etiam  &  massa,  &  mole,  adeoque  data  densitate  media  particulae  ;  potest 
haberi  ingens  discrimen  in  ipsa  figura,  sive  in  superficie  omnia  includente  puncta  &  eorum 
sequente  ductum.  Possunt  enim  in  una  particula  disponi  puncta  in  sphaeram,  in  alia  in 
pyramidem,  vel  quadratum,  vel  triangulare  prisma.  Sumatur  figura  quaecunque,  &  in 
eam  disponantur  puncta  utcunque  :  tot  erunt  ibi  distantiae,  quot  erunt  punctorum  binaria, 
qui  numerus  utique  finitus  erit.  Curva  virium  potest  habere  limites  cohaesionis  quot- 
cunque,  &  ubicunque.  Fieri  igitur  potest,  ut  limites  iis  ipsis  distantiis  respondeant,  & 
turn  eam  ipsam  formam  habebit  particula,  &  ejus  formse  poterit  esse  admodum  tenax. 
Quin  immo  per  unicam  etiam  distantiam  cum  repagulo  infinitae  resistentiae,  orto  a  binis 
asymptotis  parallelis,  &  sibi  proximis,  cum  area  hinc  attractiva,  &  inde  repulsiva  infinita, 


A  THEORY  OF  NATURAL  PHILOSOPHY  299 

any  force,  however  small  this  force  may  be.  If  these  are  sufficiently  strong,  then,  to  overcome 
all  repulsion  by  a  sufficient  great  approach,  or  all  attraction  by  a  similar  recession,  there  will 
be  required  a  force  that  is  sufficiently  great  for  the  purpose.  This  repulsion  &  attraction, 
with  me,  varies  considerably  for  different  limit-points,  both  when  the  force  itself  is  considered, 
&  when  the  magnitude  of  the  space  through  which  it  acts  is  taken  into  account ;  &  all  of 
these  things  depend  on  the  form  &  size  of  the  arcs  with  which  my  curve  of  forces  is  twined 
round  the  axis,  first  on  one  side  &  then  on  the  other.  Hence,  in  different  bodies,  there  may 
occur,  before  fracture  takes  place,  compressions  &  distensions  that  are  far  greater  or  far 
less,  &  a  force  may  be  required  for  that  fracture  that  is  far  greater  or  far  less ;  &  this  force, 
when  the  distances  are  changed,  having  overcome  the  maximum  repulsive  force  of  the 
further  arc  as  it  recedes,  would  (all  the  rest  of  the  repulsive  forces  due  to  the  first  arcs 
having  been  overcome  all  the  more  by  the  help  of  the  velocity  already  acquired  through 
the  overcoming  force,  assisted  by  the  attractive  forces  that  come  in  between)  carry  off  the 
particles  forming  the  mass  to  those  distances,  at  which  there  is  no  sensible  force,  but  the 
arc  of  exceedingly  small  amplitude  corresponding  to  gravity  is  reached. 

417.  Hence,  more  easily  in  my  Theory  than  in  the  common  theory,  because  in  mine  Hence  the  reason 

<•   A  •  j-       -i  £  i          •  i_  i"ii  \-  wny    solid     bodies 

it  follows  immediately,  we  have  an  explanation  as  to  the  reason  why  any  pillar  whatever,  win  be  broken 

made  of  a  solid  body,  is  broken  when  certain  weights  are  imposed  upon  it ;    &  also  why  under  the  pressure 

a  solid  sphere  is  crushed  when  compressed  on  both  sides.     For,  it  is  much  clearer  how  the  weight." 

texture  &  disposition  of  the  particles,  necessary  to  give  such  a  comparatively  great  sum  of 

forces,  can  be  changed,  if  all  the  points  lie  apart  from  one  another  in  a  free  vacuum, 

than  if  we  suppose  continuous  compact  parts  that  touch  one  another  ;   nor  can  I  imagine 

as  possible  any  solid  pillar  that  would  sustain  the  whole  Universe,  if  by  the  force  of  gravity 

the  whole  of  it  were  borne  in  a  given  direction  ;  &  yet  in  the  common  idea  of  continuous 

extension  of  matter  a  pillar  that  was  perfectly  solid,  of  no  matter  what  thinness,  would  be 

quite  sufficient  to  do  this. 

418.  These  matters  having  now  been  accurately  explained,  I  proceed  in  the  ordinary  There  are  many 
manner  in  all  things  that  relate  to  methods  of  experimental  investigation  of  the  different  betwee^my^heory 
force  of  cohesion  in  different  bodies,  a  mode  of  demonstration  that  Mussenbroeck  assiduously  &  the  usual  one, 
practised  with  his  usual  care ;    &  methods  of  comparing  the  resistance  to  fracture  in  the  ^fvestiga^io^of  the 
case  when  division  must  take  place  by  a  fracture  perpendicular  to  the  surfaces  to  be  broken,  forces  of  cohesion 
such  as  occur  when  a  great  weight  is  hung  beneath  a  vertical  beam,  with  the  resistance  ture^in^difie^ent 
that  is  obtained  in  the  case  when  the  surface  has  to  rotate  about  one  of  its  sides,  which  is  positions. 

torn  off,  as  happens  when  a  weight  is  hung  at  the  end  of  a  horizontal  beam.  This 
investigation,  first  started  by  Galileo,  but  without  considering  bending  or  the  compression 
of  the  fibres  that  takes  place  on  the  under  side  of  the  beam,  was  carried  on  by  several  others 
after  him  ;  &  in  all  cases  of  these  there  are  very  great  differences  to  be  found.  I  will  here 
add  but  this  one  thing  ;  it  is  possible  for  a  very  great  cohesion  to  be  acquired  by  things, 
which  of  themselves  have  no  cohesion,  by  the  interposition  of  fresh  matter.  For  instance 
in  the  case  of  ashes,  which,  after  the  oily  constituents  have  been  driven  off  by  the  action 
of  fire,  remained  inert  of  themselves ;  but,  as  soon  as  fresh  oily  constituents  have  been 
added,  become  once  more  a  coherent  mass ;  &  in  other  cases  of  like  nature.  But  this  really 
depends  on  the  distinction  between  different  kinds  of  particles  &  masses,  &  refers  to  the 
explanation  of  solidity  in  particular,  &  not  to  cohesion  in  general.  With  such  things  I 
will  now  deal,  passing  on  from  general  properties  of  bodies  to  the  multiplicity  &  variety 
of  Nature,  &  to  particular  properties  of  bodies. 

419.  The  first  thing  that  presents  itself  is  the  huge  difference,  of  many  kinds,  which  Distinction  be- 
there  can  be  amongst  different  groups  of  points  such  as  form  the  different  kinds  of  particles  kinds6 'of  particles 
of  which  bodies  are  formed.     The  first  difference  that  calls  our  attention  can  be  derived  arising    from    the 
from  the  number  of  points  that  form  the  particle  ;    this  number   can   be  quite  different  "he^their^cTume! 
within   the  same  volume.      Then   the  volume  itself  may  be  different,  as  also  may  the  their  density,  their 
density  ;  for,  of  course,  two  particles  need  not  have  either  equal  masses,  equal  volumes,  fatter  Anything  is 
or   equal   densities.     Then,  even  if  the  mass  &  the  volume  be  given,   that   is  to  say,  possible,    &    any 
the  mean  density   of   the  particle  is  given,  there  may  be    a    huge  difference    in  shape,  «"eteTadnfor°the 
that  is  to  say,  in  the  surface  enclosing  all  the  points,  &  conforming  with  them.       For,  purpose  of   main- 
the  points  in  one  particle  may  be  disposed  in  a  sphere,  in  another  in  a  pyramid,  or  a  * 

square  or  triangular  prism.  Take  any  such  figure,  &  suppose  the  points  are  disposed  in 
any  particular  manner  whatever  ;  then  there  will  be  as  many  distances  as  there  are  pairs 
of  points,  &  their  number  will  be  finite  in  every  case.  The  curve  of  forces  can  have  any 
number  of  limit-points  of  cohesion,  &  these  can  occur  anywhere  along  it.  Therefore  it 
must  be  the  case  that  limit-points  can  be  found  to  correspond  to  those  distances,  &  on 
account  of  these  the  particle  will  have  that  particular  form,  &  can  be  extremely  tenacious 
in  keeping  that  form.  Indeed,  through  a  single  distance,  with  a  restraint  of  infinite  resistance, 
arising  from  a  pair  of  parallel  asymptotes  close  to  one  another,  having  the  area  on  one  side 


300 


PHILOSOPHIC  NATURALIS   THEORIA 


potest  haberi  in  quavis  massa  cujuscunque  figurae  soliditas  etiam  infinita,  sive  vis,  quse 
impediret  dispositionis  mutationem  non  minorem  data  quacunque.  Nam  intra  illam 
figuram  [192]  posset  inscribi  continuata  series  pyramidum  juxta  num.  363  habentium 
pro  lateribus  illas  distantias  nunquam  mutandas  magis,  quam  pro  distantia  binarum  illarum 
asymptotorum,  &  positis  punctis  ad  singulos  angulos,  haberetur  massa  punctorum,  quorum 
nullum  jaceret  extra  ejusmodi  figuram,  nee  ullum  adesset  intra  illam  figuram,  vel  in  ejus 
superficie  spatii  punctum,  a  quo  ad  distantiam  minorem  ilia  distantia  data  non  haberetur 
punctum  materiae  aliquod.  Possent  autem  intra  massam  haberi  hiatus  ubicunque,  & 
quotcunque  prorsus  vacui,  inscriptis  in  solo  residue  spatio  pyramidibus  illis,  &  in  angulis 
quibusvis  posset  haberi  quivis  numerus  punctorum  distantium  a  se  invicem  minus,  quam 
distent  illae  binae  asymptoti,  &  quivis  eorum  numerus  collocari  posset  inter  latera,  &  facies 
pyramidum.  Quare  posset  variari  densitas  ad  libitum.  Sed  absque  eo,  quod  singulis 
distantiis  respondeant  in  curva  primigenia  singuli  limites,  vel  singula  asymptotorum  binaria, 
vel  ullae  sint  ejusmodi  asymptoti  praeter  illam  primam,  innumera  sunt  sane  figurarum 
genera,  in  quibus  pro  dato  punctorum  numero  haberi  potest  aequilibrium,  &  cohaesionis 
limes  per  elisionem  contrariarum  virium,  ex  solutione  problematis  indicati  num.  412. 
Hoc  discrimen  est  maxima  notatu  dignum. 


Discnmen  in  punc-  ^20.  Data  etiam  figura  potest  adhuc  in  diversis  particulis  haberi  discrimen  maximum 

torum      distribu-       11'  T       >i       •  •  n-      •  i  i 

tione  per  figuram  ob  diversam  distributionem  punctorum  ipsorum.     oic  in  eadem  sphsera  possunt  puncta 
eandem.  esse  admodum  inaequaliter  distributa  ita,  ut  etiam  paribus  distantiis  ex  altera  parte  sint 

plurima,  ex  altera  paucissima,  vel  in  diversis  locis  superficiei  ejusdem  concentricae  esse 
congeries  plurimae  punctorum  conglobatorum,  in  aliis  eorum  raritas  ingens,  &  haec  ipsa 
loca  possunt  in  diversis  a  centro  distantiis  jacere  ad  plagas  admodum  diver'sas  in  eadem 
etiam  particula,  &  in  eadem  a  centro  distantia  esse  in  diversis  particulis  admodum  diversis 
modis  distributa.  Verum  etiam  si  particulse  habeant  eandem  figuram,  ut  sphaericam,  & 
in  singulis  circumquaque  in  eadem  a  centro  distantia  puncta  aequaliter  distributa  sint ; 
ingens  adhuc  discrimen  esse  poterit  in  densitate  diversis  a  centro  distantiis  respondente. 
Possunt  enim  in  altera  esse  fere  omnia  versus  centrum,  in  altera  versus  medium,  in  altera 
versus  superficiem  extimam  :  &  in  hisce  ipsis  discrimina,  tarn  quod  pertinet  ad  loca  densi- 
tatum  earundem,  quam  quod  pertinet  ad  rationem  inter  diversas  densitates,  possunt  in 
infinitum  variari. 

Discrimen  in  vi,  qua  421.  Haec  omnia  discrimina  pertinent  ad  numerum,  &  distributionem  punctorum  in 

rJtinere :  posselsse  diversis  particulis :  sed  ex  iis  oriuntur  alia  discrimina  praecipua,  quae  maximam  corporum, 
taiem,  ut  null  a  &  phaenomenorum  varietatem  inducunt,  quae  nimirum  pertinent  ad  vires,  quibus  puncta 
l  particulam  constituentia  agunt  inter  se,  vel  quibus  tota  una  particula  agit  in  totam  alteram. 
Possunt  inprimis,  &  in  tanta  dispositionum  varietate  debent,  [193]  puncta  constituentia 
eandem  particulam  habere  vires  cohaesionis  admodum  inter  se  diversas,  ut  aliae  multo 
facilius,  alias  multo  difficilius  dispositionem  mutent  mutatione,  quae  aliquam  non  ita  parvam 
rationem  habeat  ad  totum.  Est  autem  casus,  in  quo  possint  puncta  particulae  cohserere 
inter  se  ita,  ut  nulla  finita  vi  nexus  dissolvi  possit,  ut  ubi  adsint  asymptotici  arcus  in  curva 
primitiva,  juxta  ea,  quae  persecutus  sum  num.  362. 


alia  se  ,22-  Discrimina  autem  virium,  quas  una  particula  exercet  in  aliam,  debent  esse  adhuc 

attranentes,       ana      ,'_..  •*•-.  •      i  ju 

repeiientes,  alia  plura.  Inprimis  ex  num.  222  patet,  fieri  posse,  ut  una  particula  constans  etiam  duobus 
inertes  inter  so.  punctis  tertium  punctum  in  iisdem  distantiis  collocatum  ab  earum  medio  attrahat  per 
totum  quoddam  intervallum,  vel  repellat  per  idem  intervallum  totum,  vel  nee  usquam  in 
eo  repellat,  nee  attrahat,  conspirantibus  in  primo  casu  binis  attractionibus,  in  secundo  binis 
repulsionibus  itidem  conspirantibus,  &  in  tertio  attractione,  &  repulsione  aequalibus  se 
mutuo  elidentibus.  Multo  autem  magis  summa  virium  totius  cujusdam  ^particulae  in 
aliam  totam  in  eadem  etiam  distantia  sitam,  si  medium  utriusque  spectetur,  erit  pro  diversa 
dispositione  punctorum  admodum  inter  se  diversa,  ut  nimirum  in  una  attractiones  praeva- 
leant,  in  alia  repulsiones,  in  alia  vires  oppositae  se  mutuo  elidant.  Inde  habebuntur, 
particulae  in  se  invicem  agentes  viribus  admodum  diversis,  pro  diversa  sua  constitutione 
&  particulae  ad  sensum  inertes  inter  se,  quae  quidem  persecutus  sum  ipso  num.  222. 


A  THEORY  OF  NATURAL  PHILOSOPHY  301 

attractive  &  on  the  other  side  repulsive,  there  can  be  obtained  in  any  mass  of  any  form 
whatever  a  solidity  that  is  also  infinite,  or  a  force  that  would  prevent  any  change  of  disposition 
of  the  particles  equal  to  or  greater  than  any  given  change.  For  within  that  form  there 
could  be  inscribed  a  continued  series  of  pyramids,  after  the  manner  of  Art.  363,  having 
for  sides  those  distances  which  are  never  to  be  altered  by  more  than  that  corresponding  to  the 
distance  between  the  pair  of  asymptotes.  If  the  points  are  placed  one  at  each  of  the  angles, 
there  would  be  obtained  a  mass  consisting  of  points  no  one  of  which  would  lie  outside  a 
figure  of  this  sort ;  &  no  other  point  could  get  within  that  figure  or  occupy  a  point  of  space 
on  its  surface,  from  which  there  would  not  be  some  point  oi  matter  at  a  less  distance  than  the 
given  distance.  Further,  within  the  figure,  there  may  be  any  kind  &  any  number  of  gaps 
quite  empty  of  points,  the  pyramids  being  described  only  in  the  remainder  of  the  space  ; 
&  at  the  angles  there  may  be  any  number  of  points  distant  from  one  another  less  than  the 
distance  between  the  asymptotes ;  &  there  may  be  any  number  of  them  situated  along  the 
sides  &  faces  of  the  pyramids.  Hence,  the  density  can  be  varied  to  any  extent.  But,  apart 
from  the  fact  that  to  each  distance  there  corresponds  a  limit-point  in  the  primary  curve, 
or  that  there  are  pairs  of  asymptotes,  or  any  other  asymptotes  of  the  sort  except  the  first, 
there  are  really  an  innumerable  number  of  kinds  of  figures,  in  which  with  a  given  number 
of  points  there  can  be  equilibrium,  &  a  limit-point  of  cohesion  due  to  the  cancelling  of 
equal  &  opposite  forces,  as  can  be  seen  from  the  solution  of  the  problem  indicated  in  Art. 
412.  The  following  distinction  is  especially  worth  remark. 

420.  Even  if  the  figure  is  given,  there  can  still  be  obtained  a  great  difference  between  Difference  in  the 
different  particles  on  account  of  the  different  disposition  of  the  points  that  form  it.     Thus,  p^tf  ^thin^ne 
in  the  same  sphere,  the  points  may  be  quite  unequally  distributed,  in  such  a  way  that,  same  figure, 
even  at  equal  distances,  there  may  be  very  many  in  one  part  &  very  few  in  another  ;    or 

in  different  places  on  the  same  concentric  surface  there  may  be  very  many  groups  of  points 
condensed  together,  whilst  in  others  there  are  very  few  of  them  ;  these  very  places  may 
be  at  quite  different  distances  in  different  places  even  within  the  same  particle,  &  in  different 
particles  at  the  same  distance  from  the  centre  they  may  be  distributed  in  ways  that  are 
altogether  different.  Further,  even  if  particles  have  the  same  figure,  say  spherical,  &  in 
each  of  them,  round  about,  &  at  the  same  distance  from,  the  centre  the  points  are  distributed 
uniformly  ;  yet  even  then  there  may  be  a  huge  difference  in  the  density  corresponding  to 
different  distances  from  the  centre.  For,  in  the  one,  they  may  be  all  grouped  near  the 
centre,  in  another  towards  the  middle  surface,  &  in  a  third  close  to  the  outer  surface.  In 
these  the  differences,  both  as  regards  the  positions  of  equal  density,  &  also  as  regards  the 
ratio  of  the  different  densities,  can  be  varied  indefinitely. 

421.  All  such  differences  pertain  to  the  number  &  distribution  of  points  in  the  different  Difference  in  the 
particles.     From  them  arise  the  principal  differences  that  are  left  for  consideration  ;  these  partidea  try  "to 
lead  to  the  greatest  variety  in  bodies  &  in  phenomena.    Such  as  those  that  relate  to  the  forces  conserve  their 
with  which  the  points  forming  a  particle  act  upon  one  another,  or  the  forces  with  which  beT'suc'h  that  'the 
the  whole  of  one  particle  acts  upon  the  whole  of  another  particle.     First  of  all,  the  points  particle  can  be 
forming  the  same  particle  may,  &  in  such  a  great  variety  of  distribution  must,  have  forces  of  fin^force! 
cohesion  that  are  quite  different  one  from  the  other ;    so  that  some  of  them  much  more 

easily,  &  others  with  much  more  difficulty,  change  this  distribution  with  a  change  that  bears 
a  ratio  to  the  whole  that  is  not  altogether  small.  There  is  also  the  case,  in  which  the 
points  of  a  particle  can  cohere  so  strongly  together  that  the  connection  between  them 
cannot  be  broken  by  any  finite  force ;  this  happens  when  we  have  asymptotic  arcs  in 
the  primary  curve,  as  I  showed  in  Art.  362. 

422.  Moreover  we  may  have  still  more  differences  between  the  forces  which  one  Some  particles 
particle  exerts  upon  another  particle.     First  of  all,  it  is  evident  from  Art.  222,  that  it  may  oneraanothrerJP& 
happen  that  a  particle  consisting  of  even  two  points  may  attract  a  third  point  situated  at  s?me  have  no  ac- 
the  same  distances  from  the  middle  point  of  the  distance  between  the  two  points  throughout  other.0" 

the  whole  of  a  certain  interval  of  space,  or  they  may  repel  it  throughout  the  whole  of  the 
same  interval,  or  neither  repel  or  attract  it  anywhere ;  in  the  first  case  we  have  a 
pair  of  attractions  that  are  equal  &  in  the  same  direction,  in  the  second  case  a  pair  of 
repulsions  that  are  also  equal  &  in  the  same  direction,  &  in  the  third  case  an  attraction 
&  a  repulsion  that  are  equal  to  one  another  cancelling  one  another.  Also,  to  a  far  greater 
degree,  the  sum  of  the  forces  for  the  whole  of  any  particle  upon  the  whole  of  another  particle 
even  when  situated  at  this  same  distance,  if  the  mean  for  each  is  considered,  will  be  altogether 
different  from  one  another  for  a  different  distribution  of  the  points.  Thus,  in  one  particle 
attractions  will  prevail,  in  another  repulsions,  &  in  a  third  equal  &  opposite  forces  will 
cancel  one  another.  Hence  there  will  be  particles  acting  upon  one  another  with  forces 
that  are  altogether  different,  according  to  the  different  constitutions  of  the  particles ;  & 
there  will  be  particles  that  are  approximately  without  any  action  upon  one  another,  such 
as  I  investigated  also  in  the  above-mentioned  Art.  222, 


302  PHILOSOPHIC  NATURALIS  THEORIA 

Particula:  qua:   in          423.  Aliud  discrimen  admodum  notabile  inter  ejusmodi  particularum  vires  est  illud, 

repe'iiantPUTn1Saiiis  <luod  eadem  particula  ex  altera  parte  poterit  datam  aliam  particulam  attrahere,  ex  altera 

attrahant  :     quse  rcpellere  ;  quin  immo  possunt  esse  loca  quotcunque  in  superficie  particulae  etiam  sphaericae, 

quaTTir^umqua^e  <luae  alteram  particulam  in  eadem  a  centre  distantia  sitam  attrahant,  quae  repellant,  quas 

eandem  vim  exer-  nihil  agant ;   cum  nimirum  in  iis  locis  possint  vel  plura,  vel  pauciora  esse  puncta,  quam  in 

aliis,  &  ea  ad  diversas  a  centre,  &  a  se  invicem  distantias  collocata.     Inde  autem  &  illud 

fieri  poterit,  ut,  quemadmodum  in  iis,  quae  vidimus  a  num.  231,  unum  punctum  a  duorum 

aliorum  altero  attractum,  ab  altero  repulsum,  vi  composita  urgetur  in  latus,  ita  etiam  una 

particula  ab  una  alterius  parte  attracta,  &  repulsa  ab  altera  in  altera  directione  sita,  urgeatur 

itidem  in  latus,  &  certam  assecuta  positionem  respectu  ipsius,  ad  earn  tuendam  determinetur, 

nee  consistere  possit,  nisi  in  ea  unica  positione  respectu  ipsius,  vel  in  quibusdam  determinatis 

positionibus,  ad  quas  trudatur  ab  aliis  rejecta.     Quod  si  particula  sphasrica  sit,  &  in  omnibus 

concentricis  superficiebus  puncta  aequaliter  distributa  sint,  ad  distantias  a  se  invicem  perquam 

exiguas ;    turn  ejus,  &  alterius  ejus  similis  particulae  vires  mutuas  dirigentur  ad  sensum 

ad  earum  centra,  &  fieri  poterit,  ut  in  quibusdam  distantiis  se  repellant  mutuo,  in  aliis  se 

attrahant,  quo  casu  habebitur  quidem  diffi-[i94]cultas  in  avellenda  altera  ab  altera,  sed 

nulla  difficultas  habebitur  in  altera  circa  alteram  circumducenda  in  gyrum,  sicut  si  Terrae 

superficies   horizontalis    ubique   sit,  &    egregie    laevigata ;    globus    ponderis    cujuscunque 

posset  quavis  minima  vi  rotari  per  superficiem  ipsam,  elevari  non  posset  sine  vi,  quae  totum 

ipsius  pondus  excedat. 


Quo  minojes  par-  ,2,    jn  ^ac  act}0ne  unius  particulae  in  aliam  generaliter,  quo  particulas  ipsas  minorem 

ticulae,  eo  difficihus   .     .     ~  T  .  r    .  ..  T    i  •  ...  .. 

dissoiubiies.  habuerint  molem,  eo  minus  cetens  paribus  perturbabitur  earum  respectiva  positio  ab  alia 

particula  in  data  quavis  distantia  sita  :  nam  diversitas  directionis  &  intensitatis,  quam 
habent  vires  agentes  in  diversas  ejus  partes,  quae  sola  positionem  turbare  nititur,  viribus 
asqualibus  &  parallelis  nullam  mutuae  positionis  mutationem  inducentibus,  eo  erit  minor, 
quo  distantiarum,  &  directionum  discrimen  minus  erit  :  atque  idcirco,  quemadmodum 
jam  "exposui  num.  239,  inferiorum  ordinum  particulae  difficilius  dissolvi  possunt,  quam 
particulae  ordinum  superiorum. 

Dart!cuiasa  oriri*  ex          425'  ^•sec  quidem  praecipue  notatu  digna  mihi  sunt  visa  inter  particularum  ex  homo- 

punctorum  vicima ;  geneis  etiam  punctis  compositarum  discrimina,  quae  tamen,  quod  ad  vires  pertinet,  intra 

beant°diflferre  cor-  admodum  exiguos  distantiarum  limites  sistunt  :  nam  pro  majoribus  distantiis  particularum 

pora,  qua:  ex    iis  omnium  vires  sunt  prorsus  uniformes,  uti  ostensum  jam  est  num.  212,  nimirum  attractivae 

in  ratione  reciproca   duplicata   distantiarum    ad  sensum.     Porro   hinc   illud    admodum 

evidenter  consequitur,  massas  majores  ex  adeo  diversis  particulis  compositas,  nimirum 

haec  ipsa  nostra  majora  corpora,  quas  sub  sensum  cadunt,  debere  esse  adhuc  multo  magis 

diversa  inter  sc  in  iis,  quae  ad  eorum  nexum  pertinent,  &  ad  phaenomena  exhibita  a  viribus 

se  extendentibus  ad  distantias  illas  exiguas,  licet  omnia  in  lege  gravitatis  generalis,  quae 

ad  illas  pertinet  majores  distantias,  conformia  sint  penitus,  quod  etiam  supra  num.  402 

notandum  proposui.     De  hoc  autem  discrimine,  &  de  particularibus  diversorum  corporum 

proprietatibus  ad  diversas  pertinentium  classes  jam  agere  incipiam. 

Quae  natura  soiido-          ^26.  Prima  se  mihi  offerunt  solida,  &  fluida,  quorum  discrimina  quas  sint,  &   quomodo 

q'u'i'd*  fn^soHdis  a  mea  Theoria  ortum  ducant,  est  exponendum.     Solida  ita  inter  se  connexa  sunt,  ut  quem- 

rigida.  quid  virgae  Hbet  aliquot  particularum  motum  sequantur  reliquae  :    promotae,  si  illae  promoventur  : 

qu?dICviscosa,  quid  retractae,  si  illae  retrahuntur  :    conversae  in  latus,  si  linea,  in  qua  ipsae  jacent,  directionem 

humida.  mutet  :   &  in  eo  soliditas  est  sita  :   porro  ea  dicuntur  rigida  ;   si  ingenti  etiam  adhibita  vi 

positio,  quam  habet  recta  ducta  per  duas  quasvis  particulas  massas,  respectu  rectas,  quae 

jungit  alias  quascunque,  mutari  ad  sensum  non  possit,  sed  ad  inclinandam  unam  partem 

oporteat  inclinare  totam  massam,  &  basim,  &  quanvis  ejus  rectam  eodem  angulo  ;    nam 

in  iis,  quas  flexilia  sunt,  ut  elasticas  virgae,  pars  una  directionem  positionis  mutat,  &  [195] 

inclinatur,  altera  priorem  positionem  servante  :    &    priora   ilia   franguntur,  alia   majore, 

alia    minore    vi    adhibita ;     haec    posteriora    se    restituunt.     Fluida    autem    passim    non 

utique  carent  vi  mutua  inter  particulas,  immo  pleraque  exercent,  &  aliqua  satis  magnam, 

repulsivam  vim,  ut  aer,  qui  ad  expansionem  semper  tendit,  aliqua  attractivam,  &  vel  non 

exiguam,  ut  aqua,  vel  etiam  admodum  ingentem,  ut  mercurius,  quorum  liquorum  particulae 

se  in  globum  etiam  conformant  mutua  particularum  suarum  attractione,  &  tamen  separantur 

admodum  facile  a  se  invicem  majores  eorum  massae,  ac  aliquot  partibus  motus  facile  ita 

imprimitur  :      ut  eodem  tempore  ad  remotas  satis  sensibilis  non  protendatur  ;    unde  fit 


A  THEORY  OF  NATURAL  PHILOSOPHY  303 

423.  There  is  another  difference  that  is  well  worth  while  mentioning  amongst  forces  Particles  which  at 
of  this  sort,  namely,  that  the  same  particle  in  one  part  may  exert  attraction  on  another  &eataothersnatSt"racte- 
particle,  &  repulsion  from  another  part ;    indeed,  there  may  be  any  number  of  places  in  some    which   urge 
the  surface  of  even  a  spherical  particle,  which  attract  another  particle  placed  at  the  same  °^  ^^ic^exert 
distance  from  the  centre,  whilst  others  repel,  &  others  have  no  action  at  all.     For,  at  these  the  'same  force  to 
places  there  may  be  a  greater  or  less  number  of  points  than  in  other  places,  &  these  may  Produce  rotation. 
be  situated  at  different  distances  from  the  centre  &  from  one  another.     Thus,  just  as  we 

saw  for  the  cases  considered  in  Art.  231,  that  it  may  happen  that  a  point  is  attracted 
by  one  of  two  points  &  repelled  by  the  other,  &  be  urged  to  one  side  by  the  force  that  is 
the  resultant  of  these  two,  so  also  one  particle  may  be  attracted  by  one  part  of  another 
particle,  &  repelled  by  another  part  situated  in  another  direction,  &  also  be  urged  to  one 
side  ;  &  having  gained  a  certain  position  with  respect  to  it,  is  inclined  to  preserve  that 
position ;  nor  can  it  stay  in  any  position  with  regard  to  the  other  except  the  one,  or  perhaps 
in  several  definite  positions,  to  which  it  is  forced  when  driven  out  from  others.  But  if 
the  particle  is  spherical,  &  the  points  are  equally  distributed  in  all  concentric  surfaces,  at 
very  small  distances  from  one  another ;  then  the  mutual  forces  of  it  &  another  similar 
particle  are  directed  approximately  to  their  centres ;  &  it  may  happen  that  at  certain 
distances  they  repel  one  another,  &  at  other  distances  attract  one  another  ;  &  in  the  latter 
case  there  will  be  some  difficulty  in  tearing  them  apart,  but  none  in  making  them  rotate 
round  one  another.  Just  as,  if  the  Earth's  surface  was  everywhere  horizontal,  &  perfectly 
smooth,  a  ball  of  any  weight  whatever  could  be  made  to  rotate  along  that  surface  by  using 
any  very  small  force,  whereas  it  could  not  be  lifted  except  by  using  a  force  which  exceeded 
its  own  weight. 

424.  In  general,  in  this  action  of  one  particle  on  another,  the  smaller  volume  the  The  smaller  the 
particles  have,  the  less,  other  things  being  equal,  is  their  relative  position  affected  by  another  ^ffficuity  therels  in 
particle  situated  at  any  given  distance  from  it.     For  the  differences  in  the  directions  &  breaking  it  up. 
intensities  of  the  forces  acting  on  different  parts  of  it  (which  alone  try  to  alter  their  positions, 

since  equal  &  parallel  forces  induce  no  alteration  of  mutual  position)  will  be  the  less,  the 
less  the  difference  in  the  distances  &  directions.  Hence,  just  as  I  explained  in  Art..  239, 
particles  of  lower  orders  will  be  broken  Up  with  more  difficulty  than  particles  of  higher 
orders. 

425.  The  things  given  above  seemed  to  me  to  be  those  especially  worthy  of  remark  Differences   arise 
amongst  the  differences  between  particles  formed  from  even  homogeneous  points,  which  pf  iwfaita  "^"one 
yet  remained,  as  far  as  forces  are  concerned,  within  certain  very  narrow  limits.     For,  as  another ;  how  much 
regards  greater  distances,  the  forces  of  all  the  particles  are  quite  uniform  ;    that  is  to  say,  formed  "from  the'm 
they  are  attractive  forces  varying  approximately  as  the  inverse  square  of  the  distances,  differ     from     one 
Further,  from  them  it  follows  perfectly  clearly  that  greater  masses,  formed  from  these  a 

already  composite  particles  of  different  sorts,  that  is  to  say,  the  bodies  that  lie  about  us  of 
considerable  size,  such  as  come  within  the  scope  of  our  senses,  must  be  still  much  more 
different  from  one  another  in  matters  that  have  to  do  with  the  ties  between  them,  &  with 
the  phenomena  exhibited  by  forces  extending  over  very  small  distances ;  although  all  of 
them  are  quite  uniform  as  regards  the  law  of  universal  gravitation,  which  pertains  to  greater 
distances,  a  point  to  which  I  also  called  attention  in  Art.  402.  But  I  will  now  start  to  consider 
this  difference  &  the  particular  properties  of  different  bodies  belonging  to  different  classes. 

426.  The  first  matters  that  offer  themselves  to  me  for  explanation  are  the  differences  ^hefl^re  °hSoli^ 
that  exist  between  solids  &  fluids   &  how  these  arise  according  to  my  Theory.     Solids  are  so\i^1  are  rigfd.  & 
so  connected  together  that  the  motion  of  any  number  of  the  particles  is  followed  by  the  what  elastic  rods ; 

..  .  H  ......  -•'•,  111  •  r     i_  j      what   in   fluids  are 

remaining  particles;   it  the  former  move  forward,  so  do  the  latter;  it  they  are  retracted,  viSCOUSj  &whatare 

so  are  the  rest ;   if  a  line  in  which  they  lie  changes  its  direction,  they  are  moved  to  one  watery. 

side  ;  &  in  these  facts  solidity  is  defined..     Further,  solids  are  said  to  be  rigid,  if  the  position 

of  a  straight  line  drawn  through  any  two  particles  of  the  mass  cannot  be  sensibly  changed 

with  regard  to  the  straight  line  joining  any  other  pair  of  particles  by  using  even  a  very 

large  force  ;   but  in  order  to  incline  any  one  part  of  the  mass  it  is  necessary  to  incline  the 

whole  mass,  the  base,  &  any  straight  line  in  the  mass  at  the  same  angle.     For,  in  those  that 

are  flexible,  such  as  elastic  rods,  one  part  may  change  the  direction  of  its  position  &  be 

inclined,  whilst  the  rest  maintains  its  original  position.     The  first  are  broken  by  using  in 

some  cases  a  greater,  &  in  others  a  less,  force  ;  whereas  the  latter  recover  their  form.     Now 

fluids  in  every  case  do  not  lack  mutual  force  between  their  particles  throughout ;  indeed 

very  many  of  them    exert,  &  some  of  them  a  fairly  great,  repulsive   force,    such  as   air, 

which  always  tends  to   expand  ;   whilst  others   exert  an    attractive   force,  that  is    either 

not  very  small,  as  in  the  case  of  water,  or  may  even  be  very  great,  as  in  the  case  of  mercury. 

Of  these  liquids,  the  particles  even  form  themselves  into  balls  by  the  mutual  attraction 

of  the  particles  forming  them  ;   &  yet  larger  masses  of  them  are  quite  easily  separated,  & 

motion  is  easily  given  to  any  number  of  parts  in  such  a  manner  that  the  motion  does  not 


304  PHILOSOPHIC  NATURALIS  THEORIA 

ut  fluida  cedant  vi  cuicunque  impressae,  ac  cedendo  facile  moveantur,  solida  vero  nonnisi 
tota  simul  mover!  possint,  &  viribus  impressis  idcirco  resistant  magis  :  quae  autem  resistunt 
quidem  multum,  sed  non  ita  multum,  ut  solida,  dicuntur  viscosa.  Ipsa  vero  fluida 
dicuntur  humida,  si  solido  admoto  adhaerescant,  &  sicca,  si  non  adhsereant. 

Unde  fluiditas :  tria          427.  Haec  omnia  phaenomena  praestari  possunt  per  ilia  sola  discrimina,  quas  in  diverse 

fluidorum  genera.  •      i  r     •  ,          .  TT  •  a    -j-  ...  .     -1 

particularum  textu  consideravimus.  Ut  emm  a  fluiditate  incipiamus,  mpnmis  in  ipsis 
fluidis  omnes,  particulae  in  aequilibrio  esse  debent,  dum  quiescunt,  &  si  nulla  externa  vi 
comprimantur,  vel  in  certam  dirigantur  plagam  ;  id  aequilibrium  debebit  haberi  a  solis 
mutuis  actionibus  :  sed  ejusmodi  casum  non  habemus  hie  in  nostris  fluidis,  quae  incumbentis 
massse  premuntur  pondere,  &  aliqua.  ut  aer,  etiam  continentis  vasis  parietibus  comprimuntur, 
in  quibus  idcirco  omnibus  aliqua  haberi  debet  repulsiva  vis  inter  particulas  proximas,  licet 
inter  remotiores  haberi  possit  attractio,  ut  jam  constabit.  Tria  autem  genera  fluidorum 
considerari  poterunt  :  illud,  in  quo  in  majoribus  ejus  massulis  nulla  se  prodit  mutua 
particularum  vis  :  illud,  in  quo  se  prodit  vis  repulsiva  :  illud,  in  quo  vis  attractiva  se  prodit. 
Primi  generis  fere  sunt  pulveres,  &  arenulae,  ut  illae,  ex  quibus  etiam  horologia  clepsydris 
veterum  similia  construuntur,  &  ad  fluidorum  naturam  accedunt  maxime,  si  satis  laevigatam 
habeant  superficiem,  quod  in  quibusdam  granulis  cernimus,  ut  in  milio  :  nam  plerumque 
scabritiem  habent  aliquam  &  inaequalitates,  quse  motum  difnciliorem  reddunt.  Secundi 
generis  sunt  fluida  elastica,  ut  aer  :  tertii  vero  generis  liquores,  ut  aqua,  &  mercurius. 
Porro  in  primis  ostensum  est  num.  222,  &  422,  posse  binas  particulas  eodem  etiam  punctorum 
numero  constantes,  sed  diverse  modo  dispositas,  ita  diversas  habere  virium  summas  in 
iisdem  etiam  centrorum  distantiis,  ut  aliae  se  attrahant,  aliae  se  repellant,  aliae  nihil  in  se 
invicem  agant.  Quamobrem  ejusmodi  discrimina  exhibet  abunde  Theoria.  Verum 
multa  in  singulis  diligenter  notanda  sunt ;  nam  ibi  etiam,  ubi  nulla  se  prodit  vis  attractiva, 
habetur  inter  proximas  particulas  repulsio,  ut  innui  paullo  ante,  &  jam  patebit. 


Unde  faciiis  motus  [196!  428.  Porro    in    primo    casu    statim  apparet,    unde    facilis    ille    habeatur    motus. 

in    fluidis   primi/s-  v          *••  n  ......    rr  -,  ,  , 

generis.  (Juoniam,  aucta  distantia,  nulla  sensibili  vi  se  attrahunt  particular  ;  altera  non  sequetur 

motum  alterius  ;  nisi  ubi  ilia  versus  hanc  promota  ita  accesserit,  ut  vi  repulsiva  mutua, 
quemadmodum  in  corporum  collisionibus  accidit,  cogatur  illi  loco  cedere,  quas  cessio,  si 
satis  laevigatae  superficies  fuerint,  ut  prominentes  monticuli  in  exiguos  hiatus  ingressi  motum 
non  impediant,  &  sit  locus  aliquis,  versus  quern  possint  vel  in  gyrum  actae  particulae,  vel 
elevatae,  vel  per  apertum  foramen  erumpentes,  loco  cedere ;  facile  net,  nee  alia  requiretur 
vis  ad  eum  motum,  nisi  quae  ad  inertiae  vim  vincendam  requiritur,  vel  si  graves  particulae 
sint  versus  externam  massam,  ut  hie  versus  Tellurem,  &  fluidum  motu  impresso  debeat 
ascendere,  vis,  quae  requiritur  ad  vincendam  gravitatem  ipsam  :  verum  ad  vincendam 
solam  vim  inertiae,  satis  est  quascunque  activa  vis  utcunque  exigua,  &  ad  vincendam  gravi- 
tatem, in  hoc  fluidorum  genere,  si  perfecta  sit  laevigatio  ;  satis  est  vis  utcunque  paullo  major 
pondere  massae  fluidae  ascendentis  :  quanquam  nisi  excessus  fuerit  major ;  lentissimus 
erit  motus ;  ipsum  autem  pondus  coget  particulas  ad  se  invicem  accedere  nonnihil,  donee 
obtineatur  vis  repulsiva  ipsum  elidens,  uti  supra  ostendimus  num.  348  ;  adeoque  in  statu 
aequilibrii  se  particulae,  in  hoc  -etiam  casu,  repellent,  sed  erunt  citra,  &  propre  ejusmodi 
limites,  ultra  quos  vis  attractiva  sit  ad  sensum  nulla.  Quod  si  figura  particularum  praeterea 
fuerit  sphaerica,  multo  facilior  habebitur  motus  in  omnem  plagam  ob  ipsam  circumquaque 
uniformem  figuram. 


Eadem  ratio,  &  in  429.  In.  secundo,  ac  tertio  genere  motus  itidem  habebitur  facilis,  si  particulae  sphaericae 
inter*  ipsa?~  sint,  &  paribus  a  centre  distantiis  homogeneae,  ut  nimirum  vires  dirigantur  ad  centra.  In 
ejusmodi  enim  particulis  motus  quidem  unius  particulae  circa  aliam  omni  difficultate  carebit, 
&  vires  mutuae  solum  accessum  vel  recessum  impedient.  Hinc  impresso  motu  particulis 
aliquot,  poterunt  ipsae  mover!  in  gyrum  alias  circa  alias,  &  alia  succedere  poterit  loco  ab 
alia  relicto,  quin  partes  remotiores  motum  ejusmodi  sentiant  :  quanquam  fere  semper 
fortuita  quaedam  particularum  dispositio  hiatus,  qui  necessario  relinqui  debent  inter  globos, 
&  directio  impressionis  varia  inducent  etiam  accessus  &  recessus  aliquos,  quibus  fiet,  ut 


A  THEORY  OF  NATURAL  PHILOSOPHY  305 

spread  simultaneously  in  any  sensible  degree  to  parts  further  off.  Hence  it  comes  about  that 
fluids  yield  to  any  impressed  force  whatever,  &,  in  doing  so,  are  easily  moved ;  but  solids 
cannot  be  moved  except  all  together  as  a  whole,  &  thus  offer  greater  resistance  to  an  impressed 
force.  Those  fluids  which  offer  a  considerable  resistance,  but  one  that  is  not  so  great  as  it 
is  in  the  case  of  solids,  are  called  viscous ;  again,  fluids  are  said  to  be  moist  when  they 
adhere  to  a  solid  that  is  moved  away  from  them,  &  dry  if  they  do  not  do  so. 

427.  All  these  phenomena  can  be  presented  by  means  of  the  single  difference,  which  The  origin  of  fluid- 
I  have  already  considered  in  the  different  texture  of  particles.     For,  to  begin  with  fluidity,  jp. '  three  kinds  °* 
we  have  first  of  all  that  in  fluids  all  the  particles  must  be  in  equilibrium,  whilst  they  are 

at  rest ;  &,  if  they  are  not  under  the  action  of  an  external  force,  or  driven  in  a  certain 
direction,  that  equilibrium  must  be  due  to  the  mutual  actions  alone.  But  we  do  not  have 
this  sort  of  case  here,  when  considering  the  fluids  about  us,  which  are  under  the  action  of 
the  weight  of  a  superincumbent  mass,  &  some  of  them,  like  air,  are  also  acted  upon  by  the 
walls  of  the  vessel  in  which  they  are  enclosed  ;  hence,  in  all  of  these,  there  must  be  some 
repulsive  force  between  the  particles  next  to  one  another,  although,  as  will  now  be  evident, 
there  may  also  be  an  attraction  between  more  remote  particles.  Now,  three  kinds  of 
fluids  can  be  considered  ;  one  kind,  in  which,  amongst  its  greater  parts,  no  mutual  force 
between  its  particles  is  shown  ;  another  kind,  in  which  a  repulsive  force  appears ;  &  a  third 
kind,  in  which  there  is  an  attractive  force.  Of  the  first  kind  are  nearly  all  powders  &  sands, 
such  as  those,  from  which  are  constructed  clocks  similar  to  the  clepsydras  of  the  ancients ; 
&  these  approximate  very  closely  to  the  nature  of  fluids,  if  they  have  sufficiently  polished 
surfaces,  such  as  we  see  in  some  grains,  like  millet ;  for,  the  greater  part  of  them  have  some 
roughness,  &  inequalities,  which  render  motion  more  difficult.  To  the  second  class  belong 
the  elastic  fluids,  such  as  the  air ;  &  of  the  third  kind  are  such  liquids  as  water  &  mercury. 
Further,  it  has  been  shown  particularly  in  Art.  222,  422,  that  it  is  possible  for  two  particles, 
made  up  even  of  the  same  number  of  points,  though  differently  distributed,  to  have  the  sums 
of  the  forces  corresponding  to  them  so  different,  even  at  the  same  distances  from  the  centre, 
that  some  of  them  attract,  some  repel,  &  some  have  no  action  at  all  upon  one  another  : 
hence,  my  Theory  furnishes  such  differences  in  abundance.  However,  there  are  many 
things  to  be  carefully  noted  in  each  case ;  for  even  when  no  attractive  force  is  in  evidence, 
there  is  a  repulsive  force  between  adjacent  particles,  as  I  mentioned  just  above ;  &  this 
will  be  evident  without  saying  anything  further. 

428.  Moreover,  in  the  first  case  it  is  at  once  apparent  why  there  is  easy  movement  of  The  source  of  the 
the  particles.     For,  since  when  the  distance  is  increased  the  particles  do  not  attract  one  particie'°of "fluids  of 
another  with  any  sensible  force,  the  one  does  not  follow  the  motion  of  the  other ;   except  the  first  kind, 
when  the  former  moves  towards  the  latter  &  approaches  it  to  such  an  extent  that,  just  as 

happens  in  the  cases  of  impact  of  bodies,  it  is  forced  to  give  way  to  it  by  a  mutual  repulsive 
force ;  &  this  giving  way  would  easily  take  place,  if  the  surfaces  were  sufficiently  smooth, 
so  that  the  projecting  hillocks  of  one  did  not  hinder  the  motion  by  sticking  into  the  tiny 
gaps  of  another  ;  &  if  there  were  some  place,  to  which  the  particles  could  be  forced  in  a 
curved  path,  or  elevated,  or  could  break  through  an  orifice  opened  to  them,  they  might 
give  way.  This  may  easily  happen  ;  no  other  force  would  be  required  for  the  motion  except 
that  necessary  to  overcome  the  force  of  inertia  ;  or,  if  heavy  particles  are  attracted  towards 
an  external  mass,  as  with  us  towards  the  Earth,  &  the  fluid  has  to  ascend,  then  no  other  force 
is  required  save  that  necessary  to  overcome  gravity.  But  to  overcome  the  force  of  inertia 
alone  any  active  force,  however  small,  is  sufficient ;  &  to  overcome  gravity,  in  this  kind  of 
fluids,  if  there  is  perfect  smoothness,  any  force  that  is  a  little  greater  than  the  weight  of  the 
ascending  part  of  the  fluid  will  suffice  ;  although,  unless  the  excess  were  considerable,  the 
motion  would  be  very  slow.  Moreover,  the  weight  of  the  fluid  will  force  the  particles 
somewhat  closer  together,  until  a  mutual  repulsive  force  is  produced  which  will  cancel  it,  as 
I  showed  above  in  Art.  348.  Thus,  when  in  a  state  of  equilibrium  the  particles,  even  in 
this  case,  will  repel  one  another  ;  but  they  will  lie  on  the  near  side  of,  &  close  to  such 
limit-points  as  have  the  attractive  force  on  the  far  side  of  them  practically  zero.  But  if, 
in  addition  the  shape  of  the  particles  should  be  spherical,  there  would  be  much  easier 
movement  in  all  directions  due  to  the  uniformity  of  shape  all  round. 

429.  In  the  second  &  third  classes  of  fluids  there  is  also  easy  movsment,  if  the  particles  The  same  argument 
are  spherical,  &  homogeneous  at  equal  distances  from  their  centres,  that  is  to  say,  so  that  the  other°tw°o 
the  forces  are  directed  towards  their  centres.     For,  in  the  case  of  such  particles,  the  motion  kinds  ;  differences 
of  one  particle  round  another  lacks  difficulty  of  any  sort,  &  the  mutual  forces  prevent 

approach  or  recession  only.  Hence,  if  a  motion  be  impressed  on  any  number  of  particles, 
they  could  move  in  curved  paths  round  one  another,  &  some  could  take  the  place  left  free 
by  others,  without  the  parts  further  off  feeling  the  effects  of  such  motion  ;  although  nearly 
always  the  accidental  arrangement  of  the  gaps  empty  of  particles,  which  must  of  necessity 
be  left  between  the  spheres,  &  the  varied  direction  of  the  pressures  will  lead  also  to  approach 


306  PHILOSOPHIC  NATURALIS  THEORIA 

motus  ad  remotiores  etiam  particulas  deveniat,  sed  eo  minor,  quo  major  fuerit  earum 
distantia.  Verum  hie  notandum  erit  discrimen  ingenis  inter  duos  casus,  in  quibus  partes 
fluidi  se  repellunt,  &  casus,  in  quibus  se  attrahunt. 

in  elasticis  fluidis          430.  In  primo  casu  particulae  proximae  debebunt  se  omnino  repellere,  &  vis  ex  parte 

particulas   esse  ex-      i.  v  j          •  i      *•  ,     •*  ,.  ,.,  f 

traiimitessub  altera  elidet  vim  ex  altera  ;  sed  si  repente  relmquatur  libertas  ex  parte  quavis,  sine  ulla 
arcubus  repuisivis  externa  vi,  sed  sola  ilia  particularum  actione  mutua,  recedent  reipsa  particulse  a  se  invicem, 
&  fluidum  dilatabitur ;  quin  [197]  immo  externa  vi  opus  est,  ad  continendam  in  eo  statu 
massam  ejusmodi,  uti  aerem  gravitas  superioris  atmosphaerae  continet,  vel  in  vase  occluso 
vasis  ipsius  parietes ;  &  aucta  ilia  externa  vi  comprimente  augeri  poterit  compressio, 
imminuta  imminui.  Particulae  illse  inter  se  non  erunt  in  limitibus  quibusdam  cohaesionis, 
sed  erunt  sub  repulsivo  arcu  curvae  exprimentis  vires  compositas  particularum  ipsarum. 


in  fluidis  humidis          4.31.  At  in  tertio  genere  particulas  quidem  proximae  se  mutuo  repellent,  repulsione 
fore  sequali  illi  vi,  quae  necessaria  est  ad  elidendam  vim  externam,  &  ad  elidendam  pressionem, 


proximum,    &    si  quae  oritur  a  remotiorum  attractionibus  :  verum  si  fluidum  est  parum  admodum  compres- 

debere  haberTprope  sibile,  vel  etiam  nihil  ad  sensum,  ut  aqua  ;    debent  esse  citra,  &  admodum  prope  limitem, 

vaiidissimum  arcum  ultra  quern  vel  immediate,  vel  potius,  si  id  fluidum  neque  distrahitur  (ut  nimirum  durante 

repuisivum.  gua  forma  nequeat  acquirere  spatium  multo  majus,  quod  itidem  in  aqua  accidit)  habeat 

post  limites  alios  satis  inter  se  proximos  arcum  attractivum  ad  distantias  aliquanto  majores 

protensum,  a  quo  attractio  ilia  prodeat,  quae  se  in  ejusmodi  fluidorum  massulis  prodit  ; 

licet  si  iterum  id  fluidum  ma  j  ore  vi  abire  possit  in  elasticos  vapor  es,  ut  ipsa  aqua  post  eum 

attractivum  arcum  ;   arcus  repulsivus  debeat  succedere  satis  amplus,  juxta  ea,  quse  diximus 

num.  195. 


Motus  non  obstante  ^32.  In  hoc  fluidi  genere  illud  mirum  videri  potest,  quod  ilia  attractiva  vis,  quae  in 
quad  ad  moTum  niajoribus  succedit  distantiis,  &  ille  validus  cohaesionis  limes,  qui  &  compressionem  & 
aliquot  particu-  rarcfactionem  impedit,  non  impediat  divisionem  massae,  &  separationem  unius  partis  massae 

larum  non  debeant     iv  A.  j     •  j  *_   *i     .e     •  •!•  •       o  ••          TJ-  L-T.  i 

mover!  remotae  a"  aua-  At  quomodo  id  facile  fieri  ibi  possit,  &  non  possit  in  solidis,  patebit  hoc  exemplo. 
simul  ut  in  solidis.  Concipiatur  Terrae  superficies  sphaerica  accurate,  &  bene  laevigata,  ac  gravitas  sit  eiusmodi, 

Exemplumin   qua-  •      j-  '  c  -LT  •        •  •  j-  • 

dam  hypothesi  gio-  ut  ln  distantia  perquam  exigua  hat  jam  msensibilis,  ut  vis  magnetica  in  exigua  distantia 
borum  gravium.  sensum  jam  effugit.  Sint  autem  globi  multi  itidem  laeves  mutua  attractiva  vi  praediti, 
quae  vim  in  totam  Terram  superet.  Si  quis  unum  ejusmodi  globum  apprehendat,  &  attollat ; 
secundus  ipsi  adhaerebit  relicta  Terra,  &  post  ipsum  ascendet,  reliquis  per  superficiem 
Terrae  progredientibus,  donee  alii  post  alios  eleventur,  vi  in  globum  jam  elevatum  superante 
vim  in  Terram.  Is,  qui  primum  manu  teneret  globum,  sentiret,  &  deberet  vincere  vim 
unius  tantummodo  globi  in  Terram,  quern  separat,  cum  nulla  sit  difficultas  in  progressu 
reliquorum  per  superficiem  Terrae,  quo  distantia  non  augetur,  &  globorum  jam  altiorum 
vis  in  Terram  ponatur  insensibilis.  Vinceret  igitur  aliorum  vim  post  vim  aliorum,  &  vis 
ab  eo  adhibita  major  tantummodo  vi  globi  unici  requireretur  ad  rem  praestandam.  At  si 
illi  globi  deberent  elevari  simul,  ut  si  simul  omnes  colligati  essent  per  virgas  rigidas ;  deberent 
utique  omnes  illae  vires  omnium  in  Terram  simul  superari,  &  requireretur  vis  major  omnibus 
simul.  Res  eodem  redit,  ac  ubi  fasciculus  virgarum  [198]  debeat  totus  frangi  simul,  vel 
potius  debeant  aliae  post  alias  frangi  virgae. 


Appiicatio  exempli  433.  Jd  ipsum  est  discrimen  inter  fluida  huius  generis,  &  solida.     In  his  motus  parti- 

ad  fluida,  &  sohda :        ,     ~JJ      .        r          .....  ...        J.      °  .     .  ..   * 

successiva  particu-  cularum  circa  particuias  liber  ob  earum  unitormitatem  permittit,  ut  separentur  aliae  post 
larum  separatio  in  aljas  •  dum  in  solidis  vis  in  latus,  de  qua  egimus  iam  in  pluribus  locis,  &  anguli  prominentes, 

fluidis.  r  •  1      •  •  j-  •  i*  11  •    r  • 

ac  ngurarum  irregulantas,  impediunt  ejusmodi  liberum  motum,  qui  fiat  sine  mutatione 
distantiarum,  &  cogunt  divulsionem  plurimarum  particularum  simul :  unde  oritur 
difficultas  ilia  ingens  dividend!  a  se  invicem  particulas  solidas,  quae  in  divisione  fluidorum 
est  adeo  tenuis,  ac  ad  sensum  nulla. 


a 


A  THEORY  OF  NATURAL  PHILOSOPHY  307 

&  recession  of  some  kind  ;  &  through  these  it  will  come  about  that  the  effect  of  the  motion 
will  reach  the  particles  further  off,  although  this  will  be  the  less,  the  greater  the  distance 
they  are  away.  But  here  we  have  to  notice  the  great  difference  between  the  two  cases, 
the  one,  in  which  the  parts  of  the  fluid  repel,  &  the  other,  in  which  they  attract,  one 
another. 

4.30.  In   the   first  case  adjacent  particles  must  repel  one  another,  in  every  instance,  In  e!aftic  fluids  the 

o      i        r  i-  11/-1-  -HT  TII  particles    are    out- 

&  the  force  from  one  part  must  cancel  the  force  from  another  part.     Moreover,  if  all  at  side  the  limit- 

once  freedom  of  movement  is  left  in  any  one  part,  without  any  external  force  to  prevent  it,  P?^nts-   &  under 

then  by  the  mutual  action  of  the  particles  alone,  these  particles  will  of  themselves  recede 

from  one  another  &  the  fluid  will  expand.     Indeed,  what  is  more,  there  is  need  of  an  external 

force  to  maintain  a  mass  of  this  kind  in  its  original  state,  just  as  the  gravity  of  the  upper 

atmosphere  constrains  the  air,  or  the  walls  of  a  vessel  the  air  contained  within  it.     When 

this  compressing  external  force  is  increased  the  compression  can  be  increased,  &  if  diminished 

diminished.    The  particles  themselves  will  not  be  at  distances  from  one  another  corresponding 

to  limit-points  of  cohesion  of  any  sort ;   but  these  will  correspond  to  a  repulsive  arc  of  the 

curve  that  represents  the  resultant  forces  of  the  particles. 

431.  Again,  in  the  third  kind,  adjacent  particles  must  indeed  repel  one  another,  the  in  watery  fluids  the 
repulsion  being  equal  to  that  force  that  is  necessary  to  cancel  the  external  force,  &  also  must  be™  ve'ry 
the  pressure  which  arises  from  the  attractions  of  points  further  off.     But,  if  the  fluid  is  strong  one,  of  co- 
only  very  slightly  compressible,  or  not  to  any  appreciable  extent  (like  water,  for  example),  fluid  goes  off  as  a 
then  the  particles  must  be  on  the  near  side,  &  quite  close  to,  a  limit-point ;   &  on  the  far  vapour,  there  must 
side  of  this  limit-point,  either  there  must  follow  immediately  a  comparatively  ample  attractive  sterongSrep°uisiveVarc! 
arc  ;  or,  more  strictly  speaking,  if  the  fluid  does  not  expand  (that  is  to  say,  whilst  it  maintains 

its  form,  it  cannot  acquire  much  more  space,  which  is  also  the  case  with  water),  then  it 
has,  after  several  other  limit-points  fairly  close  to  one  another,  an  attractive  arc  extending 
to  somewhat  greater  distances,  to  which  is  due  that  attraction  which  is  seen  in  small 
globules  of  fluids ;  but  if,  with  a  greater  force  applied,  the  fluid  can  after  that  go  off 
to  still  further  distances  in  the  form  of  elastic  vapours  (as  water  does),  then,  after  the 
attractive  arc  we  must  have  the  above-mentioned  comparatively  ample  repulsive  arc  ; 
as  was  shown  in  Art.  195. 

432.  In  this  kind  of  fluid  it  may  appear  strange  that  the  attractive  force  which  follows  Mutual    force  not 
at  greater  distances,  or  the  strong  limit-point  of  cohesion,  which  prevents  both  compression  „!" n ""fs  "Vasy" 
&  rarefaction,  does  not,  either  of  them,  prevent  division  of  the  mass  or  the  separation  of  because  particles 
one  part  of  it  from  the  other.     But  the  reason  why  this  can  take  place  here,  &  not  in  the  not  move^at  "the 
case  of  solids,  will  become  evident  on  considering  the  following  example.     Suppose   the  same   time,    when 
surface  of  the  Earth  to  be  perfectly  spherical,  &  quite  smooth  ;    &  suppose  gravity  to  be  al^^are  moved; 
such,  that  when  the  distance  becomes  very  small  it  becomes  insensible,  just  as  magnetic  as  is  the  case  for 
force  practically  vanishes  at  a  very  small  distance.     Then,  suppose  we  have  a  number  of  ^el  hypothesis6  of 
smooth  spheres  endowed  with  an  attractive  force  for  one  another,  which  exceeds  the  force  heavy  spheres. 
each  has  for  the  whole  Earth.     If  one  of  these  spheres  is  taken  &  lifted,  a  second  one  will 

adhere  to  it  &  leave  the  ground,  &  ascend  after  it ;  the  rest  will  move  along  the  surface 
of  the  Earth,  until  one  after  the  other  they  are  also  lifted  up,  the  attraction  towards  the 
sphere  just  lifted  exceeding  the  attraction  towards  the  Earth.  The  person,  who  took  hold 
of  the  first  sphere,  would  feel  &  would  have  to  overcome  the  force  of  only  the  one  sphere 
towards  the  Earth,  namely,  that  of  the  one  he  takes  away  ;  for  there  is  no  difficulty  about 
the  progress  of  the  rest  of  the  spheres  along  the  surface  of  the  Earth,  supposing  that  the  dis- 
tance is  not  increased,  &  assuming  that  the  force  towards  the  Earth  of  spheres  already  lifted 
is  quite  insensible.  Hence  the  force  of  one  after  that  of  another  would  be  overcome,  & 
the  whole  business  would  be  accomplished  by  his  using  a  force  that  was  just  greater  than 
the  force  due  to  a  single  sphere.  But  if  all  the  spheres  had  to  be  raised  at  once,  as  if  they 
were  all  bound  together  by  rigid  rods,  it  would  be  necessary  to  overcome  at  one  time  all 
the  forces  of  all  the  spheres  upon  the  Earth,  &  there  would  be  required  a  force  greater 
than  all  these  put  together.  It  is  just  the  same  sort  of  thing  as  when  a  whole  bundle  of  rods 
has  to  be  broken  at  the  same  time,  or  rather  the  rods  have  to  be  broken  one  after  another. 

433.  This  is  exactly  what  causes  the  difference  between  fluids  of  this  kind  &  solids.  Application  of  the 

TT7.  *•     ,         ,  ,     '  ,  .  ,      ,  .   ,  !_•      example  to  the  case 

With  the  former,   the  free   motion   of  the  particles   about  one  another,  due   to  their  of  flui<is  &  solids ; 
uniformity,  allows  them  to  be  separated  one  after  the  other.      Whilst,  with  solids,  lateral  successive    separa- 

,  .<'..,  1111.  11  ••  in-  i      •   •        tion  of  the  particles 

force,  with  which  we  have  already  dealt  m  several  places,  projecting  angles  &  irregularities  m  the  case  of  fluids. 

of  shape,  prevent  such  freedom  of  motion,  as  (with  fluids)  takes  place  without  any  change 

in  the  mutual  distances  ;  &  they  compel  us  to  tear  away  a  very  great  number  of  particles 

all  at  once.      This  is  the  cause   of  the  very  great   difficulty  in  the  way  of  dividing  the 

particles  of  solids  from  one  another  ;   &  is  the  reason  why  the  difficulty  is  very  slight,  or 

practically  nothing,  when  dividing  fluids. 


308  PHILOSOPHISE  NATURALIS  THEORIA 

Exempium     ipsius          434.  Successivam  hujusmodi  separationem  particularum  aliarum  post  alias  videmus 

tiamqin'  fluidisfTd  utique  in  ipsis  aquae  guttis  pendentibus,  quae  ubi  ita  excreverunt  :  ut  pondus  totius  guttae 

separationem    fieri  superet  vim  attractivam  mutuam  partium  ipsius ;    non  divellitur  tota  simul  ingens  cjus 

soiidis,6 si '  veiocitas  aliqua  massa,  sed  a  superiore  parte,  utut  brevissimo  tempore,  attenuatur  per  gradus ;.  donee 

debeat  esse  ingens.    illud  veluti  filum  jam  tenuissimum  penitus  superetur.     Fuerunt  prius   mille  particulae 

in  superficie,    quae    guttam  pendentem  connectebant  cum  superiore  parte  aquas,  quas 

relinquitur  adhaerens  corpori,  ex  quo  pendebat  gutta,  fiunt  paullo  post  ibi  900,  800,  700  : 

&  ita  porro  imminuto  earum  numero  per  gradus,  dum  laterales  accedunt  ad  se  invicem, 

&  attenuatur  figura  :    quarum  idcirco  resistentia  facile  vincitur,  ut  ubi  in  illo  virgarum 

fascicule  frangantur  aliae  post  alias.     At  ubi  celerrimo  motu  in  fluidum  ejusmodi  incurritur 

ita  ;    ut  non  possint  tarn  brevi  tempore  aliae  aliis  particulae  locum  dare,  &  in  gyrum  agi ; 

turn  vero  fluida  resistunt,  ut  solida.     Id  experimur  in  globis  tormentariis,  qui  ex  aqua 

resiliunt,  in  earn  satis    oblique  projecti,  ut  manente  satis  magna    horizontali  velocitate 

collisio  in  perpendicular!  fiat  more  solidorum  :    ac  eandem  quoque  resistentiam  in  aqua 

scindenda  experiuntur,  qui  se  ex  editiore  loco  in  earn  demittunt. 


Soiiditatis  causa  in          4-7  r.  Hinc  autem   pronum  est  videre,   unde  soliditatis   phaenomena   ortum   ducant. 

vi,  &  motu  in  latus :    •»-,-.     .  ,  .  •      t  c  j-        i  i_       •  i    j-       -i 

exempium  in  parai-  JNimirum  ubi  particularum  ngura  recedit  piurjmum  a  sphaenca,  vei  distnbutio  punctorum 
leiepipedis.  intra  particulam  inaequalis  est,  ibi  nee  habetur  libertas  ilia  motus  circularis,  &  omnia,  quae 

ad  soliditatem  pertinent,  consequi  debent  ex  vi  in  latus.  Cum  enim  una  particula  respectu 
alterius  non  distantiam  tantummodo,  sed  &  positionem  servare  debeat ;  non  solum,  ea 
promota,  vel  retracta,  alteram  quoque  promoveri,  vel  retrahi  necesse  est ;  sed  praeterea, 
ea  circa  axem  quencunque  conversa,  oportet  &  illam  aliam  loco  cedere,  ac  eo  abire,  ubi 
positionem  priorem  respectivam  acquirat ;  quod  cum  &  tertia  respectu  secundas  prsestare 
debeat,  &  omnes  reliquae  circunquaque  circa  illam  positae  ;  patet  utique,  non  posse  motum 
in  eo  casu  imprimi  parti  cuipiam  systematis ;  quin  &  totius  systematis  motus  consequatur 
respectivam  po-[i99]-sitionem  servantis,  quae  est  ipsa  superius  indicata  solidorum  natura. 
Res  autem  multo  adhuc  magis  manifesta  fit,  ubi  figura  multum  abludat  a  sphserica,  ut  si 
sint  bina  parallelepipeda  inter  se  constituta  in  quodam  cohaesionis  limite,  alterum  ex  adverso 
alterius.  Alterum  ex  iis  moveri  non  poterit,  nisi  vel  utrinque  a  lateribus  accedat  ad  alterum, 
vel  utrinque  recedat,  vel  ex  altero  latere  accedat,  &  recedat  ex  altero.  In  primo  casu 
imminuta  distantia  habetur  repulsiva  vis,  &  illud  alterum  progreditur  :  in  secundo,  eadem 
aucta,  habetur  attractio,  &  illud  secundum  ad  prioris  motum  consequitur  ;  in  tertio  casu, 
qui  haberi  non  potest,  nisi  per  inclinationem  prioris  parallelepipedi,  altero  latere  attracto, 
&  altero  repulso  inclinari  necesse  est  etiam  secundum  ;  quo  pacto  si  ejusmodi  parallelepi- 
pedorum  sit  series  quaedam  continua,  quse  fibram  longiorem,  vel  virgam  constituat ;  inclinata 
basi,  inclinatur  illico  series  tota  :  &  si  ex  ejusmodi  particulis  massa  constet ;  tota  moveri 
debet  ac  inclinari,  inclinato  latere  quocunque. 


om  n?ushUund  43  ^'  Quod  de  parallelepipedis  est  dictum,  id  ipsum  ad  figuras  quascunque  transferri 
discrimen  inte  potest  inaequales  utcunque,  quae  ex  altero  latere  possint  accedere  ad  aliam  particulam,  ex 
flexiiia,  &  rigida.  altero  recedere  :  habebitur  semper  motus  in  latus,  &  habebuntur  soliditatis  phasnomena, 
nisi  paribus  a  centro  distantiis  homogeneae,  &  sphaericas  formae  particulae  sint.  Verum 
ingens  in  eo  motu  discrimen  erit  inter  diversa  corpora.  Si  nimirum  vires  illae  hinc,  & 
inde  a  limite,  in  quo  particulae  constitutae  sunt,  sint  admodum  validae  ;  motus  in  latus 
fiet  celerrime,  &  nulla  flexio  in  virga,  aut  in  m'assa  apparebit ;  quanquam  erit  utique  semper 
aliqua.  Si  minores  sint  vires ;  longiore  tempore  opus  erit  ad  motum,  &  ad  positionem 
debitam  acquirendam,  quo  casu,  inclinata  parte  ima  virgas,  nondum  pars  summa  obtinere 
potest  positionem  jacentem  in  directum  cum  ipsa,  adeoque  habebitur  inflexio,  quae  quidem 
eo  erit  major,  quo  major  fuerit  celeritas  conversionis  ipsius  virgae,  uti  omnino  per  experi- 
menta  deprehendimus. 

Discrimen    inter          437.  Nee  vero  minus  facile  intelligitur  illud,  quid  intersit  inter  flexiiia  solida  corpora, 

unde!a>    &  fraglha  &  fragilia.     Si  nimirum  vires  hinc,  &  inde  ab  illo  limite,  in  quo  sunt  particular,  extenduntur 

ad  satis  magnas  distantias  eaedem,  arcu  utroque  habente  amplitudinem  non  ita  exiguam ; 


A  THEORY  OF  NATURAL  PHILOSOPHY  309 

434.  We  certainly  see  an  example  of  this  kind  of  successive  separation  of  particles,  one  Example  of  this  in 
after  the  other,  in  the  case  of  drops  of  water  hanging  suspended  ;   here,  as  soon  as  they  *£°  e^jdrtaace*to 
have  increased  up  to  a  point  where  the  weight  of  the  whole  drop  becomes  greater  than  separation  in  fluids 
the  mutual  attractive  force  of  its  parts,  any  great  part  is  not  torn  away  as  a  whole  ;   but  {^°  hfL^idf^fthe 
by  degrees,  though  in  a  time  that  is  exceedingly  short,  the  drop  is  attenuated  at  its  upper  velocity  has 'to  be 
part,  until  the  neck,  which  has  by  now  become  exceedingly  narrow,  is  finally  broken  altogether.  very  srea*- 
There  were,  say,  initially,  a  thousand  particles  in  the  surface  connecting  the  hanging  drop 

to  the  upper  part  of  the  water  which  is  left  adhering  to  the  body  from  which  the  drop 
was  suspended ;  these  a  little  afterwards  became  900,  then  800,  then  700,  &  so  on,  their 
number  being  gradually  diminished  as  the  sides  of  the  neck  approach  one  another,  &  its 
figure  is  narrowed.  Hence,  their  resistance  is  easily  overcome,  just  as  when,  in  the  bundle 
of  rods,  the  rods  are  broken  one  after  the  other.  But,  when  it  is  a  case  of  an  onset  with 
high  speed,  so  that  the  time  is  too  short  to  allow  the  particles  to  give  way  one  after  the 
other,  &  move  in  curved  paths  round  one  another  ;  then,  indeed,  fluids  resist  in  just  the 
same  way  as  solids.  This  is  to  be  observed  in  the  case  of  cannon-balls,  which  rebound 
from  the  surface  of  water,  when  projected  at  sufficiently  small  inclination  to  it ;  so  that, 
whilst  the  horizontal  velocity  remains  sufficiently  great,  the  vertical  impact  takes  place 
in  the  manner  of  that  between  solids.  Also,  those  who  dive  into  water  from  a  fairly  great 
height  will  experience  the  same  resistance  in  cle'aving  the  surface. 

435.  Further,  from  what  has  been  said,  it  can  be  se£n  without  difficulty  whence  the  The  cause  ol  solid- 
phenomena  of  solidity  defrive  their  origin.     For  instance,  when  the  shape  of  the  particles  f0rce  1CS&    motion ; 
is  very  far  from  being  spherical,  or  the  distribution  of  the  points  within  the  particle  is  not  example  ?*this  in 
uniform,  then  there  is  not  that  freedom  of  circular  motion  ;   &  all  things  that  pertain  to  pai 

solidity  must  follow  from  the  presence  of  lateral  force.  For,  since  one  particle  must  preserve 
not  only  its  distance,  but  also  its  position  with  regard  to  another  ;  not  only,  when  the 
one  is  driven  forwards  or  backwards,  must  the  other  also  be  driven  forwards  or  backwards, 
but  also  if  the  one  is  turned  about  any  axis,  it  is  necessary  that  the  other  should  give  way 
&  move  off  to  the  place  in  which  it  will  acquire  its  original  relative  position.  Since  also 
the  third  must  do  the  same  thing  with  respect  to  the  second,  &  all  the  rest  of  the  particles 
round  it  in  all  directions,  it  is  quite  clear  that  in  this  case  motion  cannot  be  imparted 
to  any  part  of  the  system,  without  a  motion  of  the  whole  system  following  it,  in  which  the 
mutual  position  is  preserved  ;  &  this  is  the  very  nature  of  solids  that  was  mentioned  above. 
Moreover,  the  matter  becomes  even  still  more  evident,  when  the  shape  differs  considerably 
from  the  spherical ;  for  instance,  if  we  have  a  pair  of  parallelepipeds  situated  with  regard 
to  one  another  at  a  distance  corresponding  to  a  limit-point  of  cohesion,  opposite  one  another. 
It  will  not  be  possible  for  one  of  them  to  be  moved,  unless  either  it  approaches  the  other 
laterally  at  both  ends,  or  recedes  at  both  ends,  or  else  approaches  at  one  end  &  recedes  at 
the  other.  In  the  first  case,  the  distance  being  diminished,  we  have  a  repulsive  force,  & 
the  second  particle  will  move  away  ;  in  the  second  case,  the  distance  being  increased,  there 
will  be  an  attraction,  &  the  second  particle  will  follow  the  motion  of  the  first.  In  the 
third  case,  which  cannot  take  place  unless  there  is  an  inclination  of  the  first  parallelepiped, 
one  end  of  the  second  being  attracted,  &  the  other  repelled,  it  is  necessary  that  the  second 
particle  should  also  be  inclined.  In  this  way,  if  there  is  a  continuous  series  of  such 
parallelepipeds,  forming  a  fairly  long  fibre  or  rod,  then,  when  the  base  is  inclined,  the 
whole  rod  must  be  inclined  along  with  it ;  &  if  a  mass  is  formed  from  such  particles,  then 
if  any  side  of  the  mass  is  inclined,  the  whole  of  the  mass  must  move  along  with  it  &  be  also 
inclined. 

436.  What  has  been  said  with  regard  to  parallelepipeds  can  be  said  also  about  any  The  same  thing  for 
figures  whatever  which  are  at  all  irregular,  if  they  can  approach  another  particle  at  one  the  ^tfleTence^be6 
side  &  recede  from  it  on  the  other  side  ;    there  will  in  every  case  be  motion  to  one  side,  tween    flexible    & 
&  the  phenomena  of  solidity  will  be  obtained,  unless  the  particles  are  homogeneous  at  n§ 

equal  distances  from  the  centre  &  spherical  in  form.  But  in  this  motion  there  is  a  very 
great  difference  among  different  bodies.  If,  for  instance,  the  forces  on  either  side  of  the 
limit-point,  in  which  the  particles  are  situated,  are  quite  strong,  the  lateral  motion  will 
be  very  swift,  &  no  bending  will  be  observed  in  the  rod  or  in  the  mass ;  although  there 
certainly  will  be  some  taking  place.  If  the  forces  are  not  so  great,  there  will  be  need  of 
a  longer  time  for  it  to  acquire  motion  &  the  proper  position  ;  &  in  this  case,  if  the  bottom 
part  of  the  rod  is  inclined,  the  top  part  of  the  rod  cannot  for  a  little  while  attain  to  a  position 
lying  in  a  straight  line  with  the  base,  &  thus  there  will  be  bending  ;  &  this  indeed  will 
be  all  the  greater,  the  greater  the  speed  with  which  the  rod  is  turned  ;  as  is  proved  by 
experiment  to  be  always  the  case. 

437.  Nor  will  it  be  less  easy  to  understand  the  reason  why  there  is  a  difference  between  The  reason  of  the 
flexible  solids  &  fragile  bodies.     For  instance,  if  the  forces  on  each  side  of  the  limit-point,  at  fl^We" 
which  the  particles  are,  are  extended  unaltered  over  sufficiently  great  distances  from  it,  &  the  bodies. 


310  PHILOSOPHIC  NATURALIS  THEORIA 

turn  vero,  vi  externa  adhibita  utrique  extreme,  vel  majore  velocitate  impressa  alteri, 
incurvabitur  virga,  atque  inflectetur,  sed  sibi  relicta  ad  positionem  abibit  suam,  &  in  illo 
inflexionis  violento  statu  vim  exercebit  perpetuam  ad  regressum,  quod  in  elasticis  virgis 
accidit.  Si  vires  illae  non  diu  durent  hinc,  &  inde  eaedem,  vel  per  satis  magnum  intervallum 
sit  ingens  frequentia  limitum ;  turn  quidem  inflexio  habebitur  sine  conatu  ad  se  restitu- 
endam,  &  sine  fractione,  tarn  vi  adhibita  utrique  extremo,  quam  ingenti  velocitate  impressa 
alteri,  ut  videmus  accidere  in  maxime  ductilibus,  [200]  velut  in  plumbo.  Si  demum 
vires  hinc,  &  inde  per  exiguum  intervallum  durent,  post  quod  nulla  sit  actio,  vel  ingens 
repulsivus  arcus  consequatur,  qui  sequentes  attractivos  superet ;  habebitur  virga  rigida, 
&  fractio,  ac  eo  major  erit  soliditas,  &  ilia,  quae  vulgo  appellatur  durities,  quo  vires  illse 
hinc  &  inde  statim  post  limites  fuerint  majores. 


Quid,  &  unde  vis-  438.  Atque  hie  quidem  jam  etiam  ad  discrimen  devenimus  inter  elastica,  &  mollia  ; 
verum  antequam  ad  ea  faciamus  gradum,  adnotabo  non  nulla,  quae  adhuc  pertinent  ad 
solidorum,  &  fluidorum  naturam,  ac  proprietates.  Inprimis  media  inter  solida,  &  fluida, 
sunt  viscosa  corpora,  in  quibus  est  aliqua  vis  in  latus,  sed  exigua.  Ea  resistunt  mutation! 
figurae,  sed  eo  majore,  vel  minore  vi,  quo  majus,  vel  minus  est  in  diversis  particularum 
punctis  virium  discrimen,  a  quo  oritur  vis  in  latus.  Viscosa  autem  praeter  tenacitatem, 
quam  habent  inter  se,  habent  etiam  vim,  qua  adhaerent  externis  corporibus,  sed  non 
omnibus,  in  quo  ad  humidos  liquores  referuntur.  Humiditas  enim  est  itidem  respectiva. 
Aqua,  quae  digitis  nostris  adhaeret  illico,  &  per  vitrum,  ac  lignum  diffunditur  admodum 
facile,  oleaginosa,  &  resinosa  corpora  non  humectat,  in  foliis  herbarum  pinguibus  extat  in 
guttulas  eminens,  &  avium  plurium  plumas  non  inficit.  Id  pendet  a  vi  inter  particulas 
fluidi,  &  particulas  extern!  corporis ;  &  jam  vidimus  pro  diversa  punctorum  distributione 
particulas  easdem  respectu  aliarum  debere  habere  in  eadem  directione  vim  attractivam, 
respectu  aliarum  repulsivam  vim  &  respectu  aliarum  nullam. 

Organicorum  cor-          439.  In  particulis  illis,  quae  ad  soliditatem  requiruntur,  invenitur  admodum  expedita 

porum     eSormatio          •        I  •      j        rj  •  •  j    TVL  j     •       • 

per  vires  in  latus  ratio  phaenomeni  ad  solida  corpora  pertmentis,  quod  Physicos  in  summam  admirationem 
versus  certa  super-  rapit,  nimirum  dispositio  quaedam  in  peculiares  quasdam  figuras,  quae  in  salibus  inprimis 

ficiei  puncta.  ,  .         ,"  .  .    ^    .    .  &          >  .     , 

apparent  admodum  constantes,  in  glacie,  &  in  mvium  stellulis  potissimum  adeo  sunt 
elegantes  etiam,  &  ad  certas  quasdam  leges  accedunt,  quas  itidem  cum  constanti  admodum 
figurarum  forma  in  gemmarum  succis  simplicibus  observamus,  quae  vero  nusquam  magis 
se  produnt,  quam  in  organicis  vegetabilium,  &  animalium  corporibus.  In  hac  mea  Theoria 
in  promptu  est  ratio.  Si  enim  particulae  in  certis  suae  superficiei  partibus  quasdam  alias 
particulas  attrahunt,  in  aliis  repellunt ;  facile  concipitur,  cur  non  nisi  certo  ordine  sibi 
adhaereant,  in  illis  nimirum  locis  tantummodo,  in  quibus  se  attrahunt,  &  satis  firmos  limites 
nancisci  possunt,  adeoque  non  nisi  in  certas  tantummodo  figuras  possint  coalescere. 
Quoniam  vero  praeterea  eadem  particula,  eadem  sui  parte,  qua  alteram  attrahit,  alteram 
pro  ejus  varia  dispositione  repellit ;  dum  massa  plurium  particularum  temere  agitata 
prastervolat ;  eae  tantummodo  sistentur,  quae  attrahuntur,  &  ad  ea  se  applicabunt  puncta, 
ad  quae  maxime  attrahuntur,  ac  in  illis  haebebunt,  in  quibus  post  accessum  maxime  tenaces 
limites  [201]  nanciscentur  ;  unde  &  secretionis,  &  nutritionis,  vegetationis,  &  certarum 
figurarum  patet  ratio  admodum  manifesta.  Et  haec  quidem  ad  nutritionem,  &  ad  certas 
figuras  pertinentia  jam  innueram  num.  222,  &  423. 


Atomistarum    sys-  440.  Quoniam  ostensum  est,  qui  fieri  possit,    ut    certam    figuram    acquirant    certa 

to'tum^x  ^a"  particularum  genera,  cujus  admodum  tenacia  sint,  si  quis  omnem  veterum  corpuscularium 
Theoria,  &  cum  sententiam,  quam  Gassendus,  ac  e  recentioribus  alii  secuti  sunt,  adhibentes  variarum 
expiicata  ^netefrea  figurarum  atomos,  ut  ad  cohaesionem  uncinatas,  ab  hac  eadem  Theoria  velit  deducere, 


. 
cohaesione  partium  is  sane  poterit,  ut  patet,  &  ejusmodi  atomos  adhibere  ad   explicationem  eorum  omnium 

phaenomenorum,  quae  pendent  a  sola  cohaesione,  &  inertia,  quae  tamen  non  ita  multa  sunt  : 
poterunt  autem  haberi  ejusmodi  atomi  cum  infinita  figurae  suae  tenacitate,  &  cohaesione 
mutua  suarum  partium  per  solas  etiam  binas  asymptotes  illas,  de  quibus  num.  419,  inter 
se  satis  proximas.  Et  si  curva  virium  habeat  tantummodo  in  minimis  distantiis  duas 
ejusmodi  asymptotes,  turn  post  crus  repulsivum  ulterioris  statim  consequatur  arcus  attrac- 
tivus,  primo  quidem  plurimum  recedens  ab  axe  cum  exiguo  recessu  ab  asymptote,  turn 


A  THEORY  OF  NATURAL  PHILOSOPHY  311 

arc  on  either  side  of  it  has  an  amplitude  that  is  not  altogether  small ;  then,  if  an  external  force 
is  applied  at  both  ends  of  the  rod,  or  a  fairly  great  velocity  is  impressed  upon  one  of  the 
two  ends,  the  rod  will  be  curved,  &  bent ;  but  if  it  is  left  to  itself  it  will  return  to  its  original 
position ;  &  whilst  in  that  violent  state  of  inflection,  it  will  continuously  exert  a  force  of 
restoration,  such  as  occurs  in  elastic  rods.  If  the  forces  do  not  continue  the  same  for  such  a 
distance  on  each  side  of  the  limit-point,  or  if  in  a  sufficiently  large  interval  there  exist  a  con- 
siderable number  of  limit-points,  then  there  will  be  bending  without  any  endeavour  towards 
restoration,  &  without  fracture,  both  when  we  apply  a  force  to  each  end,  &  when  a  great 
velocity  is  impressed  upon  one  of  them  ;  we  see  this  happen  in  solids  that  are  extremely 
ductile,  like  lead.  Finally,  if  the  forces  on  either  side  of  the  limit-point  only  continue  for 
a  very  short  space,  after  which  there  is  no  action  at  all,  or  if  a  large  repulsive  arc  follows, 
such  as  overcomes  the  attractive  arcs  that  follow  it ;  then  the  rod  will  be  rigid,  &  there 
will  be  fracture  ;  &  the  solidity,  &  what  is  commonly  called  the  hardness,  will  be  the  greater 
the  greater  the  forces  on  each  side  of  the  limit-points,  &  following  immediately  after  them. 

438.  And  now  we  come  to  the  difference  between  elastic  &  soft  bodies.     But,  before  The    nature  & 
we  pass  on  to  them,  I  will  mention  a  few  matters  that  have  to  do  with  the  nature  &  properties  source  of  viscosity, 
of  solids  &  fluids.     First  of  all,  intermediate  between  solids  &  fluids  come  viscous  bodies ; 

in  these  there  is  indeed  some  force  to  one  side,  but  it  is  very  slight.  They  resist  a  change 
of  shape ;  but,  the  force  of  resistance  is  the  greater  or  the  less,  the  greater  or  the  less  the 
difference  of  the  forces  on  different  points  of  the  particles,  from  which  arises  the  force  to 
one  side.  Viscous  bodies,  in  addition  to  the  tenacity  which  they  have  within  their  own 
parts,  have  also  another  force  with  which  they  adhere  to  outside  bodies,  but  not  to  all ; 
&  in  this  they  are  related  to  watery  liquids.  For  humidity  is  also  itself  but  relative.  Water, 
which  adheres  immediately  to  our  fingers,  &  is  quite  easily  diffused  over  glass  or  wood, 
will  not  wet  oily  or  resinous  bodies ;  on  the  greasy  leaves  of  plants  it  stands  up  in  little 
droplets ;  nor  does  it  make  its  way  through  the  feathers  of  the  greater  number  of  the  birds. 
This  depends  on  the  force  between  the  particles  of  the  fluid,  &  those  of  the  external  body ; 
&  we  have  already  seen  that,  for  a  different  distribution  of  their  points,  the  same  particles 
may  have  with  respect  to  some,  in  the  same  direction,  an  attractive  force,  with  respect 
to  others  a  repulsive  force,  &  with  respect  to  others  again  no  force  at  all. 

439.  In  particles,  such  as  are  necessary  for  solidity,  there  is  found  quite  easily  the  reason  The    formation  of 
for  a  phenomenon  pertaining  to  solid  bodies,  which  is  a  source  of  the  greatest  wonder  to  "fan^oUransverse 
physicists.     That  is,  a  disposition  in  certain  special  shapes,  which  in  salts  especially  seem  forces  directed  to- 
to  be  quite  constant ;  in  ice,  &  the  star-like  flakes  of  snow  more  especially,  they  are  wonderfully  ^fa^  s^face150"113 
beautiful ;  &  they  observe  certain  definite  laws,  such  as  we  also  see,  together  with  a  constant 

shape  of  figure,  in  the  simple  constituents  of  crystals.  But  these  are  nowhere  to  be  found 
so  frequently  as  in  the  organic  bodies  of  the  vegetable  &  animal  kingdoms.  The  reason 
for  this  comes  out  directly  in  this  Theory  of  mine.  For,  if  particles,  at  certain  parts  of  their 
surfaces,  attract  other  particles,  &  at  other  parts  repel  other  particles,  it  can  easily  be 
understood  why  they  should  adhere  to  one  another  only  in  a  certain  manner  of  arrangement ; 
that  is  to  say,  in  such  places  only  as  there  is  attraction,  &  where  there  can  be  produced 
limit-points  of  sufficient  strength ;  &  thus,  they  can  only  group  themselves  together  in 
figures  of  certain  shapes.  But  since,  in  addition  to  this,  the  same  particle,  at  the  same 
part  of  its  surface,  with  which  it  attracts  one  particle,  will  repel  another  particle  situated 
differently  with  respect  to  it ;  whilst  the  mass  of  the  great  number  of  particles,  set  in 
motion  at  random,  will  slip  by,  those  only  will  stay,  which  are  attracted  ;  &  they  will  attach 
themselves  to  the  points  to  which  they  are  most  attracted,  &  will  adhere  to  those  points 
in  which,  after  approach,  limit-points  of  the  greatest  tenacity  are  produced.  From  this 
the  reason  for  secretion,  nutrition,  the  growth  of  plants,  &  fixity  of  shape,  is  perfectly  evident. 
I  have  indeed  already  remarked  on  these  matters,  as  far  as  they  pertain  to  nutrition  &  fixity 
of  shape,  in  Arts.  222  &  423. 

440.  Since  it  has  been  shown  how  it  may  be  possible  for  certain  kinds  of  particles  to  The  whole  of  the 
acquire  certain  definite  shapes,  of  which  they  are  quite  tenacious ;    if  anyone  should  wish  to  bySttthe  f°™omists 
derive  from  this  same  theory  the  whole  idea  of  the  ancient  corpuscularians,  such  asGassendi  can  be  derived 
&  others  of  the  more  modern  philosophers  have  followed,  employing  atoms  of  various  shapes,  wTTh^whi^h0?! 
hooked  together  for  cohesion  ;  he  will  certainly  be  able,  as  is  evident,  to  use  atoms  of  this  sort  agrees   very  well ; 
to  explain  all  these  phenomena  that  depend  upon  cohesion  alone,  &  inertia  ;  but  the  number  cohesfon^o"'   the 
of  these  is  not  very  great.    Moreover,  atoms  of  this  sort  can  be  had  with  an  infinite  tenacity  parts  of  their 
of  shape,  &  mutual  cohesion  of  their  parts,  by  even  the  sole  assumption  of  those  pairs  of  **°™s  IS  exPlamed 
asymptotes  sufficiently  close  to  one  another,  of  which  I  spoke  in  Art.  419.     Even  if  the 
curve  of  forces  should  have  at  very  small  distances  two  such  asymptotes  only,  &  then 
immediately  after  the  repulsive  arc  of  the  far  one  of  these  there  should  follow  an  attractive 
arc,  such  as  first  of  all  recedes  a  great  distance  from  the  axis  whilst  it  recedes  only  slightly 
from  the  asymptote,  &  then  returns  towards  the  axis  &  approximates  immediately  to  the 


3i2  PHILOSOPHISE  NATURALIS  THEORIA 

ad  axem  regrediens,  &  accedens  statim  ad  formam  gravitati  exhibendae  debitam ;  haberentur 
per  ejusmodi  curvam  atomi  habentes  impenetrabilitatem,  gravitatem,  &  figurae  suse 
tenacitatem  ejusmodi,  ut  ab  ea  discedere  non  possent  discessu  quantum  libuerit  parvo  ; 
cum  enim  possint  illse  duae  asymptoti  sibi  invicem  esse  proximo  intervallo  utcunque  parvo, 
posset  utique  ita  contrahi  intervallum  istud,  ut  figurae  mutatio  aequalis  datae  cuicunque 
utcunque  parvae  mutationi  eviteatur.  Ubi  enim  cuicunque  figurae  inscripta  est  series 
continua  cubulorum,  &  puncta  in  singulis  angulis  posita  sunt,  mutari  non  potest  figura 
externorum  punctorum  ductum  sequens  mutatione  quadam  data,  per  quam  quaedam 
puncta  discedant  a  locis  prioribus  per  quaedam  intervalla  data,  manentibus  quibusdam, 
ut  manente  basi,  nisi  per  quaedam  data  intervalla  a  se  invicem  recedant,  vel  ad  se  invicem 
accedant  saltern  aliqua  puncta,  cum,  data  distantia  puncti  a  tribus  aliis,  detur  etiam  ejus 
positio  respectu  illorum,  quae  mutari  non  potest,  nisi  aliqua  ex  iisdem  tribus  distantiis 
mutetur,  unde  fit,  ut  possit  data  quaevis  positionis  mutatio  impediri,  impedita  mutatione 
distantiae  per  intervallum  ad  earn  mutationem  necessarium.  Quod  si  illae  binae  asymptoti 
essent  tantillo  remotiores  a  se  invicem,  turn  vero  &  mutatio  distantiae  haberi  posset  tantillo 
major,  &  idcirco  singulis  distantiis  illata  vi  aliqua  posset  figura  non  nihil  mutari,  &  quidem 
exigua  mutatione  distantiarum  singularum  posset  in  ingenti  serie  punctorum  haberi  inflexio 
figurae  satis  magna  orta  ex  pluribus  exiguis  flexibus.  Sic  &  spirales  atomi  efformari  possent, 
quarum  spiris  per  vim  contractis  sentiretur  ingens  elastica  vis,  sive  determinatio  ad 
expansionem,  ac  per  hujusmodi  atomos  possent  iti-[202J-dem  plurima  explicari  phsenomena, 
ut  &  nexus  massarum  per  uncos  uncis,  vel  spiris  insertos,  quo  pacto  explicari  itidem  posset 
etiam  illud,  quomodo  in  duabus  particulis,  quarum  altera  ad  alteram  cum  ingenti  velocitate 
accesserit,  oriatur  ingens  nexus  novus,  nimirum  sine  regressu  a  se  invicem,  unco  nimirum 
alterius  in  alterius  foramen  injecto,  &  intra  illud  converso  per  virium  inaequalitatem  in 
diversas  unci  partes  agentium,  ut  jam  prodire  non  possit ;  nam  unci  cavitas,  &  foramen, 
seu  porus  alterius  particulae,  posset  esse  multo  amplior,  quam  pro  exigua  ilia  distantia 
insuperabili,  ut  idcirco  inseri  posset  sine  impedimento  orto  a  viribus  agentibus  in  minore 
distantia.  Eaedem  autem  atomi  haberi  possunt,  etiam  si  curva  habeat  reliquos  omnes 
flexus,  quos  habet  mea,  quo  pacto  ad  alia  multo  plura,  ut  ad  fermentationes  inprimis,  ac 
vaporum,  &  luminis  emissionem  multo  aptiores  erunt ;  &  sine  asymptoticis  arcubus,  qui 
vires  exhibeant  extra  originem  abscissarum  in  infinitum  excrescentes,  idem  obtineri  poterit 
per  solos  limites  cohaesionis  admodum  validos  cum  tenacitate  figurae  non  quidem  infinita, 
sed  tamen  maxima,  ubi,  quod  illi  veteres  non  explicarunt,  cohaesio  partium  atomorum 
inter  se,  adeoque  atomorum  soliditas,  ut  &  continuata  impenetrabilitatis  resistentia,  & 
gravitas,  ex  eodem  general!  derivaretur  principio,  ex  quo  &  reliqua  universa  Natura.  Illud 
unum  hie  notandum  superest,  ejusmodi  atomos  habituras  necessario  ubique  distantiam 
a  se  invicem  majorem,  quam  pro  ilia  insuperabili  distantia,  ad  quam  externa  puncta  devenire 
ibi  non  possunt. 


Cur  non  omnia          441.  Hue  etiam  pertinet  solutio  hujusmodi  difficultatis,  quae  sponte  se  objicit  :    si 

lice^omnra*  puncta  omnia  niateriae  puncta  simplicia  sunt,  &  vires  in  quavis  directione  circumquaque  exercent 

sint  circumquaque  easdem  ;    omnia  corpora  ex  iis  utique  composita  erunt  fluida  multo  potiore  jure,  quam 

ejusdem  vis.  fluida  esse  debeant,  quae  globulis  constent  easdem  in  omni  circum  directione  vires  exercen- 

tibus.     Huic  difficultati  hie  facile  occurritur  :    si  particularum  puncta  possent  vi  adhibita 

mutare  aliquanto  magis    distantias  inter  se,  nam  aliqua  etiam  ad  circulationem  exigua 

mutatio  requiritur  ;    posset  autem  imprimi  exiguo  numero  punctorum  constituentium 

unam  e  particulis  primorum  ordinum,  quin  imprimatur  simul  omnibus  ejusmodi  punctis, 

vel  satis  magno  eorum  numero,  motus  ad  sensum  idem ;    turn  utique  haberetur  idem, 

quod  habetur  in  fluidis,  &  separates  aliis  punctis  post  alia,  motus  facilis  per  omnes  omnium 

corporum  massas  obtineretur.     At  particulae  primi  ordinis  ab  indivisibilibus  punctis  ortae, 

ut  &  proximorum  ordinum  particulae  ortae  ab  iis,  sua  ipsa  parvitate  molis  tueri  possunt 

juxta  num.  424  formam  suam,  &  positionem  punctorum  :   nam  differentia  virium  exercit- 

arum  in  diversa  earum  puncta  potest  esse  perquam  exigua,  summa  virium  prohibente 

tantum  accessum  unius  particulae  ad  alteram,  quo  tamen  accessu  inaequalitas  virium,  & 


A  THEORY  OF  NATURAL  PHILOSOPHY  313 

form  proper  to  represent  gravitation  ;  by  such  a  curve  we  should  get  atoms  having 
impenetrability,  gravitation,  &  tenacity  of  shape  of  such  a  kind  that  they  would  not  be 
able  to  depart  from  this  shape  by  any  small  amount  we  wish  to  assign.  For,  since  the 
two  asymptotes  can  be  very  close  together,  distant  from  one  another  by  any  interval  no  matter 
how  small,  this  interval  can  in  every  case  be  contracted  to  such  an  extent,  that  the  change 
of  shape  will  be  just  less  than  any  given  change  no  matter  how  small.  For,  if  within  any 
figure  there  is  inscribed  a  continuous  series  of  little  cubes,  &  points  are  situated  at  each 
of  their  corners,  the  figure  cannot  be  changed,  following  the  lead  of  external  points,  by 
any  given  change  through  which  certain  points  depart  from  their  original  positions  through 
certain  given  intervals,  whilst  others  stay  where  they  are,  i.e.,  whilst  the  base,  say,  stays 
where  it  was  ;  unless  they  recede  from  one  another  through  a  certain  given  interval,  or 
approach  one  another,  or  some  of  the  points  do  so  at  least.  For,  if  the  distances  of  a  point 
from  three  other  points  are  given,  its  position  with  regard  to  them  is  also  given  ;  &  this 
cannot  be  changed  without  altering  some  one  of  the  three  distances ;  hence,  any  change 
of  position  can  be  prevented  by  preventing  the  change  of  distance  through  any  interval 
that  is  necessary  to  such  a  change  of  position.  But  if  the  pair  of  asymptotes  were  just  a 
little  further  away  from  one  another,  then  in  truth  there  would  be  possibility  of  getting  a 
change  of  distance  that  was  also  just  a  little  greater;  &  thus,  a  force  being  produced  at  each 
distance,  the  figure  might  suffer  some  change ;  &  by  a  very  slight  change  of  each  of  the 
distances  in  a  very  long  series  of  points  there  might  be  obtained  a  bending  of  the  figure  of  com- 
paratively large  amount,  due  to  a  large  number  of  these  slight  bendings.  In  such  a  way  atoms 
might  be  formed  like  spirals ;  &,  if  these  spirals  were  compressed  by  a  force,  there  would  be 
experienced  a  very  great  elastic  force  or  propensity  for  expansion  ;  also  by  means  of  atoms 
of  this  nature  an  explanation  could  be  given  of  a  very  large  number  of  phenomena,  such 
as  the  connection  of  masses  by  means  of  hooks  inserted  into  hooks  or  coils ;  &  in  this  way 
also  an  explanation  could  be  given  of  the  reason  why,  in  the  case  of  two  particles  of  which 
one  has  approached  the  other  with  a  very  great  velocity,  there  arises  a  fresh  connection 
of  great  strength,  that  is,  one  so  strong  that  there  is  no  rebound  of  the  particles  from  one 
another.  For  instance,  it  may  be  said  that  the  hook  of  the  one  is  introduced  into  an  opening 
in  the  other,  &  twisted  within  it  by  the  inequality  of  the  forces  acting  on  different  parts 
of  the  hook,  so  that  it  cannot  get  out  again.  For  the  concavity  of  the  hook,  &  the  opening 
or  pore  of  the  second  particle,  may  be  much  wider  than  that  corresponding  to  that  very  - 
slight  distance  limiting  nearer  approach  ;  &  thus  the  hook  can  be  inserted  without  hindrance 
due  to  forces  acting  at  those  very  small  distances.  These  same  atoms  might  be  obtained, 
even  if  the  curve  had  all  the  inflected  arcs  that  are  present  in  mine  ;  &  then  such  atoms 
would  be  much  more  suitable  to  explain  fermentations  especially,  as  well  as  the  emission 
of  vapours  &  of  light.  If  there  were  no  asymptotic  arcs  representing  indefinitely  increasing 
forces  beyond  the  origin  of  abscissae,  the  same  result  could  be  obtained  by  means  of  limit- 
points  of  cohesion  alone  ;  with  tenacity  of  figure,  not  indeed  infinite,  but  still  very  great  if 
these  were  very  powerful.  In  this  case,  there  could  be  derived  from  the  same  general 
principle,  from  which  is  derived  the  whole  of  Nature  in  general,  an  explanation  of  the 
cohesion  of  the  parts  of  the  atoms  (which  the  ancients  did  not  explain),  &  therefore  of  their 
solidity ;  &  also  the  continued  resistance  of  impenetrability,  &  gravitation  too.  There 
remains  but  one  thing  for  me  to  mention  ;  namely,  that  atoms  of  this  kind  will  necessarily 
keep  to  a  greater  distance  from  one  another  than  that  corresponding  to  the  distance  limiting 
further  approach,  beyond  which  external  points  cannot  come. 

441.  Here  also  is  the  place  to  solve  a  difficulty  that  spontaneously  presents  itself.     If  The  reason  why  all 
all  points  of  matter  are  simple,  &  if  they  exert  the  same  forces  in  all  directions  round  bodies  are  not  fluid, 

,  .     .    ,  ,      J  11  i      T         i  r         i     although  all  points 

themselves  ;   then  it  is  far  more  natural  to  expect  that  all  bodies  that  are  composed  of  such  in     ail    directions 
points  would  be  fluid  than  that  those,  which  consist  of   little  spheres  exerting   the  same  !?und    arf    under 

f  n     T  i  i         n     •  i          r™  i  •         i-rr-       i  •       tne  same  force- 

forces  in  all  directions  around,  are  bound  to  be  fluid.  The  answer  to  this  difficulty  is 
easily  given ;  if  the  points  of  particles  can,  by  application  of  force,  increase  their  mutual 
distances  by  a  fair  amount  (for  some  slight  change  is  necessary  even  for  circulation),  and  if 
further  it  were  possible  to  impress  a  practically  equal  motion  on  a  very  small  number  of 
points  forming  one  of  the  particles  of  the  first  order,  without  at  the  same  time  giving  this 
motion  to  all  such  points,  or  even  to  any  considerable  number  of  them ;  in  that  case  we 
certainly  should  obtain  the  same  effect  as  is  obtained  in  the  case  of  fluids ;  &  the  points  being 
separated  one  after  the  other,  an  easy  movement  would  be  obtained  throughout  all  masses 
of  all  bodies.  But,  particles  of  the  first  order,  formed  from  indivisible  points,  as  also  those 
of  the  next  orders  formed  from  the  first,  can,  owing  to  their  very  smallness  of  volume, 
preserve  their  form  &  the  mutual  arrangement  of  their  points,  as  was  shown  in  Art.  424. 
For,  the  difference  between  the  forces  acting  on  different  points  of  them  may-  be  extremely 
small,  since  the  sum  of  the  forces  prevents  too  close  an  approach  of  one  particle  to  the  other  ; 
&  yet  by  this  approach  an  inequality  in  the  forces  &  an  obliquity  in  their  directions  is  obtained, 


3H  PHILOSOPHIC  NATURALIS  THEORIA 

obliquitas  directionum  ha-[203]-beatur  adhuc  satis  magna  ad  vincendas  vires  mutuas, 
mutandam  positionem,  qua  positione  manente,  manet  injequalitas  virium,  quas  diversa 
puncta  ejus  particulae  exercent  in  aliam  particulam.  Ea  inaequalitas  itidem  potest  non 
esse  satis  magna,  ut  possit  illius  mutuas  vires  vincere,  &  textum  dissolvere,  sed  esse  tanta, 
ut  motum  inducat  in  latus,  ac  ejus  motus  obliquitas,  &  virium  inaequalitas  eo  deinde  erit 
major,  quo  ad  altiores  ascenditur  particularum  ordines,  donee  deveniatur  ad  corpora, 
quae  a  nobis  sentiuntur. 
Difficuitas  deter-  Ai2.  Solida  externum  corpus  ad  ea  delatum  intra  suam  massam  non  recipiunt,  ut 

mmandi      resisten-       •  T  n    •  i          v  i  •  •  •  i         •  •        T>      •  • 

tiam    fluidorum :  vidimus  :  at  liuida  solidum  intra  se  moven  permittunt,  sed.  resistunt  motui.     Kesistentiam 
method;    indirectae  eiusmodi    accurate  comparare,  &  eius  leges  accurate  definire,  est    res  admodum  ardua. 

idprsestandi  eaedem    fi  .  ,r.          ,  J ,  °  .  „     ,.          .  . 

in  hac  Theoria  ac  Oporteret  nosse  ipsamvinum  legem  determinate,  &  numerum,  &  dispositionem  punctorum, 
m  communi.  ac  habere  satis  promotam  Geometriam,  &  Analysin  ad  rem  praestandam.     Sed  in  tanta 

particularum,  &  virium  multitudine,  quam  debeat  esse  res  ardua,  &  humano  captu  superior 
determinatio  omnium  motuum,  satis  constat  ex  ipso  problemate  trium  corporum  in  se 
mutuo  agentium,  quod  num.  204  diximus  nondum  satis  generaliter  solutum  esse.  Hinc 
alii  ad  alias  hypotheses  confugiunt,  ut  rem  perficiant,  &  omnes  ejusmodi  methodi  asque 
cum  mea,  ac  cum  communi  Theoria,  consentire  possunt. 

fontes  &r!rtriusltue          443'  ^  tamen  aliquid  innuam  etiam  de  eo  argumento,  duplex  est  resistentiae  fons 
lex.  in  fluidis ;   primo  quidem  oritur  resistentia  ex  motu  impresso  particulis  fluidi ;   nam  juxta 

leges  collisionis  corporum,  corpus  imprimens  motum  alteri,  tantundem  amittit  de  suo. 
Deinde  oritur  resistentia  a  viribus,  quas  particulae  exercent,  dum  alias  in  alias  incurrunt, 
quae  earum  motum  impediunt,  quo  casu  comprimuntur  non  nihil  particulae  ipsae  etiam 
in  fluidis  non  elasticis  egressae  e  limitibus,  &  aequilibrio  :  acquirunt  autem  motus  admodum 
diversos,  gyrant,  &  alias  impellunt,  quae  a  tergo  urgent  non  nihil  corpus  progrediens, 
quod  potissimum  a  fluidis  elasticis  a  tergo  impellitur,  dilatato  ibi  fluido,  dum  a  fronte  a 
fluido  ibi  compresso  impeditur  :  sed  ea  omnia,  uti  diximus,  accurate  comparare  non  licet. 
Illud  generaliter  notari  potest  :  resistentia,  quae  provenit  a  motu  communicate  particulis 
fluidi,  &  quae  dicitur  orta  ab  inertia  ipsius  fluidi,  est  ut  ejus  densitas,  &  ut  quadratum 
velocitatis  conjunctim  :  ut  densitas  quia  pari  velocitate  eo  pluribus  dato  tempore  particulis 
motus  idem  imprimitur,  quo  densitas  est  major,  nimirum  quo  plures  in  dato  spatio 
occurrunt  particulae  :  ut  quadratum  velocitatis,  quia  pari  densitate  eo  plures  particulas 
dato  tempore  loco  movendae  sunt,  quo  major  est  velocitas,  nimirum  quo  plus  spatii  percur- 
ritur,  &  eo  major  singulis  imprimitur  motus,  quo  itidem  velocitas  est  major.  Resistentia 
autem,  quae  oritur  a  viribus,  quas  in  se  exercent  particulae,  si  vis  ea  esset  eadem  in  singulis, 
quacunque  velocitate  [204]  moveatur  corpus  progrediens,  esset  in  ratione  temporis,  sive 
constans  :  nam  plures  quidem  eodem  tempore  particulae  earn  vim  exercent,  sed  breviore 
tempore  durat  singularum  actio,  adeoque  summa  evadit  constans.  Verum  si  velocitas 
corporis  progredientis  sit  major ;  particulae  magis  compinguntur,  &  ad  se  invicem  accedunt 
magis,  adeoque  major  est  itidem  vis.  Quare  ejusmodi  resistentia  est  partim  constans, 
sive,  ut  vocant,  in  ratione  momentorum  temporis,  &  partim  in  aliqua  ratione  itidem 
velocitatis. 


Quam  legem  vide-  AAA_  Porro  ex  expenmentis  nonnulhs  videtur  erui,  resistentiam  in  nonnullis  fluidis 

antur   innuere    ex-  '  .  •  j       v  i      •  ...  .,... 

perimenta:  in  vis-  esse  partim  m  ratione  duplicate  velocitatum,  partim  in  ratione  earum  simplici,  &  partim 
co sis  resistentiam  constantem,  sive  in  ratione  momentorum  temporis,  quanvis  ubi  velocitas  est  ingens, 
deprehendatur  major  :  &  ubi  fluiditas  est  ingens,  ut  in  aqua,  ut  secundum  resistentiae 
genus,  quod  est  magis  irregulare,  &  incertum,  fit  respectu  prioris  exiguum,  satis  accedit 
resistentia  ad  rationem  duplicatam  velocitatum.  Sed  &  illud  cum  Theoria  conspirat, 
quod  viscosa  fluida  multo  magis  resistunt,  quam  pro  ratione  suae  densitatis,  &  velocitate 
corporis  progredientis  :  nam  in  ejusmodi  fluidis,  quae  ad  solida  accedunt,  illud  secundum 
resistentiae  genus  est  multo  majus,  quod  quidem  in  solidis  usque  adeo  crescit  :  quanquam 
&  in  iis  intrudi  per  ingentem  vim  intra  massam  potest  corpus  extraneum,  ut  clavus  in  murum, 
vel  in  metallum,  quae  tamen,  si  fragilia  sunt,  &  sensibilem  compressionem  non  admittant, 
diffringuntur. 

Probiemata  alia  ad  44.5.  Jam  vero  quaecunque  a  Newtono  primum,  turn  ab  aliis  demonstrata  sunt  de 

nTn^fa^Hid^m   rnotu  corporum,  quibus  resistitur  in  variis  rationibus  velocitatum,  ea  omnia  consentiunt 
communia  huic  itidem  cum  mea  Theoria,  &  hujus  sunt  loci,  ac  ad  illam  pertinent  Mechanicae  partem, 

Q  motu  solidorum  per  fluida.       Sic  etiam  determinatio  figurae,  cui  minimum 


A  THEORY  OF  NATURAL  PHILOSOPHY 

which  is  sufficiently  great  to  overcome  the  mutual  forces  &  to  alter  their  position  ;  &  when 
this  position  stays  as  it  was,  so  also  does  the  inequality  between  the  forces,  which  the  different 
points  of  the  particle  exert  upon  another  particle.  Again,  this  inequality  may  not  be  great 
enough  to  overcome  the  mutual  forces  of  that  particle,  &  break  up  its  formation  ;  but 
yet  great  enough  to  induce  lateral  motion  ;  the  obliquity  of  this  motion,  &  the  inequality 
of  forces  will  therefore  be  so  much  the  greater,  the  further  we  ascend  in  the  orders  of  the 
particles,  until  we  finally  reach  such  bodies  as  affect  our  senses. 

442.  As  we  see,  solids  do  not  receive  within  their  mass  an  external  body  that  is  brought 
close  up  to  them  ;   but  fluids  allow  a  solid  to  be  moved  within  their  mass,  resisting  however 
the  motion.     To  find  such  resistance  accurately,  &  to  make  out  the  laws  which  govern 
it,  is  a  matter  of  great  difficulty.     It  would  be  necessary  to  know  the  law  of  forces  exactly, 
the  number  &  arrangement  of  the  points,  &  to  be  in  possession  of  fairly  advanced  geometry 
&  analysis  to  accomplish  a  solution.     But,  when  dealing  with  such  a  great  number  of  points 
&  forces,  how  difficult  the  matter  is  bound  to  be  can  be  fairly  seen  by  reference  to  that 
problem  of  the  three  bodies  acting  upon  one  another,  which  I  said,  in  Art.  204,  had  not 
yet  been  solved  at  all  generally.      Hence,  others  resort  to  other  hypotheses  for  their  purposes ; 
all  such  methods  can  be  reconciled  as  well  with  my  theory  as  with  the  common  one. 

443.  So  that  I  may  not  leave  the  point  altogether  untouched,  I  will  just  remark  that 
the  source  of  resistance  in  fluids  is  twofold.     First,  we  have  resistance  due  to  the  motion 
impressed  on  the  particles  of  the  fluid  ;  for,  according  to  the  laws  of  the  impact  of  bodies, 
the  body  which  impresses  the  motion  on  the  other  will  lose  just  as  much  of  its  own  motion. 
Secondly,  there  is  resistance  due  to  the  forces  exerted  by  the  particles,  as  they  approach 
one  another,  which  hinders    their  motion  ;    &  in  this  case,  the  particles  themselves  are 
compressed  to  some  extent,  even  in  non-elastic  fluids,  as  they  go  beyond  the  limit-points 
&  equilibrium.     Moreover  they  acquire  different  motions,  they  gyrate  &  drive  off  others 
that  are  driving  the  moving  body  to  some  extent  from  the  back ;   &  especially  in  the  case 
of  elastic  fluids  we  have  this  force  at  the  back  of  the  body,  owing  to  the  fluid  being  there 
dilated,  whilst  at  the  same  time  there  is  a  hindering  force  at  the  front  due  to  the  fluid  being 
compressed  there.     But  all  these  things,  as  I  have  said,  cannot  be  accurately  determined. 
It  can,  however,  be  in  general  noted  that  the  resistance  due  to  the  motion  communicated 
to  the  particles  of  a  fluid,  which  is  said  to  arise  from  the  inertia  of  the  fluid,  varies  as  its 
density  &  the  squares  of  the  velocities  j  ointly.     As  the  density,  because  in  the  same  time, 
for  equal  velocities,  the  same  motion  is  impressed  upon  a  number  of  particles  which  is  the 
greater,  the  greater  the  density,  i.e.,  the  greater  the  number  of  particles  occupying  the 
same  space.     As  the  squares  of  the  velocities,  because  in  the  same  time,  for  equal  densities, 
the  number  of  particles  to  be  moved  in  position  is  the  greater,  the  greater  the  velocity, 
that  is  to  say,  the  greater  the  space  to  be  traversed  ;    &  the  motion  that  is  impressed  on 
each  point  is  the  greater,  the  greater  the  velocity.     Again,  the  resistance  that  is  due  to 
the  forces  which  the  particles  exert  on  one  another,  if  the  force  is  the  same  for  each  of  them, 
with  whatever  velocity  the  moving  body  proceeds,  would  be  in  proportion  to  the  time, 
or  constant.     For,  it  is  true  that  a  large  number  of  particles  exert  this  force  in  the  same 
time,  but  the  action  of  each  only  lasts  for  a  quite  short  time  ;   &  thus  the  sum  turns  out 
to  be  constant.     If  the  velocity  of  the  moving  body  is  greater,  the  particles  are  driven 
together  more  closely,  &  approach  one  another  more  nearly,  &  so  also  the  force  is  greater. 
Hence  this  kind  of  resistance  is  partly  constant,  or,  as  it  is  usually  termed,  proportional 
to  instants  of  time,  &  partly  in  some  way  proportional  to  the  velocity  as  well. 

444.  Further  the  results  of  some  experiments  seem  to  indicate  that  the  resistance 
in  some  fluids  is  partly  as  the  squares  of  the  velocities,  partly  as  the  velocities  simply,  &  partly 
constant,  or  as  the  instants  of  time,  although  where  the  velocity  is  very  great,  it  is  found 
to  be  greater.     Also  when  the  fluidity  is  great,  as  in  the  case  of  water,  the  second  kind  of 
resistance,  which  is  the  more  irregular  &  uncertain  of  the  two,  becomes  exceedingly  small 
compared  with  that  of  the  first  kind,  &  the  total  resistance  approaches  fairly  closely  to  a 
variation  as  the  squares  of  the  velocities.     It  is  also  in  agreement  with  the  Theory  that 
the  resistance  for  viscous  fluids  is  much  greater  than  that  corresponding  to  the  ratio  of 
densities  &  the  velocities  of  the  moving  bodies.      For,  in  such  fluids,  which  are  a  near 
approach  to  solids,  the  second  kind  of  resistance  is  by  far  the  greater,  &  indeed  increases  to 
so  great  an  extent  as  in  solids.     Although,  in  solids  also,  an  extraneous  body  can  be  introduced 
within  their  mass  by  means  of  a  very  great  force,  just  as  a  nail  may  be  driven  into  a  wall,  or 
into  metal ;  yet  if  these  are  fragile  &  do  not  admit  of  sensible  compression,  they  are  broken. 

445.  But  there  are  several  other  things,  first  demonstrated  by  Newton,  &  afterwards 
by  others,  concerning  the  motion  of  bodies,  under  a  resistance  varying  as  different  powers 
of  the  velocity ;   &  all  of  these  are  also  in  agreement  with  my  Theory,  &  come  in  in  this 
connection  ;    they  belong  also  to  that  part  of  Mechanics  which  deals  with  the  motion  of 
solids  through  fluids.     So  also  the    determination  of    the  figure  of   least  resistance,  the 


The  difficulty  of 
determining  the  re- 
sistance of  fluids ; 
the  indirec  t 
methods  for  accom- 
plishing this  are  tho 
same  in  my  Theory 
as  in  the  usual  one. 


Two  sources  of 
resistance,  &  the 
laws  of  each. 


The  law  that  ex- 
periments seem  to 
indicate  :  the  resist- 
ance is  greater  in 
viscous  fluids. 


Other  problems 
relating  to  resist- 
ance that  are 
common  also  to  this 
Theory. 


3i6  PHILOSOPHIC  NATURALIS  THEORIA 

resistitur,  determinatio  vis  fluid!  solidum  impellentis  directionibus  quibuscunque,  mensura 
velocitatis  inde  oriundae  per  corporum  objectorum  resistentiam  observatione  definitam, 
innatatio  solidorum  in  fluidis,  ac  alia  ejusmodi,  &  mihi  communia  sunt  :  sed  oportet 
rite  distinguere,  quae  sunt  hypothetica  tantummodo,  ab  iis,  quas  habentur  reapse  in 
Natura. 

Alia  pertinentia  446.  Ad  fluida  &  solida  pertinent  itidem,  quaecunque  in  parte  secunda  demonstrata 
in"  %rtePCsrcunda^  sunt  de  pressione  fluidorum,  &  velocitate  in  efHuxu,  quaecumque  de  aequilibrio  solidorum, 
discrimen  inter  de  vecte,  de  centro  oscillationis,  &  percussionis,  quas  quidem  in  Mechanica  pertractari 
eiastica,  &  moiha.  solent.  Illud  unum  addo,  ex  motu  facili  particularum  fluidi  aliarum  circa  alias,  &  irregulari 
earum  congestione,  facile  deduci,  debere  pressionem  propagari  quaquaversus.  Sed  de 
his  jam  satis,  quas  ad  soliditatem,  &  fluiditatem  pertinent  :  illud  vero,  quod  pertinet  ad 
discrimen  inter  eiastica,  &  mollia,  brevi  expediam.  Eiastica  sunt,  quae  post  mutationem 
[205]  figurae  redeunt  ad  formam  priorem  ;  mollia,  quae  in  nova  positione  perseverant. 
Id  discrimen  Theoria  exhibet  per  distantiam,  vel  propinquitatem  limitum,  juxta  ea,  quae  dicta 
num.  199.  Si  limites  proximi  illi,  in  quo  particular  coherent,  hinc,  &  inde  plurimum  ab 
eo  distant,  imminuta  multum  distantia,  perstat  semper  repulsiva  vis ;  aucta  distantia,  perstat 
vis  attractiva.  Quare  sive  comprimatur  plus  aequo,  sive  plus  aequo  distrahatur  massa,  ad 
figuram  veterem  redit ;  ubi  rediit,  excurrit  ulterius,  donee  contraria  vi  elidatur  velocitas 
concepta,  ac  oritur  tremor,  &  oscillatio,  quae  paullatim  minuitur,  &  extinguitur  demum, 
partim  actione  externorum  corporum,  ut  per  aeris  resistentiam  sistitur  paullatim  motus 
penduli,  partim  actione  particularum  minus  elasticarum,  quae  admiscentur,  &  quae  possunt 
tremorem  ilium  paullatim  interrumpere  frictione,  ac  contrariis  motibus,  &  sublapsu,  quo 
suam  ipsam  dispositionem  nonnihil  immutent.  Si  autem  limites  sint  satis  proximi ;  causa 
externa,  quae  massam  comprimit,  vel  distrahit,  posteaquam  adduxit  particulas  ab  uno 
cohaasionis  limite  ad  alium,  ibi  eas  itidem  cogit  subsistere,  quae  ibidem  semel  constitutae 
itidem  in  aequilibrio  sunt,  &  habetur  massa  mollis. 


Fluida     eiastica,  447.  Quaedam  eiastica  fluida  non  habent  particulas  positas  inter  se  in  limitibus  cohae- 

suTT\nPTimitibus  sionis,  sed  in  distantiis  repulsionum,  &  quidem  ingentium,  ut  aer  :  sed  vel  incumbente 
cohaesionis ;  omnia  pondere,  vel  parietibus  quibusdam  impeditur  recessus  ille,  &  sunt  quodammodo  ibidem 
&  solida,  &  fluida  jn  statu  violento ;  licet  semper  puncta  singula  in  aequilibrio  sint,  oppositis  repulsionibus 

eiastica     esse,     ocu  *         i  „         i  Pi  n    •  \  •  i  •      • 

non  dici,  quia  sensi-  se  mutuo  elidentibus.  Omnia  autem  &  solida,  &  fluida,  quae  videntur  nee  comprimi,  nee 
r  u^as  nabere  vires  mutuas  inter  particulas,  sed  in  limitibus  esse,  adhuc  eiastica  sunt,  sive 
vim  repulsivam  exercent  inter  particulas  proximas,  saltern  quse  sensibili  gravitate  sunt 
prasdita,  quae  nimirum*  vis  repulsiva  vim  gravitatis  elidat.  Verum  ea  distantias  parum 
admodum  mutant,  mutatione,  quae  idcirco  sensum  omnem  effugiat ;  quod  accidit  in 
aqua,  quae  in  fundo  putei,  &  prope  superficiem  supremam  habet  eandem  ad  sensum  densi- 
tatem,  &  in  metallis,  &  in  marmoribus,  &  in  solidis  corporibus  passim,  quaa  pondere  majore 
imposito  nihil  ad  sensum  comprimuntur.  Sed  ea  idcirco  appellari  non  solent  eiastica,  & 
ad  ejusmodi  appellationem  non  sufficit  vis  repulsiva  etiam  ingens  inter  particulas  proximas  : 
sed  etiam  requiritur  mutatio  sensibilis  distantiae  respectu  distantiae  totalis  respondens 
sensibili  mutationi  virium. 


Dura    nuiia    esse  :  448.  Dura  corpora  in  eo  sensu,  in  quo  a  Physicis  duritiei  nomen  accipitur,  ut  nimirum 

unde  Ira^uitas" r&  figuram  nihil  prorsus  immutent,  nulla  sunt  in  mea  Theoria,  ut  &  nulla  compacta  penitus, 

ductmtas.  ac  plane  solida,  quemadmodum  diximus  etiam  num.  266 ;    sed  dura  vocat  vulgus,  quae 

satis  magnam  exercent  vim,  ne  figuram  mutent,  sive  eiastica  sint,  sive  fragilia,  sive  mollia. 

Fragilitas,  unde  ortum  ducat,  expositum  est  paullo  su-  [206]  -perius  num.  437,  &  inde  etiam 

quid  ductilitas,  ac  malleabilitas  sit,  facile  intelligitur.     Ductilia  nimirum  a  mollibus  non 

differunt,  nisi  in  majore,  vel  minore  yi,  qua  figuram  tuentur  suam  :  ut  enim  mollia  pressione 

tenui,  &  ipsis  digitis  comprimuntur,  vel  saltern  figuram  mutant,  sed  mutatam  retinent, 

ita  ductilia  ictu  validiore  mallei  mutant  itidem  figuram  suam  veterem,  &  retinent   novam, 

quam  acquirunt. 

Superiora    omnia  449.  Atque  hoc  demum  pacto  quaecunque  pertinent  ad  fluidorum,  &  solidorum diversa 

Theori"  ^ejus  foe*  genera,  nam  &  eiastica,  mollia,  ductilia,  fragilia  eodem  referuntur,  invenimus  omnia  in 

cunditas :  ilia  omnia  illo  particularum  discrimine  orto  ex  sola  diversa  combinatione    punctorum,  quam   nobis 

ajJensitate  non  pen-  fheoria  rite  applicata  exhibuit,  in  quibus  omnibus  immensa  varietas  itidem  haberi  poterit, 


A  THEORY  OF  NATURAL  PHILOSOPHY  317 

determination  of  the  force  of  a  fluid  driving  a  solid  in  any  directions,  the  measurement  of 
the  velocity  arising  thence  by  means  of  the  observed  resistance  of  bodies  placed  in  the  way, 
the  flotation  of  bodies  in  fluids,  &  other  things  of  the  same  kind,  are  all  common  to  my 
Theory.  But  it  is  necessary  to  distinguish  which  of  them  are  only  hypothetical  &  which 
of  them  really  occur  In  Nature. 

446.  To  fluids  &  solids  are  to  be  referred  all  those  matters,  which  in  the  second  part  other  matters  that 
were  demonstrated  with  regard  to  pressure  of  fluids,  &  velocity  of  efflux ;    &  all   matters  ^eere  SecoCndSedpart 
relating  to  equilibrium  of  solids,  the  lever,  the  centre  of  oscillation,  &  the  centre  of  percussion  ;  really   pertain    to 
all  of  which  indeed  are  usually  considered  in  connection  with  Mechanics.     I  will  but  add  distincton^tween 
that,  from  the  ease  of  movement  of  the  particles  of  a  fluid  about  one  another,  &  from  their  elastic  &  soft 
irregular  grouping,  it  readily  follows  that  in  them  pressure  must  be  propagated  in  every 

direction.  But  I  have  now  said  enough  about  those  matters  that  refer  to  solidity  &  fluidity  ; 
however,  I  will  make  a  few  remarks  on  matters  that  relate  to  the  distinction  between  elastic 
&  soft  bodies.  Those  bodies  are  elastic,  which  after  change  of  shape  return  to  their  original 
form  ;  &  those  are  soft,  which  remain  in  their  new  state.  This  distinction  my  Theory 
shows  to  be  consequent  upon  the  distance  or  closeness  of  the  limit-points ;  as  I  said  in 
Art.  199.  If  the  limit-points,  that  are  next  to  the  one  in  which  the  particles  cohere,  are 
far  distant  from  it  on  either  side,  then,  when  the  distance  is  much  diminished,  there 
will  still  be  a  repulsive  force  all  the  time ;  &  if  the  distance  is  increased  there  will  be  a 
similar  attractive  force.  Hence,  whether  the  mass  is  compressed  more  than  is  natural,  or 
expanded  more  than  is  natural,  it  will  return  to  its  original  form.  When  it  has  returned  to 
its  original  form,  it  will  go  beyond  it,  until  the  velocity  attained  is  cancelled  by  the  opposite 
force  ;  and  a  tremor,  or  oscillation,  will  be  produced,  which  will  be  gradually  diminished  and 
ultimately  destroyed,  partly  by  the  action  of  external  bodies,  just  as  the  motion  of  a  pendulum 
is  stopped  by  the  resistance  of  the  air,  &  partly  by  the  action  of  less  elastic  particles  which 
are  interspersed,  which  can  gradually  break  down  the  oscillation  by  their  friction,  &  also 
by  contrary  motions,  &  a  relapse  by  which  they  change  their  own  distribution  somewhat. 
But  if  these  limit-points  are  fairly  close,  the  external  cause,  which  compresses  or  expands 
the  mass,  after  that  it  has  brought  the  particles  from  one  limit-point  of  cohesion  to  another, 
will  force  them  also  to  stay  at  the  latter  ;  &  these,  when  once  grouped  in  this  position,  will 
also  be  in  equilibrium,  &  a  soft  mass  will  be  the  result. 

447.  The  particles  of  some  elastic  fluids  are  not  at  limit-points  of  cohesion  with  respect  Elastic    fluids 

r  ,  ,.  ,.  -\-O-L  whose  particles  are 

to  one  another,  but  are  at  distances  corresponding  to  repulsions,  &  these  too  very  great ;  not  at  limit-points 
for  instance,  air.     But  recession  is  prevented  either  by  superincumbent  weight,  or  by  of  cohesion.     AH 

,      .  .  <•      •    i  j.   •  i  j-  111      solids   &  fluids  are 

enclosing  walls ;   these  are  in  some  sort  of  violent  condition  at  these  distances,  although  really  elastic,  but 
each  point  is  always  in  equilibrium,  due  to  the  opposite  repulsions  cancelling  one  another.  are  not  ,called  so- 

,,  ,,        ,.',      „     n^.  ,          1-1  •  i  a  •  i  because  they  do  not 

Moreover,  all  solids  &  fluids,  which  appear  neither  to  suffer  compression,  nor  to  have  any  suffer    sensible 

mutual  forces  between  their  particles,  but  to  be  at  limit-points,  are  however  elastic  ;   that  compression. 

is  to  say,  they  exert  a  repulsive  force  between  their  adjacent  particles ;   at  least  those  do 

which  are  possessed  of  sensible  gravitation,  for  it  is  this  repulsive  force  that  cancels  the  force 

of  gravity.     The  distances  are  in  fact  changed  very  slightly,  the   change  being   therefore 

one  that  is  beyond  the  scope  of  our  senses.     This  is  the  case  for  water  ;  with  it,  the  density  is 

practically  the  same  at  the  bottom  of  a  well  as  it  is  at  the  upper  surface  ;    the  same  thing 

happens  in  the  case  of   metals   &  marbles  &  in  all  solid  bodies,  in  which  if  a  fairly  large 

weight  is  superimposed  there  is  no  sensible  compression.     But  such  things  are  not  usually 

termed  elastic,  for  the  reason  that  a  repulsive  force  between  adjacent  particles,  even  if  it 

is  very  great,  is  not  sufficient  for  such  an  appellation ;  in  addition,  there  is  required  to  be  a 

sensible  change  of  distance,  compared  with  the  whole  distance,  to  correspond  with  a  sensible 

change  in  the  forces. 

448.  There  are  in  my  Theory  none  of  those  bodies,  that  are  hard  in  the  sense  in  which  Then5  are  no  hard 
hardness  is  accepted  by  Physicists,  namely  such  as  do  not  suffer  the  slightest  change  of  shape  ;  bodies   are    called 
&  also  there  are  none  that  are  perfectly  compact,  or  quite  solid,  as  I  said  in  Art.  266.     But  haFd  '•    henc.e.  fra- 
those  are  usually  termed  hard,  which  exert  a  fairly  great  force  to  prevent  change  of  form  ;  ^ 

they  may  be  either  elastic,  fragile  or  soft.  The  source  of  fragility  has  been  explained  just 
above,  in  Art.  437  ;  &  from  this  also  the  nature  of  ductility  &  malleability  can  be  easily 
understood.  For  instance,  ductile  &  malleable  solids  only  differ  from  one  another  in  the 
greater  or  less  strength  with  which  they  preserve  their  form  ;  for,  just  as  soft  bodies  under 
slight  pressure,  even  of  the  fingers,  are  compressed,  or  change  their  form,  but  retain  the  form 
thus  changed  ;  so  ductile  bodies  under  the  stronger  force  of  a  blow  with  a  'mallet  also 
change  their  original  shape,  &  retain  the  new  form  that  they  acquire. 

449.  Finally,  in  this  way,  whatever  properties  there  may  be  relating  to  different  kinds  A11  *he  above  pro- 
of fluids  &  solids  (for  elastic,  soft,  ductile  &  fragile  bodies  all  come  to  the  same  thing),  we  fro'm^in^Tifeory ; 
have  made  them  all  out  from  the  difference  between  particles  that  is  produced  by  the  a11  of  them  do  n°t 
difference  in  the  combination  of   the  points  alone  ;    this  will  be  shown  by  my  Theory  if  sity6"' 


3i8  PHILOSOPHIC  NATURALIS  THEORIA 

&  debebit ;  si  curva  primigenia  ingentem  habeat  numerum  intersectionum  cum  axe,  & 
particulse  primi  ordinis,  ac  reliquas  ordinum  superiorum  dispositiones,  quae  in  infinitum 
variari  possunt,  habuerint  plurimas,  &  admodum  diversas  inter  se,  ac  eas  inprimis,  quae  ad 
haec  ipsa  figurarum,  &  virium  discrimina  requiruntur.  Illud  unum  hie  diligenter  notandum 
est,  quod  ipsam  Theoriam  itidem  commendat  plurimum,  hasce  proprietates  omnes  a  densitate 
nihil  omnino  pendere.  Fieri  enim  potest,  uti  num.  183  notavimus,  ut  curva  virium 
primigenia  limites,  &  arcus  habeat  quocunque  ordine  in  diversis  distantiis  permixtos 
quocunque  numero,  ut  validiores,  &  minus  validi,  ac  ampliores,  &  minus  ampli  commis- 
ceantur  inter  se  utcunque,  adeoque  phenomena  eadem  figurarum,  &  virium  aeque  inveniri 
possint,  ubi  multo  plura,  &  ubi  multo  pauciora  puncta  massam  constituunt. 

Communia  quatuor  450.  Jam  vero  ilia,  qua;  vulgo  elcmenta  appellari  solent,  Terra,  Aqua,  Aer,  Ignis, 

eiementa  quid  smt.  njj1jj  aliud  mihi  sunt,  nisi  diversa  solida,  &  fluida,  ex  iisdem  homogeneis  punctis  composita 
diversimode  dispositis,  ex  quibus  deinde  permixtis  alia  adhuc  magis  composita  corpora 
oriuntur.  Et  quidem  Terra  ex  particulis  constat  inter  se  nulla  vi  conjunctis,  quae  solidi- 
tatem  aliarum  admixtione  particularum  acquirunt,  ut  cineres  oleorum  ope,  vel  etiam 
aliqua  mutatione  dispositionis  internas,  ut  in  vitrificatione  evenit,  quae  transformationes 
quo  pacto  accidant,  dicemus  postremo  loco.  Aqua  est  fluidum  liquidum  elasticitate  carens 
cadente  sub  sensum  per  compressionem  sensibilem,  licet  ingentem  exerceant  repulsivam 
vim  ejus  particulae,  sustinentes  velexternae  vis,  vel  sui  ipsius  ponderis  pressionem  sine  sensibili 
distantiarum  imminutione.  Aer  est  fluidum  elasticum,  quern  admodum  probabile  est 
constare  particulis  plurimorum  generum,  cum  e  plurimis  etiam  fixis  corporibus  generetur 
admodum  diversis,  ut  videbimus,  ubi  de  transformationibus  agendum  erit,  ac  propterea 
continet  vapores,  &  exhalationes  plurimas,  &  heterogenea  corpuscula,  quae  in  eo  innatant  : 
sed  ejus  particulae  satis  magna  vi  se  repellunt,  [207]  &  ea  repulsiva  particularum  vis 
imminutis  distantiis  diu  perdurat,  ac  pertinet  ad  spatium,  quod  habet  ingentem  rationem 
ad  earn  tanto  minorem  distantiam,  ad  quam  compressione  reduci  potest,  &  in  qua  adhuc 
ipsa  vis  crescit,  arcu  curvae  adhuc  recedente  ab  axe  :  is  vero  arcus  ad  axem  ipsum  deinde 
debet  ruere  prasceps,  ut  circa  proximum  limitem  adhuc  ingentes  in  eo  residue  spatio 
variationes  in  arcubus,  &  limitibus  haberi  possint.  Porro  extensionem  tantam  arcus  repulsivi 
evincit  ipsa  immanis  compressio,  ad  quam  ingenti  vi  aer  compellitur,  qui  ut  habeat  com- 
pressiones  viribus  prementibus  proportionales,  debet,  ut  monuimus  num.  352,  habere 
vires  repulsivas  reciproce  proportionales  distantiis  particularum  a  se  invicem.  Is  autem 
etiam  in  fixum  corpus,  &  solidum  transire  potest,  quod  qua  ratione  fieri  possit,  dicam  itidem, 
ubi  de  transfoimationibus  agemus  in  fine.  Ignis  etiam  est  fluidum  maxime  elasticum, 
quod  violentissimo  intestine  motu  agitatur,  ac  fermentationem  excitat,  vel  etiam  in  ipsa 
fermentatione  consistit,  emittit  vero  lucem,  de  quo  pariter  agemus  paullo  inferius,  ubi  de 
fermentatione,  &  emissione  vaporum  egerimus  inter  ea,  quae  ad  Chemicas  operationes 
pertinet,  ad  quas  jam  progredior. 


Chemicarum  opera-          4.151.  Chemicarum  operationum  principia  ex  eodem  deducuntur  fonte,  nimirum  ex 

ducTTacffe^ex  uio  ^°  particularum  discrimine,  quarum  aliae  inter  se,  &  cum  quibusdam  aliis  inertes,  alias 

particularum    dis-  ad  se  attrahunt,  alias  repellunt  constanter  per  satis  magnum  intervallum,  ubi  attractio 

ium'efiectuunfcau-  ipsa  cum  aliis  est  major,  cum  aliis  minor,  aucta  vero  satis  distantia,  evadit  ad  sensum  nulla  ; 

sas  singuiares  non  quarum  itidem  aliae  respectu  aliarum  habent  ingentem  virium  alternationem,  quam  mutato 

mente  humana"    *  nonnihil  textu  suo,  vel  conjunctae,   &   permixtae   cum   aliis    mutare   possunt,   succedente 

pro  particulis  compositis  alia  virium  lege  ab  ea,  quae  in  simplicibus  observabatur.     Hasc 

omnia  si  habeantur  ob  oculos  ;    mihi  sane  persuasum  est,  facile  inveniri  posse  in  hac  ipsa 

Theoria  rationem  generalem  omnium  Chemicarum  operationum  :    nam  singuiares  deter- 

minationes  effectuum,  qui  a  singulis  permixtionibus  diversorum  corporum,  per  quas  unice 

omnia  prasstantur  in  Chemia,  sive  resolvantur  corpora,  sive  componantur,  requirerent 

intimam  cognitionem  textus  particularum  singularum,  &  dispositionis,  quam  habent  in 

massis  singulis,  ac  prasterea  Geometriae,  &  Analyseos  vim,  quae  humanae  mentis  captum 

excedit  longissime.     Verum  illud  in  genere  omnino  patet,  nullam  esse  Chemiae  partem, 

in  qua  praeter  inertiam  massae,  &  specificam  gravitatem,  alia  virium  mutuarum  genera 

inter  particulas  non  ubique  se  prodant,  &  vel  invitis  incurrant  in  oculos,  quod  quidem 

vel  in  sola  postrema  quaestione  Opticae  Newtoni  abunde  patet,  ubi  tam  multa  &  attractionum, 


A  THEORY  OF  NATURAL  PHILOSOPHY  319 

properly  applied,  &  in  all  such  things  also  an  immense  variety  can  &  must  be  produced. 
Provided  that  the  primary  curve  has  a  number  of  intersections  with  the  axis,  &  provided 
that  particles  of  the  first  order,  &  the  rest  of  the  higher  orders,  have  arrangements  (which 
indeed  can  be  infinitely  varied)  that  are  great  in  number  &  all  different  from  one  another  ; 
&  those  especially  that  are  required  for  these  differences  in  shape  &  forces.  Now,  one 
thing  is  at  this  point  to  be  noted  carefully,  one  that  also  supports  the  Theory  itself  very 
strongly,  namely,  that  all  these  properties  are  totally  independent  of  density.  For  it  is 
possible  that,  as  I  mentioned  in  Art.  183,  the  primary  curve  of  forces  may  have  limit-points 
&  arcs  mixed  together  in  any  order  at  different  distances,  and  there  may  be  any  number 
of  either  ;  so  that  stronger  &  weaker  limit-points,  more  &  less  ample  arcs  may  be  intermingled 
in  any  manner  amongst  themselves  ;  &  thus  the  same  phenomena  of  shapes  &  forces  can  be 
met  with  when  the  number  of  points  constituting  a  mass  is  much  larger  or  much  smaller. 

450.  Now  those  things,  which  are  commonly  called  the  Elements,  Earth,  Water,  Air  The  nature  of  the 
&  Fire,  are  nothing  else  in  my  Theory  but  different  solids  &  fluids,  formed  of  the  same 
homogeneous  points  differently  arranged  ;    &  from  the  admixture  of  these  with  others,  called. 

other  still  more  compound  bodies  are  produced.  Indeed  Earth  consists  of  particles  that 
are  not  connected  together  by  any  force  ;  &  these  particles  acquire  solidity  when  mixed 
with  other  particles,  as  ashes  when  mixed  with  oils  ;  or  even  by  some  change  in  their  internal 
arrangement,  such  as  comes  about  in  vitrification  ;  we  will  leave  the  discussion  of  the 
manner  in  which  these  transformations  take  place  till  the  end.  Water  is  a  liquid  fluid  devoid 
of  elasticity  such  as  comes  within  the  scope  of  the  senses  through  a  sensible  compression  ; 
although  there  is  a  strong  repulsive  force  exerted  between  its  particles,  which  is  sufficient 
to  sustain  the  pressure  of  an  external  force  or  of  its  own  weight  without  sensible  diminution 
of  the  distances.  Air  is  an  elastic  fluid,  which  in  all  probability  consists  of  particles  of 
very  many  different  sorts  ;  for  it  is  generated  from  very  many  totally  different  fixed  bodies, 
as  we  shall  see  when  we  discuss  transformations.  For  that  reason,  it  contains  a  very  large 
number  of  vapours  &  exhalations,  &  heterogeneous  corpuscles  that  float  in  it.  Its  particles, 
however,  repel  one  another  with  a  fairly  large  force  ;  &  this  repulsive  force  of  the  particles 
lasts  for  a  long  while  as  the  distances  are  diminished,  &  pertains  to  a  space  that  bears  a  very 
large  ratio  to  the  so  much  smaller  distance,  to  which  it  can  be  reduced  by  compression  ; 
&  at  this  distance  too  the  force  still  increases,  the  arc  of  the  curve  corresponding  to  it  still 
receding  from  the  axis.  But  after  that,  the  curve  must  return  very  steeply,  so  that  in 
the  neighbourhood  of  the  next  limit-point  there  may  yet  be  had  in  the  space  that  remains 
great  variations  in  the  arcs  &  the  limit-points.  Further  such  great  extension  of  the  repulsive 
arc  is  indicated  by  the  great  compression  induced  by  the  pressure  due  to  a  large  force  ; 
&  this,  in  order  that  the  compression  may  be  proportional  to  the  impressed  force,  shows, 
as  we  pointed  out  in  Art.  352,  that  there  must  be  repulsive  forces  inversely  proportional 
to  the  distances  of  the  particles  from  one  another.  Moreover  it  can  pass  into  &  through 
a  fixed  &  solid  body  ;  &  the  reason  of  this  also  I  will  state  when  I  deal  with  transformations 
towards  the  end.  Fire  is  also  a  highly  elastic  fluid,  which  is  agitated  by  the  most  vigorous 
internal  motions  ;  it  excites  fermentations,  or  even  consists  of  this  very  fermentation  ;  it 
emits  light,  with  which  also  we  will  deal  a  little  later,  when  we  discuss  fermentation  & 
emission  of  vapours  amongst  other  things  referring  to  chemical  operations  ;  to  these  we 
will  now  pass  on. 

451.  The  principles  of  chemical  operations  are  derived  from  the  same  source,  namely,  Th«  different  kinds 
from  the  distinctions  between  particles  ;  some  of  these  being  inert  with  regard  to  themselves  tions6™!^  readily 
&  in  combination  with  certain  others,  some  attract  others  to  themselves,  some  repel  others  derived  from    the 
continuously  through  a  fairly  great  interval  ;    &  the  attraction  itself  with  some  is  greater,  p^tTcTe  sTTh  e 
&  with  others  is  less,  until  when  the  distance  is  sufficiently  increased  it  becomes  practically  special    causes   of 
nothing.     Further,  some  of  them  with  respect  to  others  have  a  very  great  alternation  of 


forces  ;   &  this  can  vary  if  the  structure  is  changed  slightly,  or  if  the  particles  are  grouped  intelligence  of  the 

&  intermingled  with  others  ;  in  this  case  there  follows  another  law  of  forces  for  the  compound  h 

particles,  which  is  different  to  that  which  we  saw  obeyed  by  the  simple  particles.     If  all 

these  things   are  kept   carefully  in   view,  I  really  think   that  there  can  be  found  in  this 

Theory  the  general  theory  for  all  chemical  operations.      For  the  special  determination  of 

effects  that    arise  from  each  of  the  different  mixtures  of  the  different   bodies,   through 

which  alone  all  effects  in  chemistry  are  produced,  whether  the  bodies  are  resolved  or 

compounded,  would  require  an  intimate  knowledge  of  the  structure  of  each  kind  of  particle, 

&  the  arrangement  of  these  in  each  of  the  masses  ;    &,  in  addition,  the  whole  power  of 

geometry  &  analysis,  such  as   exceeds  by  far  the  capacity  of  the  human  mind.     But  in 

general  it   is   quite  evident  that  there  is  no  part   of  chemistry,  in  which,   in  addition  to 

inertia  of  mass,  &  specific  density,  there  are  not  everywhere  produced  other  kinds  of  mutual 

forces  between  the  particles  ;    &  these  will  meet  our  eyes  without  our  looking  for  them, 

as  is  indeed  abundantly  evident  in  the  single  question  that  comes  last  at  the  end  of  Newton's 


320  PHILOSOPHISE  NATURALIS  THEORIA 

&  vero  etiam  repulsionum  indicia,  atque  argumenta  proferuntur.  Omnia  etiam  genera 
eorum,  quae  ad  Chemiam  pertinent,  singillatim  persequi,  infinitum  essct  :  evolvam 
speciminis  loco  praecipua  quaedam. 

Quid  sint :  dissolu-  [208]  45  2.  Primo  loco  se  mihi  offerunt  dissolutio,  &  ipsi  contraria  praecipitatio.  Immissis  in 
pnma  quomodo  fiati  qusedam  fluida  quibusdam  solidis,  cernimus,  mutuum  nexum,  qui  habebatur  inter  eorum 
&  quae  sit  ejus  particulas,  dissolvi  ita,  ut  ipsa  jam  nusquam  appareant,  qua;  tamen  ibidem  adhuc 
manere  in  particulas  perquam  exiguas  redacta,  &  dispersa,  ostendit  praecipitatio.  Nam 
immisso  alio  corpore  quodam,  decidit  ad  fundum  pulvisculus  tenuissimus  ejus 
substantiae,  &  quodammodo  depluit.  Sic  metalla  in  suis  quasque  menstruis  dissolvuntur, 
turn  ope  aliarum  substantiarum  praecipitantur  :  aurum  dissolvit  aqua  regia,  quod  immisso 
etiam  communi  sale  praecipitatur.  Rei  ideam  est  admodum  facile  sibi  efformare  satis 
distinctam.  Si  particulae  solidi,  quod  dissolvitur,  majorem  habent  attractionem  cum 
particulis  aquae,  quam  inter  se  ;  utique  avellentur  a  massa  sua,  &  singulae  circumquaque 
acquirent,  fluidas  particulas,  quae  illas  ambiant,  uti  limatura  ferri  adhaeret  magnetibus, 
ac  fient  quidam  veluti  globuli  similes  illi,  quern  referret  Terra  ;  si  ei  tanta  aquarum  copia 
affunderetur,  ut  posset  totam  alte  submergere,  vel  illi,  quern  refert  Terra  submersa  in 
acre  versus  earn  gravitante.  Si,  ut  reipsa  debet  accidere,  ilia  vis  attractiva  in  distantiis 
paullo  majoribus  sit  insensibilis ;  ubi  jam  erit  ad  illam  distantiam  saturata  eo  fluido  particula 
solidi,  ulterius  fluidum  non  attrahet,  quod  idcirco  aliis  eodem  pacto  particulis  solidi  immersi 
affundetur.  Quare  dissolvetur  solidum  ipsum,  ac  quidam  veluti  globuli  terrulas  suas 
cum  ingenti  affusa  marium  vi  exhibebunt,  quae  terrulae  ob  exiguam  molem  effugient  nostros 
sensus,  nee  vero  decident  sustentatae  a  vi,  quae  illas  cum  circumfuso  mari  conjungit  :  sed 
globuli  illi  ipsi  constituent  quandam  veluti  continui  fluidi  massam.  Ea  est  dissolutionis 
idea. 


Quomodo  fiat  pr«-  453.  Quod  si  jam  in  ejusmodi  fluidum  immittatur  alia  substantia,  cujus  particulae 

qUa  l  particulas  fluidi  ad  se  magis  attrahant,  &  fortasse  ad  majores  etiam  distantias,  quam 
attrahuntur  ab  illis ;  dissolvetur  utique  hasc  secunda  substantia,  &  circa  ipsius  particulas 
affundentur  particulae  fluidi,  quae  prioris  solidi  particulis  adhaerebant,  ab  illis  avulsae,  & 
ipsis  ereptae.  Illae  igitur  nativo  pondere  intra  fluidum  specifice  levius  depluent,  &  habebitur 
praecipitatio.  Pulvisculus  autem  ille  veterem  particularum  suarum  nexum  non  acquiret 
ibi  per  sese,  vel  quia  &  gluten  fortasse  aliquod  admodum  tenue,  quo  connectebantur  invicem, 
dissolutum  simul  jam  deest  in  superficiebus  illis,  quarum  separatio  est  facta,  vel  potius 
quia,  ut  ubi  per  limam,  per  tunsionem,  vel  aliis  similibus  modis  solidum  in  pulverem 
redactum  est,  vel  utcunque  confractum,  juxta  ea,  quae  diximus  num.  413,  non  potest  iterum 
solo  accessu,  &  appressione  deveniri  ad  illos  eosdem  limites,  qui  prius  habebantur. 


piuviam      fortasse  ACA    Hoc  pacto  dissolutionis,  &  praecipitationis  acquiritur  idea  admodum  distincta  ; 

esse  quoddam  pra-    „    r  -i  jj  •    :  .  , 

cipitationis  genus :  &  fortassc  etiam  pluvia  est  quoddam  praecipitationis  genus,  nee  provenit  e  sola  unione  par- 
mira   phenomena  f20Ql-ticularum  aquas,  quae  prius  tantummodo  dispersae  temere    fuerint,  &    ob    solam 

commixtionum    *  •       f.  •  •  A  -i  •         • 

quomodo  expiicen-  tenuitatem  suam  sustentatae    ac  suspenses  innatavermt.     Apparet  ibi  etiam,  qua  ratione 
tur-  binae  substantiae  commisceri  possint,  &  in  unam  massam  coalescere.     Id  quidem  in  fluido 

commixto  cum  solido  patet  ex  ipso  superiore  exemplo  solutionis.  In  binis  fluidis  facile 
admodum  fit,  &  si  sint  ejusdem  ad  sensum  specificae  gravitatis,  solo  motu,  &  agitatione 
impressa  fieri  quotidie  cernimus,  ut  in  aqua,  &  vino,  sed  etiam  si  sint  gravitatum  admodum 
diversarum,  attractione  particularum  unius  in  particulas  alterius  fieri  potest  unius  dissolutio 
in  altero,  &  commixtio.  Fieri  autem  potest,  ut  ejusmodi  commixtione  e  binis  etiam  fluidis 
oriatur  solidum,  cujusmodi  exempla  in  coagulis  cernimus  :  &  in  Physica  illud  quoque 
observatur  quandoque,  binas  substantias  commixtas  coalescere  in  massam  unicam  minorem 
mole,  quam  fuerit  prius,  cujus  phaenomeni  prima  fronte  admodum  miri  in  promptu  est 
causa  in  mea  Theoria  :  cum  particulae,  quae  nimirum  se  immediate  non  contingcbant, 
aliis  interpositis  possint  accedere  ad  se  magis,  quam  prius  accesserint.  Sic  si  haberetur 
massa  ingens  elastrorum  e  ferro  distractorum,  quorum  singulis  inter  cuspides  adjungerentur 
globuli  magnetici  ;  hac  nova  accessione  materiae  minueretur  moles,  victa  repulsione  mutua 


A  THEORY  OF  NATURAL  PHILOSOPHY  321 

Optics,  where  there  are  many  indications  of  both  attractions  &  repulsions  as  well,  &  arguments 
are  brought  forward  with  regard  to  them.  Further,  to  investigate  separately  all  matters 
that  relate  to  chemistry  would  be  an  endless  task  ;  so  I  will  discuss  certain  of  the  more 
important,  by  way  of  example. 

452.  In  the  first  place  there  occur  to  me  solution  &  its  converse,  precipitation.    When  The  nature  of  soiu- 

TJ        ,.  ,  .     r,        .   ,  •       n     •  i  i  i  •  i'ii  tlon    &     precipita- 

certam  solids  are  mixed  with  certain  nuids,  we  see  that  the  mutual  connection  which  there  tion ;  how  the  first 

used  to  be  between  the  particles  of  each  is  dissolved  in  such  a  way  that  the  solids  are  no  comes  about,  &  its 

longer  visible  ;    &  yet  that  they  are  still  there,  reduced   to  extremely  small  particles  & 

dispersed,  is  shown  by  precipitation.     For,  if  a  certain  other  body  is  introduced,  there  falls 

to  the  bottom  an  extremely  fine  powder  of  the  original  solid,  as  if  it  rained  down.     So  metals, 

each  in  its  own  solvent,  dissolve,  &  with  the  help  of  other  substances  are  precipitated. 

"Aqua  regia  "  dissolves  gold;  &  this,  on  the  addition  of  common  salt,  is  precipitated.     It 

is  quite  easy  to  get  a  clear  idea  of  the  matter.     Suppose  that  the  particles  of  the  solid  have 

a  greater  attraction  for  the  particles  of  the  water  than  for  one  another  ;    then  they  will 

certainly  be  torn  away  from  their  own  mass,  &  each  of  them  will  gather  round  itself  fluid 

particles,  which  will  surround  it,  in  the  same  manner  as  iron  filings  adhere  to  a  magnet ;  & 

each  would  become  something  in  the  nature  of  little  spheres  similar  to  what  the  Earth 

would  resemble,  if  a  sufficiency  of  water  were  to  be  poured  over  it  to  submerge  it  deeply, 

or  to  what  the  Earth  does  resemble,  submerged  as  it  is  in  the  air  gravitating  towards  it. 

If,  as  is  bound  to  happen,  the  attractive  force  becomes  insensible  at  distances  a  little  greater, 

then,  when  a  particle  of  a  solid  has  become  saturated  to  that  distance  with  the  fluid,  it 

will  no  longer  attract  the  fluid ;   &  therefore  the  latter  will  surround  other  particles  of 

the  immersed  solid  in  the  same  manner.     Hence  the  solid  will  be  dissolved,  &  each  of 

the  little  spheres,  so  to  speak,  would  represent  a  little  earth  with  its  great  abundance  of  sea 

surrounding  it ;    &  these  little  earths,  on  account  of  their  exceedingly  small  volume  will 

escape  our  notice  ;    &  they  cannot  fall,  sustained  as  they  are  by  the  force  that  connects 

them  with  the  sea  which  surrounds  them.     Now  these  little  globes  themselves  form  a 

certain  mass  of  as  it  were  continuous  fluid  ;  hence  we  get  an  idea  of  the  nature  of  solution. 

453.  If  now  another  substance  is  introduced  into  a  fluid  of  this  kind,  the  particles  The   manner  in 
of  which  attract  the  particles  of  the  fluid  to  themselves  with  a  stronger  force,  &  perhaps  occurs ;  sfits  cause. 
too  at  greater  distances,  than  they  are  attracted  by  the  particles  of  the  first  solid ;    then 

this  second  solid  will  be  dissolved  in  every  case,  &  its  particles  will  be  surrounded  by  the 
particles  of  the  fluid,  which  formerly  adhered  to  the  particles  of  the  first  solid,  being  torn 
away  from  the  latter  &  seized  by  the  particles  of  the  second  solid.  The  particles  of  the 
first  solid  will  then  rain  down  on  account  of  their  own  weight  within  the  fluid  which 
is  specifically  lighter,  &  there  will  be  precipitation.  Further,  the  fine  powder  will  not 
of  itself  then  acquire  the  former  connection  between  its  particles ;  this  may  be  because 
a  sort  of  very  thin  cement,  by  which  the  particles  were  connected  together,  has  perhaps 
been  at  the  same  time  dissolved,  &  this  is  now  absent  from  the  surfaces  which  have  been 
separated  ;  but  more  probably  it  is  because,  just  as  when,  by  means  of  a  file  or  a  hammer 
or  the  like,  a  solid  has  been  reduced  to  powder,  or  broken  up  in  any  manner,  it  cannot  by 
mere  approach  &  pressing  together  get  back  once  more  to  the  same  limit-points  as  before, 
as  I  said  in  Art.  413. 

454.  In  this  way  a  perfectly  clear  idea  of  solution  &  precipitation  is  acquired.     Perhaps  Perhaps   rain  is 

,       TJ.T  .  f     f  •   i      .  r     ,  ^  .  ,  .   ,r     some  sort  of  preci- 

also  ram  is  some  sort  of  precipitation,  &  does  not  merely  come  Irom  the  union  oi  particles  pitation;    how 
of  water  which  previously  had  been  only  dispersed  at  random,  &  had  floated,  sustained  &  certain    wonderful 

,     ,    .        .  r      .  J.  .     .          J          r  .          ,  A  i  i  phenomena  in  con- 

suspended  in  the  air,  owing  to  their  extreme  tenuity  alone.  Also,  we  can  now  see  how  nection  with  mix- 
two  substances  can  be  mixed  together  to  coalesce  into  a  single  mass.  This  indeed,  in  the  ^ures  are  exPlained- 
case  of  a  fluid  mixed  with  a  solid,  is  evident  from  the  example  of  solution  given  above. 
It  takes  place  quite  easily  in  the  case  of  two  fluids,  &,  if  they  are  practically  of  the  same 
specific  gravity,  we  see  it  happening  every  day  by  mere  motion  &  the  agitation  impressed  ; 
as  in  the  case  of  water  &  wine.  But  even  if  their  specific  gravities  are  quite  different, 
by  the  attraction  of  the  particles  of  the  one  upon  the  particles  of  the  other,  there  may 
be  solution  of  the  one  in  the  other,  &  thus  a  mixture  of  the  two.  Further,  it  may  happen 
that  from  a  mixture  of  this  kind,  even  of  two  fluids,  there  may  be  produced  a  solid ;  we 
see  examples  of  such  a  thing  in  rennet.  In  Physics  also,  it  is  observed  sometimes  that  two 
substances  mixed  together  coalesce  into  a  single  mass  having  a  smaller  volume  than  before  ; 
the  cause  of  this  phenomenon,  which  at  first  sight  appears  wonderful,  is  to  be  found 
immediately  with  my  Theory.  For,  the  particles,  which  originally  did  not  immediately 
touch  one  another,  when  others  are  interposed,  may  approach  nearer  to  one  another  than 
they  did  before.  Thus,  if  we  have  a  large  heap  of  springs  made  of  iron,  &  to  them  we 
add  a  number  of  little  magnetic  spheres,  placing  one  between  the  tips  of  each  spring  ; 
then,  with  this  fresh  addition  of  matter,  the  whole  volume  is  diminished,  the  mutual 


322  PHILOSOPHIC  NATURALIS  THEORIA 

per  attractionem  magneticam,  qua  cuspides  elastrorum  ad  se  invicem  accederent. 
Cur  ad  commix-  455.  Ubi  solidum  cum  solido  commiscendum  est,  ut  fiat  unica  massa,  ibi  quidem 
requiratur80  'cent™  oportet  solida  ipsa  prius  contundere,  vel  etiam  dissolvere,  ut  nimirum  exiguae  particulae 
sio :  quid  ad  earn  seorsim  possint  ad  exiguas  alterius  solidi  accedere,  &  cum  iis  conjungi.  Id  autem  fit 
"e-  potissimum  per  ignem,  cujus  vehementi  agitatione,  &  vero  etiam  fortasse  actione  ingenti 
mutua  inter  ejus  particulas,  &  inter  quaedam  peculiaria  substantiarum  genera,  ut  olea, 
&  sulphur,  quas  ut  gluten  quoddam  conjungebant  inter  se  vel  inertes  particulas,  vel  etiam 
mutua  repulsione  prasditas,  dissolvit  omnium  corporum  nexus  mutuos,  &  massas  omnes 
demum,  si  satis  validus  sit,  cogit  liquari,  &  ad  naturam  fluidorum  accedere.  Dissolutarum, 
ac  liquescentium  massarum  particulae  commiscentur,  &  in  unam  massam  coalescunt  :  ubi 
autem  sic  coaluerunt,  possunt  iterum  saepe  dissimiles  separari  eadem  actione  ignis,  qui 
aliquas  prius,  alias  posterius,  cogit  minore  vi  abire  per  evaporationem,  &  maxime  fixas 
majore  vi  reddit  volatiles.  Inaequalibus  ejusmodi  diversarum  substantiarum  attractionibus, 
&  inaequalibus  adhaesionibus  inter  earum  particulas,  omnis  fere  nititur  ars  separandi  metalla 
a  terris,  cum  quibus  in  fodinis  commixta  sunt,  &  alia  aliorum  ope  prius  uniendi,  turn  etiam 
a  se  invicem  separandi,  quas  omnia  singillatim  persequi  infmitum  foret.  Generalis  omnium 
explicatio  facile  repetitur  ab  ilia,  quam  exposui,  particularum  diversa  constitutione,  quarum 
alias  respectu  aliarum  inertes  sunt,  respectu  aliarum  activitatem  habent,  sed  admodum 
diversam,  turn  [210]  quod  pertinet  ad  directionem,  turn  quod  ad  intensitatem  virium. 


TO?ataVza!tionem  45^  ^e  Liquatione,  &  volatilizatione  dicam  illud  tantummodo,  eas  fieri  posse  etiam 
fieri  posse  per  agita-  sola  ingenti  agitatione  particularum  fluidi  cujuspiam  tenuissimi,  cujus  particulae  ad  solidi, 
p'ar  ticuTfr'um1  ^  ^x*  corPoris  particulas  accedant  satis,  &  inter  ipsarum  etiam  intervalla  irrumpant ;  qui 
Prima quomodo  fiat]  motus  intestinus,  unde  haberi  possit,  jam  exponam,  ubi  de  fermentatione  egero,  &  effer- 
vescentia.  Nam  inprimis  ea  intestina  agitatione  induci  potest  in  particulas  corporis  solidi, 
&  fixi  motus  quidam  circa  axes  quosdam,  qui  ubi  semel  inductus  est,  jam  illae  particulas 
vim  exercent  circunquaque  circa  ilium  axem  ad  sensum  eandem,  succedentibus  sibi  invicem 
celerrime  punctis,  &  directionibus,  in  quibus  diversae  vires  exercentur,  qui  etiam  axes  si 
celerrime  mutentur,  irregulari  nimirum  impulsu,  habebitur  in  iis  particulis  id,  quod 
asquivaleat  sphaericitati  &  homogeneitati  particularum,  ex  qua  fluiditatem  supra  repetivimus, 
atque  hujus  ipsius  rei  exemplum  habuimus  num.  237  in  motu  puncti  per  peripheriam 
ellipseos,  cujus  focos  bina  alia  puncta  occupent.  Haec  fluiditas  erit  violenta,  &  desinente 
tanta  ilia  agitatione,  ac  cessante  vi,  quae  agitationem  inducebat,  cessabit,  ac  fluidum  etiam 
sine  admixtione  novas  substantiae  poterit  evadere  solidum.  Poterit  autem  paullatim 
cessare  motus  ille  rotationis  tarn  per  inasqualitatem  exiguam,  quae  semper  remanet  inter 
vires  in  diversis  locis  particulas  diversas,  &  obsistit  semper  nonnihil  rotationi,  quam  per 
ipsam  expulsionem  illius  agitatae  substantiae,  ut  igneas,  &  per  resistentiam  circumjacentium. 


Aiialiquationis  .57.  Deinde  haberi   etiam   poterit   liquatio   per   subtractionem   heterogenearum,   & 

ratio     per    separa-     , .  „  T-> '.  .      ,  *  .    ,  /     ,       ,        .   .  .   '  , 

tionem  partium  dirtormium  particularum,  quae  magis  nomogeneas,  &  ad  spnasricitatem  accedentes  particulas 
heterogenearum.  alligabant  quodammodo  impedito  motu  in  gyrum.  Id  sane  videtur  accidere  in  pluribus 
substantiis,  quae  quo  magis  depurantur,  &  ad  homogeneitatem  reducuntur,  eo  minus 
tenaces  evadunt,  &  viscosae.  Sic  viscositas  est  minima  in  petroleo,  major  in  naphtha,  & 
adhuc  major  in  asphalto,  aut  bitumine,  in  quibus  substantiis  Chemia  ostendit,  eo  majorem 
haberi  viscositatem,  quo  habetur  major  compositip. 

taizatic?"  &Kitio0l&  ^'  Quoc^  s^  Pri°re  modo  liquatio  accidat,  &  in  eo  motu  particulae  a  limitibus 
voiatiiizatio  aeri's.  cohaesionis,  in  quibus  erant,  abeant  ad  distantias  paullo  majores,  in  quibus  habeatur  ingens 
repulsivus  arcus,  se  repente  fugient,  quo  pacto  corpus  fixum  evadet  volatile.  Eandem 
autem  volatilitatem  acquiret ;  si  particulae  quae  fixum  corpus  componebant,  erant  quidem 
inter  se  in  distantiis  repulsionum  validissimarum,  sed  per  interjacentes  particulas  alterius 
substantias  cohibebatur  ilia  repulsiva  vis  superata  ab  attractione,  quam  exercebat 
in  eas  nova  intrusa  particula  :  si  enim  haec  agitatione  ilia  excutiatur,  vel  ab  alia,  quas 
ipsam  attrahat  magis,  praetervolante  ad  exiguam  distantiam  abripia-[2ii]-tur  ;  turn 
vero  repulsiva  vis  particularum  prioris  substantiae  reviviscit  quodammodo,  &  agit, 
ac  ipsa  substantia  evadit  volatilis,  quae  iterum  nova  earundem  particularum  intrusione 
figitur.  Id  sane  videtur  accidere  in  acre,  qui  potest  ad  fixum  redigi  corpus,  &  Halesius 


A  THEORY  OF  NATURAL  PHILOSOPHY  323 

repulsion  being  overcome  by  the  magnetic  attraction,  with  which  the  tips  of  the  springs 
would  approach  one  another. 

455.  When  a  solid  has  to  be  mixed  with  a  solid  to  form  a  single  mass,  it  is  necessary  Why    crushing    is 
to  first  of  all  crush  the  solids,  or  even  to  dissolve  them,  so  that  the  exceedingly  small  particles  fixture  'o^soHds* 
of  the  one  can  separately  approach  those  of  the  other  solid,  &  combine  with  them.     Now  the  effect  of  fire  in 
this  especially  takes  place  in  the  case  of  fire  ;   by  its  vigorous  internal  movement,  &  perhaps  thefartof separating 
too  through  a  very  great  mutual  attraction  between  its  particles  &  those  of  certain  particular  metals, 
kinds  of  substance,  like  oils  &  sulphur,  these  two  causes  acting  as  a  sort  of  cement  to  join 
together  either  inert  particles,  or  even  particles  possessed  of  a  mutual  repulsion,  fire  dissolves 
the  mutual  connections  of  all  bodies  &  finally  forces,  if  it  is  sufficiently  powerful,  all  masses 
to  melt,  &  to  approach  fluids  in  their  natures.     The  particles  of  the  masses  thus  dissolved 
&  in  a  molten  condition  mingle  together  &  coalesce  into  one  single  mass.     Moreover,  after 
they  have  thus  coalesced,  the  dissimilar  substances  can  once  more  be  separated  by  the 
same  action  of  fire,  which  forces,  some  at  first  &  others  later,  the  particles  to  go  off,  with 
a  smaller  force  through  evaporation,  &  renders  volatile  the  most  refractory  particles  when 
the  intensity  is  greater.     Upon  the  unequal  attractions  of  different  substances  of  this  kind, 
&  upon  the  unequal  adhesions  between  their  particles,  depends  almost  entirely  the  art 
of  separating  metals  from  the  earths  with  which  they  are  mixed  in  the  ores ;  &  some  metals 
from  others,  by  means  of  first  uniting  them  &  then  separating  them  once  more ;    but  to 
investigate  all  these  matters  singly  would  be  an  endless  task.     The  general  explanation 
of  them  all  is  easily  derived  from  that  diverse  constitution  of  the  particles  that  I  have 
expounded  ;   namely,  that  some  particles  are  inert  with  respect  to  others,  &  have  activity 
with  respect  to  yet  others ;  where  this  activity  is  altogether  varied,  both  as  regards  the 
directions,  &  as  regards  the  intensities,  of  the  forces. 

4156.  With  regard  to  liquefaction  &  volatilization,  I  will  only  say  this  :    that  these  Liquefaction  &  voi- 

1  •        i      .  i  i  •   i  • .  t  n    •  J     atilization  can  take 

phenomena  can  take  place  simply  through  a  violent  agitation  of  some  very  tenuous  fluid,  piace  owjng  to  a 

whose  particles  approach  sufficiently  close  to  the  particles  of  the  solid  fixed  body,  &  push  very  great  agitation 

into  the  intervals  between  them.     How  this  internal  motion  can  happen  I  will  explain,  manner3  in6 which 

when  I  discuss  fermentation  &  effervescence.     First  of  all,  owing  to  the  internal  agitation,  the  first  happens. 

there  can  be  induced  in  the  particles  of  the  solid  fixed  body  motions  about  certain  axes ; 

&  when  these  motions  have  once  been  set  up,  the  particles  will  exert  a  rotary  force  about 

the  axis  which  is  practically  uniform,  the  points  following  one  another  extremely  quickly, 

&  also  the  directions  in  which  the  different  forces  are  exerted  ;    &  if  these  axes  are  also 

changed  very  rapidly,  due,  say  to  an  irregular  impulse,  we  shall  have  in  the  particles  what 

is  equivalent  to  the  sphericity  &  homogeneity  of  particles,  from  which  we  have  derived 

fluidity  in  a  preceding  article ;   we  had  also  an  example  of  this  kind  of  thing,  in  Art.  237, 

in  the  motion  of  a  point  along  the  perimeter  of  an  ellipse,  of  which  two  other  points  occupied 

the  foci.     This  fluidity  will  be  very  violent,  &,  as  soon  as  the  great  agitation  ends  &  the 

force  which  caused  the  agitation  ceases,  the  agitation  will  cease  as  well,  &  the  fluid  will 

be  able  to  become  solid  once  more,  without  the  admixture  of  any  fresh  substance.     Further, 

this  motion  of  rotation  may  gradually  cease,  owing  not  only  to  the  slight  inequality  that 

will  always  remain  between  the  different  forces  at  different  places  of  a  particle,  ever  tending 

to  hinder  the  rotation  to  some  extent,  but  also  to  the  expulsion  of  the  substance  in  agitation 

(fire,  say),  &  through  the  resistance  of  the  particles  lying  in  the  neighbourhood. 

4157.  Secondly,  there  may  be  liquefaction  through  the  subtraction  of  heterogeneous  Another    reason 

10 '      . ,.  '  '    .   ,  i  •   i     i       ^    i  i     6  i  •   i  i  •   i_f°r   liquefaction    is 

&  non-uniform  particles,  which  bound  together  the  more  homogeneous  particles  which  through  the  separa- 
approximate  to  sphericity,  in  such  a  way  as  to  hinder  their  rotary  motion.     This  is  in  tlon  of  heterogene- 

i  •  i  i'ii  i  •  -          •     •  •  i_      ous  parts. 

fact  seen  to  happen  m  several  substances,  which  become  less  tenacious  &  viscous,  the 
more  they  are  purified  &  reduced  to  homogeneity.  Thus  the  viscosity  is  very  small  in 
rock-oil,  greater  in  naphtha,  still  greater  in  asphalt  or  bitumen  ;  &,  in  these  substances, 
chemistry  shows  that  the  viscosity  is  the  greater,  the  more  compound  the  substance. 

458.  But  if  liquation  should  take  place  in  the  first  manner,  &  due  to  the  motion  the  How  volatilization 

.',        i        i  i  n-  r  ii-''  i'ii  T  vi  takes    place  ;  fixa- 

particles  should  go  on  from  the  limit-points  at  which  they  were  to  distances  a  little  greater,  tjon  &  volatilization 

&  if  for  these  distances  there  should  be  a  very  large  repulsive  arc,  then  the  particles  will  fly  off  °J  air- 

with  great  speed  ;  &  in  this  way  a  fixed  body  will  become  volatile.     Moreover  it  will  acquire 

the  same  volatility,  if  the  particles  which  form  the  body  were  at  such  distances  from  one 

another  as  correspond  to  very  strong  repulsions,  but  are  held  together  by  intervening 

particles  of  another  substance,  the  repulsive  force  being  overcome  by  the  attractions  exerted 

upon  them  by  the  new  particles  that  have  been  introduced  between  them.     For,  if  these  are 

displaced  by  the  agitation,  or  are  seized  by  others,  which  attract  them  more  strongly,  as 

they  fly  past  at  a  slight  distance,  then  the  repulsive  force  of  the  first  substance  will  revive, 

as  it  were,  &  come  into  action  ;    &  the  substance  will  become  volatile,  &  will  once  again 

become  fixed  on  a  fresh  introduction  of  the  same  intervening  particles.     This  in  fact  is 

seen  to  happen  in  the  case  of  air,  which  can  be  reduced  to  a  fixed  body.     Hales  has  proved 


324  PHILOSOPHIC  NATURALIS  THEORIA 

demonstravit  per  experimenta,  partem  ingentem  lapidum,  qui  in  vesica  oriuntur,  & 
calculorum  in  renibus  constare  puro  acre  ad  fixitatem  reducto,  qui  deinde  potest  iterum 
statum  volatilem  recuperare  :  ac  halitus  inprimis  sulphurei,  &  ipsa  respiratio  animalium 
ingentem  aeris  copiam  transf  ert  a  statu  volatili  ad  fixum.  Ibi  non  habetur  aeris  compressio 
sola  facta  per  cellularum  parietes  ipsum  concludentes  ;  ii  enim  disrumperentur  penitus,  cum  aer 
in  ejusmodi  fixis  corporibus  reducatur  ad  molem  etiam  millecuplo  minorem,  in  quo  statu,  si  in- 
tegras  haberet  elasticas  vires,  omnia  sane  repagula  ilia  diffringeret.  Halesius  putat,  eum  in  illo 
statu  amittere  elasticitatem  suam,  quod  fieret  utique,  si  particulse  ipsius  ad  earn  inter  se  dis- 
tantiam  devenirent,  in  qua  jam  vis  repulsiva  nulla  sit,  sed  potius  attractiva  succedat  :  sed  fieri 
itidem  potest,  ut  vim  quidem  repulsivam  adhuc  ingentem  habeant  illae  particulae,  sed 
ab  interposita  sulphurei  halitus  particula  attrahantur  magis,  ut  paullo  ante  vidimus  in 
elastris  a  globulo  magnetico  cohibitis,  &  constrictis.  Turn  quidem  elasticitas  in  aere  ad 
fixitatem  redacto  maneret  tola,  sed  ejus  effectus  impediretur  a  prasvalente  vi.  Atque  id 
quidem  animadverti,  &  monui  ante  aliquot  annos  in  dissertatione  De  Turbine,  in  qua  omnia 
turbinis  ipsius  phenomena  ab  hac  aeris  fixatione  repetii. 

459    P°rro  agitatio  ilia  particularum  in  igne,  ac  in  fermentationibus,  &  effervescentiis, 
igne,      fermenta-  unde  oriatur,  facile  itidem  est  in  mea  Theoria  exponere.     Ut  primum  crus  meae  curvse 
^   impenetrabilitatem    exhibuit,    postremum    gravitatem,  intersectiones    autem    varia 
contorsione    curvae  cohaesionum   genera  ;     ita    alternatio    arcuum    jam    repulsivorum,    jam    attractivorum, 
fermentationes  exhibet,  &  evaporationes  variorum  generum,  ac  subitas  etiam  deflagrationes, 
&  explosiones,  illas,  quae  occurrunt  in  Chemia  passim,  &  quam  in  pulvere  pyrio  quotidie 
intuemur.     Quas  autem  hue  ex  Mechanica  pertinet,  jam  vidimus  num.   199.     Dum  ad 
se  invicem  accedunt  puncta  cum  velocitate  aliqua,  sub  omni  arcu  attractive  velocitatem 
augent,  sub  omni  repulsive  minuunt  :    contra  vero  dum  a  se  invicem  recedunt,  sub  omni 
repulsive  augent,  sub   omni    attractive    minuunt,    donee    in    accessu    inveniant    arcum 
repulsivum,  vel  in  recessu  attractivum  satis  validum  ad  omnem  velocitatem  extinguendam. 
Ubi  eum  invenerint,  retro  cursum  reflectunt,  &  oscillant  hinc,  &  inde,  in  quo  itu,  &  reditu 
perturbato,  ac  celeri,  fermentationis  habemus  ideam  satis  distinctam. 


cr          46°-  Et  in  accessu  quidem  semper  devenitur  ad  arcum  repulsivum  aliquem  parem 

semper  .•  .        .J.  r        .  >j.j 

sisti  a  primo  crure  extinguendas    velocitati    cuinbet    utcun-[2i2j-que    magnae  ;    devenitur    enim   saltern  ad 

cessuS1Vbinipr<casus"  Primum  asymptoticum  crus,  quod  in  infinitum  protenditur  :   at  pro  recessu  duo  hie  casus 

in     primo    cruris  occurrunt   potissimum  considerandi.     Vel  enim  etiam  in  recessu   devenitur  ad  aliquod 

UcTsemperasStiPre-  crus  asymptoticum  attractivum  cum  area  infinita,  de  cujusmodi  casibus  egimus  jam  num. 

cessum  etiam.  195,  vel  devenitur  ad  arcum  attractivum  recedentem  longissime,  &  continentem  aream 

admodum  ingentem,  sed  finitam.     In  utroque  casu  actio  punctorum,  quae  extra  massam 

sunt  sita,  aliorum  punctorum  massse  intestine  illo  motu  agitatae  oscillationem    augebit 

aliorum  imminuet,  &  puncta  alia  post  alia  procurrent  ulterius  versus  asymptotum,  vel  limitem 

terminantem  attractivas  vires  :    quin  etiam  actiones  mutuae  punctorum  non  in  directum 

jacentium   in   massa    multis   punctis   constante,    mutabunt   sane  singulorum    punctorum 

maximos  excursus  hinc,  &  inde,  &  variabunt  plurimum  accessus  mutuos,  ac  recessus,  qui 

in  duobus  punctis  solis  motum  habentibus  in  recta,  quae   ilia    conjungit,  deberent,  uti 

monuimus  num.   192,  sine  externis  actionibus  esse  constantis  semper  magnitudinis.     In 

accessu  tamen  in  utroque  casu  ad  compenetrationem    sane    nunquam    deveniretur  :    in 

recessu  vero  in  primo  casu  cruris  asymptotici,  &  attractionis  in  infinitum  crescentis  cum 

area  curvae  in  infinitum  aucta,  itidem  nunquam  deveniretur  ad  distantiam  illius  asymptoti. 

Quare  in  eo  primo  casu  utcunque  vehemens  esset  interna  massae  fermentatio,  utcunque 

magnis  viribus,  ab  externis  punctis  in  majore  distantia  sitis  perturbaretur  eadem  massa, 

ipsius  dissolutio  per  nullam  finitam  vim,  aut  velocitatem  alteri  parti  impressam  haberi 

unquam  posset. 

in    secundo    casu  tfa    At  in  secundo  casu,  in  quo  arcus  attractivus  ille  ultimus  ems  spatii  ingens  esset, 

arcus      attractivi         ,    ,7  .  .  ..,,..  J         r. 

ingentis,  sed  finiti  sea  nnitus,  posset  utique  quorundam  punctorum  in  ilia  agitatione  augen  excursus  usque 

egressus      partis  ac[  limitem,  post  quern  limitem  succedente  repulsione,  iam  illud  punctum  a   massa  ilia 
punctorum     excus-  ,r  .  .  11  n-  «•     • 

sorum  e  fine  oscil-  quodammodo  velut  avulsum  avolaret,  &  motu  accelerate  recederet.     01  post  eum  limitem 

latioms    sine     re-  summa  arearum  repulsivarum  esset  maior,  quam  summa  attractivarum,  donee  deveniatur 
gressu.  ,  ...  r.  .  J  .       ?•  . 

ad  arcum  mum,  qui  gravitatem  exprirmt,  in  quo  vis  jam  est  perquam  exigua,  &  area 
asymptotica  ulterior  in  infinitum  etiam  producta,  est  finita,  &  exigua  ;  turn  vero  puncti" 
elapsi  recessus  ab  ilia  massa  nunquam  cessaret  actione  massse  ipsius,  sed  ipsum  punctum 
pergeret  recedere,  donee  aliorum  punctorum  ad  illam  massam  non  pertinentium  viribus 
sisteretur,  vel  detorqueretur  utcunque.  In  fortuita  autem  agitatione  interna,  ut  &  in 


A  THEORY  OF  NATURAL  PHILOSOPHY 


325 


by  means  of  experiments  that  the  great  part  of  stones,  that  are  produced  in  the  bladder, 
&  of  the  small  ones  in  the  kidneys,  consists  of  pure  air  reduced  to  fixation  ;  &  that  this 
can  once  again  recover  its  volatile  state.  In  this  case  the  compression  of  the  air  is  not 
obtained  simply  by  the  boundaries  that  enclose  it  ;  for  these  would  be  completely  broken 
down,  since  the  air  in  such  fixed  solids  is  reduced  to  a  volume  that  is  even  a  thousand  times 
less  ;  &  in  this  state,  if  the  elastic  forces  still  were  unimpaired,  all  restraints  would  be  easily 
overcome.  Hales  thought  that,  when  in  this  state,  it  loses  its  elasticity  ;  &  this  would 
indeed  happen  if  its  particles  attained  that  distance  from  one  another,  in  which  there  is  no 
repulsive  force,  but  rather  an  attractive  force  succeeds  the  repulsive  force.  It  might  also 
happen  that  these  particles  still  possess  a  very  large  repulsive  force,  but  by  the  interposition 
of  particles  of  a  sulphurous  vapour  they  are  attracted  to  a  greater  extent  than  they  are 
repelled  ;  as  just  above  we  saw  was  the  case  for  springs  restrained  &  constricted  by  little 
magnetic  spheres.  Then,  indeed,  the  elasticity  in  air  reduced  to  fixity  would  remain 
unaltered,  but  its  effect  would  be  prevented  by  a  superior  force.  I  considered  this  point 
of  view  &  mentioned  it  some  years  ago  in  my  dissertation  De  Turbine,  in  which  all  the 
phenomena  of  the  whirlwind  are  derived  from  this  fixation  of  the  air. 

459.  Further,  the  source  of  the  agitation  of  the  particles  in  fire,  fermentation,  &  Cause  of  the  agita- 
effervescence  is  also  easily  explained  by  my  Theory.     Just  as  the  first  branch  of  my  curve  *ion  °f  th<?  particles 

,  .,.     J  /         ',  . 7     .  J       „       ,  .  .,','.      m    fire,     fermenta- 

gives  me  impenetrability,  &  the  last  branch  gravitation,  &  the  intersections  with  the  axis  tions,   &    efferves- 
the  various  kinds  of  cohesions ;    so  also  the  alternation  of  the  arcs,  now  repulsive,  now  fence-  derived  from 

..  .  ,....'  t,  '          *      the    contortions   of 

attractive,  represent  fermentations  &  evaporations   of    various  kinds,  as  well  as  sudden  the     curve    round 

conflagrations  &  explosions ;   such  things  as  occur  everywhere  in  chemistry,  &  what  we  see  the  axls- 

every  day  in  the  case  of  gunpowder.     Those  things  from  Mechanics  that    belong   here 

we  have  already  seen  in  Art.  199.     So  long  as  points  approach  one  another  with  any  velocity, 

they  increase  the  velocity  under  every  attractive  arc,  &  diminish  it  under  every  repulsive 

arc.     On  the  other  hand,  so  long  as  they  recede  from  one  another,  they  increase  the  velocity 

under  every  repulsive  arc  &  increase  it  under  every  attractive   arc ;   until,  in  approach, 

they  come  to  a  repulsive  arc,  or  in  recession,  to  an  attractive  arc,  which  is  sufficiently  strong 

to  destroy  the  whole  of  the  velocity.     When  they  have  reached  this,  they  retrace  their 

paths,  &  oscillate  backwards  &  forwards ;   &  in  this,  the  backward  &  forward  motion  being 

perturbed  &  rapid,  we  have  a  sufficiently  clear  notion  of  what  fermentation  is. 

460.  Now,  on  approach,  there  is  always  reached  some  repulsive  arc  or  other,  which  Oscillations  on  ap- 
is capable  of  destroying  any  velocity  however  great ;  for  at  least  finally  the  first  asymptotic  stopped  aib  y^hat 
branch,  which  goes  off  to  infinity,  is  reached.     But  on  recession,  there  are  two  cases  met  first   repulsive 
with,  which  have  to  be  considered  in  this  connection.     For,  on  recession,  either  there  is  cession'  ^here^are 
reached  an  asymptotic  attractive  branch  having  an  infinite  area,  cases  of  which  kind  I  two  cases,     in  the 
dealt  with  in  Art.  194 ;    or  else  we  come  to  an  attractive  arc  receding  very  far  from  the  an^^nrptot^at- 
axis,  &  containing  an  exceedingly  great  but  finite  area.     In  either  case,  the  action  of  points  tractive      branch, 
situated  outside  the  mass  will  increase  the  oscillation  of  some  of  the  points  of  the  mass  aiw^TsWped    1S 
that  is  agitated  by  the  internal  motion,  &.  will  diminish  that  of  other  points ;  &  one  point 

after  another  will  go  off  beyond  the  mass  towards  the  asymptote,  or  the  limit-point  bounding 
the  attractive  forces.  Moreover,  the  mutual  actions,  of  points  not  lying  in  the  same  straight 
line  in  a  mass  consisting  of  many  points,  will  change  considerably  the  largest  oscillations 
of  each  of  the  points ;  especially  will  they  alter  their  mutual  approach  &  recession,  which 
for  two  points  only,  having  a  motion  in  the  straight  line  joining  them,  must  be,  except 
for  external  action,  always  of  constant  magnitude,  as  I  remarked  in  Art.  192.  On  approach, 
however,  in  either  case,  the  position  corresponding  to  compenetration  can  never  really 
be  reached.  But,  on  recession,  in  the  first  case,  where  there  is  an  asymptotic  branch,  &  an 
attraction  indefinitely  increased  along  with  an  area  of  the  curve  also  increasing  indefinitely, 
in  this  case  also  it  can  never  attain  the  distance  of  that  asymptote.  Hence,  in  the  first 
case,  however  fierce  the  internal  fermentation  of  the  mass  may  be,  no  matter  with  how 
great  forces  from  external  points  situated  further  off  the  mass  may  be  affected,  its  dissolution 
can  never  be  effected  by  any  finite  force,  or  velocity  impressed  on  any  one  part  of  it. 

461.  Now,  in  the  second  case,  in  which  the  attractive  arc  at  the  end  of  the  space  is  in  the  second  case, 
very  large,  but  finite,  it  will  indeed  be  possible  for  the  motion  of   some    points  in  the  veiy  great ebutfiriite 
agitation  to  be  increased  right  up  to  the  limit-point  ;  &,  as  repulsion  follows  the  limit-point  attractive  arc,  there 
that  point  of  the  mass  will  now  be  as  it  were  torn  off,  &  it  will  fly  away  &  leave  the  mass  Wf  som^T^'he 
with  accelerated  motion.     If  after  the  limit-point,  the  sum  of  the  repulsive  areas  should  points  at  the  end 
be  greater  than  the  sum  of  those  that  are  attractive,  that  is,  until  that  arc  is  reached  which  these"  °wui  atflyn'ofE 
represents  gravity,  where  the  force  then  becomes  exceedingly  small,  &  the  asymptotic  area,  without  returning, 
when  produced  still  further,  is  finite  &  very  small ;   then  indeed  the  recession  of  the  point 

that  has  left  the  mass  will  never  cease  owing  to  any  action  of  the  mass  itself,  but  the  point 
will  go  on  receding,  until  it  is  stopped  by  the  forces  from  other  points  not  belonging  to  that 
mass,  or  its  path  is  contorted  in  some  manner.  Moreover,  in  irregular  internal  agitation, 


326  PHILOSOPHISE  NATURALIS  THEORIA 

externa  perturbatione  fortuita,  illud  accidet,  quod  in  omnibus  fortuitis  combinationibus 
accidit,  ut  numerus  casuum  cujusdam  dati  generis  in  dato  ingenti  numero  casuum  aeque 
possibilium  dato  tempore  recurrat  ad  sensum  idem,  adeoque  effluxus  eorum  punctorum, 
si  massa  perseveret  ad  sensum  eadem,  erit  dato  tempore  ad  sensum  idem,  vel,  massa  multum 
imminuta,  imminuetur  in  aliqua  ratione  [213]  massae,  cum  a  multitudine  punctorum 
pendeat  etiam  casuum  possibilium  multitudo. 


Jam  P^UI"ima  considerari  possent.  &  casuum  differentium,  ac  combinationum 
evaporatio  lenta.  numerus  in  immensum  excrescit  ;  sed  pauca  quaedam  adnotabimus.  Ubi  intervallum, 
quod  massam  claudit  inter  limites  accessus,  &  recessus,  est  aliquanto  majus,  &  posteriorum 
arearum  repulsivarum  summa  non  multum  excedit  summam  attractivarum,  fiet  paullatim 
lenta  quasdam  evaporatio  :  puncta  quae  in  fortuita  agitatione  ad  eum  finem  deveniunt, 
erunt  pauca  respectu  totius  massas,  quae  tamen  in  ingenti  massa,  &  eodem  fermentationis 
statu  erunt  eodem  tempore  ad  sensum  aequali  numero,  ac,  massa  imminuta,  imminuetur 
&  is  numerus,  massa  autem  diu  perseverabit  ad  sensum  nihil  mutata.  Habebitur  ibi  quaedam 
velut  ebullitio,  &  vaporum  quantitas,  ac  vis  in  egressu  in  diversis  substantiis  variari  plurimum 
poterit,  cum  pendeat  a  situ,  in  quo  ilia  puncta  collocata  sint  intra  curvam  :  nam  possunt 
in  aliis  substantiis  esse  citra  alios  ingentes  arcus  attractivos,  quorum  posteriores  vel  sint 
prioribus  minus  validi,  vel  arcus  repulsivos  se  subsequentes  minus  validos  habeant. 


Vel  subita  explosio,  ,g,    gecj  sj  intervallum,  quod  massam  claudit  inter  limites  accessus,  &  recessus,  sit 

&    deflagratio ;     ac  T   J  * .  .     .  ' 

transformationes  perquam  exiguum,  arcus  attractivus  postremus  non  sit  ita  validus,  &  succedat  arcus  repui- 
vajiz>.  avolante  sivus  validissimus ;  fieri  utique  poterit,  ut  massa,  quae  respective  quiescebat,  adveniente, 
exiguo  motu  a  particulis  externis  satis  proxime  accedentibus,  ut  possint  inaequalem  motum 
imprimere  punctis  particularum  massse,  agitatio  ejusmodi  in  ipsa  massa  oriatur,  qua 
brevissimo  tempore  puncta  omnia  transcendant  limitem,  &  cum  ingenti  repulsiva  vi,  ac 
velocitate  a  se  invicem  discedant.  Id  videtur  accidere  in  explosione  subita  pulveris  pyrii, 
qui  plerumque  non  accenditur  contusione  sola  ;  sed  exigua  scintilla  accedente  dissilit  fere 
momento  temporis,  &  tanta  vi  repulsiva  globum  e  tormento  ejicit.  Idem  apparet  in  iis 
phosphoris,  quae  deflagrant  solo  aeris  contactu  :  ac  nemo  non  videt,  quanta  in  iis  omnibus 
haberi  possunt  discrimina.  Possunt  nimirum  alia  facilius,  alia  difficilius  deflagrare,  alia 
serius,  alia  citius  :  potest  sine  lenta  evaporatione  solvi  tota  massa  tempore  brevissimo ; 
potest,  ubi  massa  fuerit  heterogenea,  avolare  unum  substantiae  genus  aliis  remanentibus. 
&  interea  possunt  ex  iis,  quae- remanent,  fieri  alia  mixta  admodum  diversa  a  praecedentibus, 
mutato  etiam  textu  particularum  altiorum  ordinum  per  id,  quod  plures  particulas  ordinum 
inferiorum,  quas  pertinebant  ad  diversas  particulas  superiorum,  coalescant  in  particulam 
ordinis  superioris  novi  generis  :  hinc  tarn  multae  compositiones,  &  transformationes  in 
Natura,  &  in  Chemia  inprimis  :  hinc  tarn  multa,  tarn  diversa  vaporum  genera,  &  in  aere 
elastico  a  tam  diversis  corporibus  fixis  genito  tantum  discrimen.  Patet  ubique  immensus 
excursui  campus  :  sed  eo  relicto  [214]  progredior  ad  alia  nonnulla,  quae  ad  fermentationes, 
&  evaporationes  itidem  pertinent. 


4^4"  Substantia,  quae  fuerat  dissoluta,  non  solum  per  praecipitationem  colligitur  iterum, 
figur»  residui,  ut  in  ut  ubi  metalla  cadunt  suo  pondere  in  tenuem  pulvisculum  redacta  ;  sed  etiam  per  evapor- 
sahbus.  ationem,  ut  diximus,  in  salibus,  qui  evaporato  illo  fluido,  in  quo  fuerant  dissoluti,  remanent 

in  fundo.  Et  quidem  sales  non  remanent  sub  forma  tenuis  pulvisculi,  particulis  minutissimis 
prorsus  inertibus,  sed  colliguntur  in  massulas  grandiusculas  habentes  certas  figuras  quae 
in  aliis  salibus  aliae  sunt,  &  angulosas  in  omnibus,  ac  in  maxime  corrosivis  horrendum  in 
modum  cuspidatae,  ac  serratse,  unde  &  sapores  salium  acutiores,  &  aliquorum  ex  iis,  quas 
corrosiva  sunt,  fibrillarum  tenuium  in  animantibus  proscissio,  ac  destructio  organorum 
necessariorum  ad  vitam.  Quo  autem  pacto  eas  potissimum  figuras  induere  possint,  id 
patet  ex  num.  439,  ut  &  figuras  crystallorum  &  succorum,  ex  quibus  gemmae,  &  duri  lapides 
fiunt  ubi  simplices  sunt,  &  suam  quique  figuram  affectant,  ac  aliorum  ejusmodi,  quae  post 
evaporationem  concrescunt,  haberi  utique  possunt,  ut  ibidem  ostensum  est,  per  hoc,  quod 
in  certis  tantummodo  lateribus,  &  punctis  particulae  alias  particulas  positas  ad  certas 
distantias  attrahant,  adeoque  sibi  adjungant  certo  illo  ordine,  qui  respondet  illis  punctis, 
vel  lateribus. 


A  THEORY  OF  NATURAL  PHILOSOPHY  327 

just  as  also  in  irregular  external  perturbation,  the  same  thing  happens,  as  always  does  happen 
in  irregular  combinations  ;  namely,  out  of  a  given  very  large  number  of  cases  of  a  given 
kind,  all  equally  possible,  the  same  number  of  cases  will  recur  in  any  given  interval  of  time. 
Hence,  so  long  as  the  mass  remains  practically  the  same,  there  will  be  the  same  number 
of  points  going  off  ;  £  when  the  mass  is  much  diminished  this  number  will  also  be  diminished 
in  some  way  proportional  to  the  mass ;  for  on  the  number  of  points  depends  also  the  number 
of  possible  cases. 

462.  We  may  now  consider  a  very  large  number  of  matters;    &  indeed  the  number  Hence  from  a  differ 
of  different  cases  &  combinations  increases  immensely  ;    but  we  will  only  mention  just  a  come 

few  of  them.  When  the  interval,  which  encloses  the  mass  between  limits  of  approach  ation. 
&  recession,  is  somewhat  large,  &  the  sum  of  the  later  repulsive  areas  does  not  greatly  exceed 
that  of  the  attraction,  then  a  slow  evaporation  will  take  place.  Points  which,  in  the  irregular 
agitation,  arrive  at  the  outside,  will  be  few  in  comparison  with  the  whole  mass ;  &  yet 
these,  in  a  very  large  mass,  in  the  same  state  of  fermentation,  will  be  practically  of 
the  same  number  in  the  same  time ;  &  this  number  will  be  diminished  if  the  mass  is 
diminished,  but  the  mass  itself  will  remain  for  a  long  time  practically  unaltered.  Then 
there  will  be  a  sort  of  ebullition ;  &  the  amount  of  the  vapour,  &  the  force  on  egress 
may  be  very  different  in  different  substances ;  for  it  will  depend  on  the  position  at  which 
the  points  are  situated  within  the  curve.  In  some  substances  they  may  be  on  the  near 
side  of  some,  &  in  others  of  other,  very  great  attractive  arcs ;  &  of  these  the  later  arcs  may 
be  either  less  powerful  than  those  in  front,  or  they  may  have  less  powerful  repulsive  arcs 
following  them. 

463.  But  if  the  interval,  which  encloses  the  mass  between  limits  of  approach  &  recession  °r   there  may  be 
should  be  exceedingly  small,  the  last  attractive  arc  may  not  be  so  very  strong,  &  a  very  ^  deflagration^* 
strong  repulsive  arc  may  follow  it.     Then  indeed,  it  may  happen  that,  as  the  mass,  which  various     transfor- 
was  in  a  state  of  relative  rest,  coming  up  to  the  limit  with  but  a  slight  motion  due  to  oTthTmLxLre  flies 
external  points  approaching  close  enough  to  it  to  be  capable  of  impressing  a  non-uniform  off- 
motion  on  the  points  of  the  particles,  an  agitation  within  the  mass  will  be  produced  of 

such  a  kind  that  owing  to  it  all  the  points  in  an  extremely  short  time  will  cross  the  limit, 
&  then  they  will  fly  off  from  one  another  with  a  huge  repulsive  force  &  a  high  velocity. 
This  kind  of  thing  is  seen  to  take  place  in  the  sudden  explosion  of  gunpowder,  which 
commonly  is  not  set  on  fire  by  a  blow  alone  ;  but  on  contact  with  the  smallest  spark  goes 
off  almost  at  once,  &  with  a  very  great  repulsive  force  drives  out  the  ball  from  the  cannon. 
The  same  thing  is  seen  in  phosphorous  substances,  which  go  on  fire  merely  on  contact  with  the 
air  ;  &  nobody  can  fail  to  see  the  differences  that  may  exist  in  all  these  things.  Thus, 
some  of  them  go  on  fire  comparatively  easily,  others  with  greater  difficulty,  some  slowly  & 
others  more  suddenly  ;  the  whole  of  the  mass  may  be  broken  up  without  any  slow  evapora- 
tion in  an  exceedingly  short  time.  If  the  mass  was  originally  heterogeneous,  one  part  may 
fly  off  while  the  rest  remains  ;  &  while  this  happens,  the  parts  that  remain  may  form  fresh 
mixtures  altogether  different  from  the  original,  the  structure  of  the  particles  of  the  higher 
orders  even  being  altered  ;  owing  to  the  fact  that  several  particles  of  lower  orders,  which 
originally  belonged  to  different  particles  of  higher  orders,  now  coalesce  into  a  particle  of  a 
higher  order  of  a  fresh  kind.  From  this  we  get  such  a  large  number  of  compositions  & 
transformations  in  Nature,  &  more  especially  in  chemistry ;  hence  we  get  such  a  large 
number  of  different  kinds  of  vapours,  &  the  great  differences  in  elastic  air,  which  is  formed 
from  such  different  fixed  bodies.  An  immense  field  for  inquiry  is  laid  open  ;  but  I  must 
leave  it  &  go  on  to  some  other  matters,  which  also  refer  to  fermentations  &  evaporations. 

464.  A  substance,  which  has  been  dissolved,  can  be  once  more  obtained,  not  only  by  Concretions,    after 
precipitation,  as  when  metals  fall  by  their  own  weight  reduced  to  the  form  of  an  impalpable  ^^^"de  finite 
powder,  but  also  by  evaporation,  as  we  have  said,  in  the  case  of  salts,  which,  on  the  fluid  shapes  in  the  resi- 
in  which  they  were  dissolved    being  evaporated,  remain  behind  at   the  bottom.     Nor  fnu^'It^formstance 
indeed  do  salts  remain  behind  in  the  form  of  a  fine  powder,  with  their  minutest  particles 

quite  inert ;  but  they  are  grouped  together  in  fairly  large  masses  having  definite  shapes, 
which  differ  for  different  salts ;  these  are  angular  in  all  salts,  &  fearfully  pointed  &  jagged 
in  those  salts  of  a  particularly  corrosive  nature.  In  consequence,  the  salts  are  rather 
sharp  to  the  taste  ;  &  with  some  of  them,  which  are  corrosive,  there  is  a  power  of  cutting 
the  slender  fibres  of  living  things,  &  of  destroying  the  organs  that  are  necessary  to  life. 
The  manner  in  which  they  can  acquire  these  shapes  especially  is  clear  from  Art.  439 ;  as 
also  the  shapes  of  crystals  &  those  jellies  from  which  are  formed  gems  &  hard  stones,  when 
they  are  simple,  &  each  adheres  to  its  own  shape  ;  &  also  of  some  of  the  same  kind,  which 
take  form  after  evaporation  ;  &  in  every  case  this  possibility  is  explained,  as  was  also  shown 
in  the  same  article,  from  the  fact  that  particles  attract  other  particles  situated  at  certain 
distances  only  at  certain  of  their  sides  &  points ;  &  thus  they  will  only  attach  them  to 
themselves  in  a  certain  definite  manner  that  corresponds  to  the  particular  points,  or  sides. 


328  PHILOSOPHIC  NATURALIS  THEORIA 

Quomodo   possit  ^gr    Fermentatio  paullatim  minuitur,  &  demum  cessat,  cuius  imminuti  motus  causas 

fermentatio   cessar.          .   7   J.      ..         .  '  .  ...   J.        .  ...          . 

attigi  plunbus  locis,  ut  num.  197.  Modern  autem  pertmet  illud  etiam,  quod  mnui  num. 
440.  Irregularitas  particularum,  ex  quibus  corpora  constant,  &  inaequalitas  virium, 
plurimum  confert  ad  imminuendum,  &  demum  sistendum  motum.  Ubi  nimirum  aliquae 
particulae,  vel  totse  irruerunt  in  majorum  cavitates,  vel  ubi  suos  uncos  quosdam  aliarum 
uncis,  vel  foraminibus  inseruerunt,  explicari  non  possunt,  &  sublapsus  quidam,  &  compres- 
siones  particularum  accidunt  in  massa  temere  agitata,  quse  motum  imminuunt  &  ad 
sensum  extinguunt,  quo  &  in  mollibus  sisti  motus  potest  post  amissam  figuram.  Multum 
itidem  potest  ad  minuendum,  ac  demum  sistendum  motum  sola  asperitas  ipsa  particularum, 
ut  motus  in  scabro  corpore  sistitur  per  frictionem  ;  multum  incursus  in  externa  puncta, 
ut  aer  pendulum  sistit  :  multum  particulae,  quae  emittuntur  in  omnes  plagas,  ut  in 
evaporatione,  vel  ubi  corpus  refrigescit,  excussis  pluribus  igneis  particulis,  qua;  dum  evolant 
actione  paticularum  massae,  ipsis  massae  particulis  procurrentibus  motum  in  partes  contrarias 
imprimunt,  &  dum  illae,  quas  oscillationem  auxerant,  aliae  post  alias  aufugiunt,  illae,  quae 
remanent,  sunt,  quae  oscillationes  ipsas  internis,  &  externis  actionibus  minuebant. 


Cur  quaedam   sub-          466.  Porro  non  omnes  substantiae  cum  omnibus   fermentant,  sed  cum  quibusdam 

cifnf1qeuibeusdamfI&  tantummodo  :  acidacum  alcalinis ;  &  [215]  quod  quibusdam  videtur  mirum,  sunt  quaedam, 

non  cum  aliis ;  cur  qU3e  apparent  acida  respectu    unius  substantiae,  &  alcalina  respectu  alterius.     Ea  omnia 

mentent'    "debeant  'in  mea  Theoria  facilem  admodum  explicationem  habent  :  nam  vidimus,  particulas  quasdam 

contundi.  respectu  quarundam  inertes  esse,  cum  quibus  commixtae  idcirco  non  fermentant,  respectu 

aliarum  exercere  vires  varias  :   adeoque  si  respectu  quarundam  habeant  pro  variis  distantiis 

diversas  vires,  &  alternationem  satis  magnam  attractionum,  ac  repulsionum  ;    statim,  ac 

satis  prope  ad  ipsas  accesserint,  fermentant.     Sic  si  limatura  ferri  cum  sulphure  commisce- 

atur,  &  inspergatur  aqua,  oritur  aliquanto  post  ingens  fermentatio,  quae  &  inflammationem 

parit,  ac  terraemotuum  exhibet  imaginem  quandam,  &  vulcanorum.     Oportuit  ferrum  in 

tenues  particulas  discerpere,  ac  ad  majorem  mixtionem  adhuc  adhibere  aquam. 


ignem  esse  fennen-  467.  Ignem  ego  itidem  arbitror  esse  quoddam  fermentatioms  genus,  quod  acquirat 

tationis       genus:         ,      h    '.     .  6  111  •  t 

quomodo  excitetur  ve*  potissimum,  vel  etiam  sola  sulphurea  substantia,  cum  qua  iermentat  materia  lucis 
tanta    fermentatio  vehementissime,  si  in  satis  magna  copia  collecta  sit.     Ignem  autem  voco  eum,  qui  non 

ab  exieua  scintilla  r      •  i    «         i    r      •       «    i  •     i     i  • 

tantum  rareiacit  motu  suo,  sed  &  caleiacit,  &  meet,  quae  omma  habentur,  quando  matena 
ilia  sulphurea  satis  fermentescit.  Porro  ignis  comburit,  quia  in  substantiis  combustibilibus 
multum  adest  substantiae  cujusdam,  quae  sulphure  abundat  plurimum,  &  quae  idcirco 
sulphurea  appellari  potest,  quas  vel  per  lucem  in  satis  magna  copia  collectam,  vel  per  ipsam 
jam  fermentescentem  sulphuream  substantiam  satis  praegnantem  ipsa  lucida  materia  sibi 
admotam  fermentescit  itidem,  &  dissolvitur,  ac  avolat.  Is  ingens  motus  intestinus 
particularum  excurrentium  fit  utique  per  vires  mutuas  inter  particulas,  quae  erant  in 
aequilibrio  :  sed  mutatis  parum  admodum  distantiis  exigui  etiam  punctorum  numeri  per 
exiguum  unius  scintillae,  vel  tenuissimorum  radiorum  accessum,  jam  aliae  vires  succedunt, 
&  per  earum  reciprocationem  perturbatur  punctorum  motus,  qui  cito  per  totam  massam 
propagatur. 

Exempium  avicuiae          468.  Imaginem  rei  admodum  vividam  habere    possumus   in   sola   etiam    gravitate. 

djmota  arenula    m   -,-,  °  .         . .       , .  ...  ,.r..  r       j  • 

summo  monte  de-  Emergat  e  man  satis  editus  mons,  per  cujus  latera  dispositae  smt  versus  iundum  mgentes 
jicientis      lapiiios,  lapidum  praegrandium  moles,  turn  quo  magis  ascenditur,  eo  minores ;   donee  versus  apicem 

saxa,     rupes,     &   ,    r.1v      .    r     ,°  .  ^      ,         °.  .      ,          .   '  .,.,     .  \ 

excitantis  in  man  lapilli  smt,  &  in  summo  monte  arenulae:  smt  autem  omnia  tere  in  asquiliDno  pendentia 
subjecto  undas  Jta,  ut  vi  respectu  molis  exigua  devolvi  possint.  Si  avicula  in  summo  monte  commoveat 
arenulam  pede  ;  haec  decidit,  &  lapiiios  secum  dejicit,  qui,  dum  ruunt,  majores  lapides 
secum  trahunt,  &  hi  demum  ingentes  illas  moles  :  fit  ruina  immanis,  &  ingens  motus,  qui, 
decidentibus  in  mare  omnibus,  mare  ipsum  commovet,  ac  in  eo  agitationem  ingentem,  & 
undas  immanes  ciet,  motu  aquarum  vehementissimo  diutissime  perdurante.  Avi-[2l6]-cula 
aequilibrium  arenulae  sustulit  vi  perquam  exigua  :  reliquos  motus  gra vitas  edidit,  quae 
occasionem  agendi  est  nacta  ex  illo  exiguo  motu  avicuiae.  Haec  imago  quaedam  est  virium 
intestinarum  agentium,  ubi  cum  vires  crescere  possint  in  immensum,  mutata  utcunque 
parum  distantia  ;  multo  adhuc  major  effectus  haberi  potest,  quam  in  casu  gravitatis,  quae 


A  THEORY  OF  NATURAL  PHILOSOPHY  329 

465.  The  fermentation  diminishes  gradually,  &  at  length  ceases ;    I  have  touched  The  manner  in 
upon  the  causes  of  this  diminished  motion  in  several  places,  for  instance,  in  Art.  197.     The  may  cease!16 
remarks  I  made  in  Art.  440  also  refer  to  the  same  thing.     The  irregularity  of  the  particles, 

from  which  the  bodies  are  formed,  &  the  inequality  of  the  forces,  especially  contribute 
to  the  diminution  &  final  stoppage  of  the  motion.  Thus,  when  certain  particles,  or  the 
whole  of  them  enter  cavities  in  larger  particles,  or  when  they  insert  their  hooks  into  the 
hooks  or  openings  of  others,  these  cannot  be  disentangled,  &  certain  relapses  &  compressions 
of  the  particles  happen  in  a  mass  irregularly  agitated,  which  diminish  the  motion  & 
practically  destroy  it  altogether  ;  &  due  to  this  the  motion  even  in  soft  bodies  can  be 
stopped  after  a  loss  of  shape.  Also  the  roughness  of  the  particles  alone  may  do  much  toward 
diminishing  &  finally  stopping  the  motion ;  just  as  motion  in  a  rough  body  is  stopped  by 
friction.  Impact  with  external  bodies  has  a  great  effect,  e.g.,  the  air  stops  a  pendulum. 
Much  may  be  due  to  the  emission  of  particles  in  all  directions,  as  in  evaporation  ;  or  when 
a  body  freezes,  many  igneous  particles  being  driven  off  in  the  process ;  &  as  these  particles 
fly  off  by  the  action  of  the  particles  of  the  mass,  impress  a  motion  in  the  opposite  direction 
on  those  particles  as  they  move  ;  &  while  those  that  had  increased  the  oscillation,  one 
after  the  other  fly  off,  those  that  are  left  are  such  as  were  diminishing  these  oscillations 
by  internal  &  external  actions. 

466.  Further,  all  substances  do  not  ferment  with  every  substance,  but  with  some  of  The   reason    why 
them  only.     Thus,  acids  ferment-only  with  alkalies ;  &,  what  to  some  seems  to  be  wonderful,  £5°™^  S^J^^.S 
there  are  some  substances  that  appear  to  be  acid  with  respect  to  one  substance,  &  alkaline  tain  substances  & 
with  respect  to  another.     Now,  all  these  things  have  a  perfectly  easy  explanation  in  my  ^  s^g  ^ust  fbe 
Theory.     For,  we  have  seen  that  certain  particles  are  inert  with  regard  to  certain  other  powdered  before 
particles,  &  therefore  when  these  are  mixed  together  there  will  be  no  fermentation.     With  they  wU1  ferment- 
regard  to  ethers,  again,  they  exert  various  forces ;  hence,  if  with  respect  to  certain  of  them 

they  have  different  forces  for  different  distances,  &  a  sufficiently  great  alternation  of 
attractions  &  repulsions,  they  will  immediately  ferment  on  being  brought  into  sufficiently 
close  contact  with  them.  Thus,  if  iron-filings  are  mixed  with  sulphur,  &  moistened  with 
water,  there  will  be  produced  in  a  little  time  a  great  fermentation  ;  &  this  also  produces 
inflammation,  &  exhibits  phenomena  akin  to  earthquakes  &  volcanoes.  It  is  necessary, 
however,  that  the  iron  should  be  powdered  very  finely,  &  that  water  should  be  used  to 
give  a  still  closer  mingling  of  the  particles. 

467.  I  believe  also  that  fire  itself  is  some  kind  of  fermentation,  which  is  acquired,  either  Fire  is  some  sort  of 
more  especially,  or  even   solely  by  some   sulphurous  substance,  with  which  the  matter  mlmner^'wh'ich*^ 
forming  light  ferments  very  vigorously,  if  it  is  concentrated  in  sufficiently  great  amount,  great  a  fermenta- 
Moreover  I  apply  the  term  fire  to  that  which  not  only  rarefies  through  its  own  motion,  ^"the^Ugntest^of 
but  also  produces  heat  &  light ;    &  all  these  conditions  are  present  when  the  sulphurous  sparks, 
substance  ferments  sufficiently.     Further,  fire  burns,  because  in  combustible  substances 

there  is  present  much  of  a  substance  largely  consisting  of  something  like  sulphur,  for  which 
reason  it  may  be  termed  a  sulphurous  substance.  Such  a  substance,  either  by  contact  with 
light  concentrated  in  sufficiently  great  amount,  or  by  contact  with  the  already  fermenting 
sulphurous  substance  which  is  charged  with  the  matter  of  light  to  a  sufficient  degree, 
will  also  ferment,  &  be  broken  up,  &  fly  off.  The  very  great  internal  motion  of  the  particles 
flying  off  is  in  every  case  due  to  the  mutual  forces  between  the  particles,  which  originally 
were  in  equilibrium  ;  but,  the  distances  of  even  a  very  small  number  of  points  being  changed 
ever  so  little,  by  the  slightest  accession  of  a  spark,  or  of  its  feeblest  rays,  other  forces  then 
take  their  place,  the  motion  of  the  points  is  also. disturbed  by  their  oscillations,  &  this  is 
quickly  propagated  throughout  the  whole  of  the  mass. 

468.  We  can  obtain  a  really  vivid  picture  of  the  matter,  even  in  the  case  of  gravity  As  example,  in  the 
alone.     Suppose  that  from  the  sea  there  rises  a  mountain  of  considerable  height,  &  that  t^movfng^sin^e 
along  the  sides  of  it  there  lie  immense  masses  of  huge  stones,  &  the  higher  one    goes,  grain  of   sand  on 
the  smaller  the  stones  are  ;   until  towards  the  top  the  stones  are  quite  small,  &  at  the  very  *h?  toP  °.f.  a  ™oun- 

.  , ,  11          i  M>I     •  tain,  hurling  down 

summit  they  are  mere  grains  of  sand.     Also  suppose  that  all  of  these  are  just  in  equilibrium,  stones,    rocks, 
so  that  they  can  be  rolled  down  by  a  very  slight  force  compared  with  their  whole  volume,  j^e^aws^^the 
If,  now, a  little  bird  on  the  top  of  the  mountain  moves  with  his  foot  just  one  grain  of  the  sand,  sea  that  lies  at  the 
this  will  fall,  &  bring  down  with  it  the  small  stones ;    these,  as  they  fall,  will  drag  with  *°9*  of  the  moun' 
them  the  larger  stones,  &  these  in  their  turn  will  move  the  huge  boulders.     There  will 
be  an  immense  collapse  &  a  huge  motion  ;  &,  as  all  these  stones  fall  into  the  sea,  the  motion 
will  communicate  itself  to  the  sea  &  cause  in  it  a  huge  agitation  &  immense  waves,  &  this 
vigorous  motion  of  the  water  will  last  for  a  very  considerable  time.     The  little  bird  disturbed 
the  equilibrium  of  the  grain  of  sand  with  a  very  slight  force  ;  gravity  produced  the  remaining 
motions,  &  it  obtained  its  opportunity  for  acting  through  the  slight  motion  of  the  little 
bird.     This  is  a  kind  of  picture  of  the  internal  forces  that  act,  when,  owing  to  the  possibility 
of  the  forces  increasing  indefinitely,  on  the  distance  being  changed  ever  so  slightly,  a  much 


330  PHILOSOPHISE  NATURALIS  THEORIA 

quidem  perseverat  eadem,  aucta  tantummodo  velocitate  descensus  per  novas  accelera- 
tiones. 

?uSX  mateHa  PSU!-          4^9*  Quo&  s'1  ^8™  excitatur  tantummodo  per  sulphureae  substantias  fermentationem  ; 

phura:,    ab   igne  ubi  nihil  adsit  ejus  substantiae,  nullus  crit  metus  ab  igne.     Videmus  utique,  quo  minus 

Sine  fortasse  inlpso  ejusm°di  substantiae  corpora  habeant,  eo  minus  igni  obnoxia  esse,  ut  ex  amianto  &  telas 

Sole  posse  manere  fiant,  quas  igne  moderate  purgantur,  non  comburuntur.     Censeo  autem  idcirco    nostras 

substantias  Utesas.   nasce  terrestres  substantias  ab  igne  satis  intense  dissolvi  omnes,  &  inflammari,  quod  omnes 

ejusmodi  substantias  aliquid  admixtum  habeant,  quod  nectat  etiam  inter  se  plurimas  inertes 

particulas.     At  si  corpora  haberentur  aliqua,  quae  nihil  ex  ejusmodi  substantia  haberent 

admixtum  ;   ea  in  medio  igne  vehementissimo  illaesa  perstarent,  nee  ullum  motum  acqui- 

rerent,  quern  nimirum  nostra  haec  corpora  acquirunt  ab  igne  non  per  incursum,  sed  per 

fermentationem  ab  internis  viribus  excitatam.     Hinc  in  ipso  Sole,  &  fixis,  ubi  nostra  corpora 

momento  fere  temporis  conflagrarent,  &  in  vapores  abirent  tenuissimos,  possunt  esse  corpora 

ea  substantia  destituta,  quae  vegetent,  &  vivant  sine  ulla  organici  sui  textus  laesione  minima. 

Videmus  certe  maculas  superficiei  Solis  proximas  durantes  aliquando  per  menses  etiam 

plures,  ubi  nostrae  nubes,  quibus  eae  videntur  satis  analogae,  brevissimo  tempore  dissiparentur. 


Exempium  fermen-          470.  Id  mirum  videbitur  hommi  prasjudicns  praeoccupato  ;    nee  mtelliget,  qui  fieri 

tationis,     quam  .T/  •  v       -j    •       c    i      •  •  •  i   i  •  •    '  i  • 

cum  aceto  habent  possit,  ut  vivat  aliquid  in  bole  ipso,  in  quo  tanto  major  esse  debet  vis  ustoria,  dum  me 
aiiquae  terrae,  aiiis  exiguus  radiorum  solarium  numerus  majoribus  cavis  speculis,  vel  lentibus  collectus  dissolvit 
omnia.  At  ut  evidenter  pateat,  cujusmodi  praejudicium  id  sit  :  fingamus  nostra  corpora 
compacta  esse  ex  illis  terris,  quas  bolos  vocant,  quae  a  diversis  aquis  mineralibus  deponuntur, 
quas  cum  acidis  fermentant,  ac  omnia  corpora,  quas  habemus  prae  manibus,  vel  ex  eadem 
esse  terra,  vel  plurimum  ex  ea  habere  admixtum.  Acetum  nobis  haberetur  loco  ignis  : 
quascunque  corpora  in  acetum  deciderent,  ingenti  motu  excitato  dissolverentur  citissime, 
&  si  manum  immitteremus  in  acetum  :  ea  ipsa  per  fermentationem  exortam  amissa, 
protinus  horrore  concuteremur  ad  solam  aceti  viciniam,  &  eodem  modo  videretur  nobis 
absurdum  quoddam,  ubi  audiremus,  esse  substantias,  quae  acetum  non  metuant,  &  in  eo 
diu  perstare  possint  sine  minimo  motu,  atque  sui  textus  laesione,  quo  vulgus  rem  prorsus 
absurdam  censebit,  si  audiat,  in  medio  igne,  in  ipso  Sole,  posse  haberi  corpora,  quas  [217] 
nullam  inde  laesionem  accipiant,  sed  pacatissime  quiescant,  &  vegetent,  ac  vivant. 


Deiumme.  senten-          ^yj.  Hasc  quidem  de  igne;    jam  aliquid  de  luce,  quam   ignis  emittit,  &  quas  satis 

tiam  de    emissione         „    T'      .  .  i  ca  j  j  •     • 

luminis     prseferen-  collecta  ipsum  excitat.     Ipsa  lux  potest  esse  emuvium  quoddam  tenuissimum,  &  quasi 

dam  omnino  undis  vapor  fcrmentatione  ignea  vehementi  excussus.     Et  sane  validissima,  meo  quidem  iudicio, 
fluidi  elastici.  1-11  j  i 

argumenta  sunt,  contra  omnes  alias  hypotheses,  ut  contra  undas,  per  quas  onm  pnasnomena 
lucis  explicare  conatus  est  Hugenius,  quam  sententiam  diu  consepultam  iterum  excitare 
conati  sunt  nuper  summi  nostri  asvi  Geometras,  sed  meo  quidem  judicio  sine  successu  (r)  : 
nam  explicarunt  illi  quidem,  &  satis  aegre,  paucas  admodum  luminis  proprietates,  aliis 
intactis  prorsus,  quas  sane  per  earn  hypothesim  nullo  pacto  explicari  posse  censeo,  &  quarum 
aliquas  ipsi  arbitror  omnino  opponi  :  sed  earn  sententiam  impugnare  non  est  hujus  loci, 
quod  quidem  alibi  jam  prasstiti  non  semel.  Mirum  sane,  quam  egregie  in  effluviorum 
emanantium  sententia  ex  mea  Theoria  profluant  omnes  tarn  variae  lucis  proprietates,  quam 
explicationem  fuse  persecutus  sum  in  secunda  parte  dissertationis  De  Lumine :  prascipua 
capita  hie  attingam ;  interea  illud  innuam,  videri  admodum  rationi  consentaneam 
ejusmodi  sententiam  materiae  effluentis,  vel  ex  eo,  quod  in  Ingenti  agitatione,  quam  habet 
ignis,  debet  utique  juxta  id,  quod  vidimus  num.  195,  evolare  copia  quasdam  particularum, 
ut  in  ebullitionibus,  effervescentiis,  fermentationibus  passim  evaporationes  habentur. 


6 


Proprietates   lumi-  4.72.  Praecipuas  proprietates  luminis  sunt  ejus  emissio  constans,  &  ab  aequali  massa, 

ut  ab  eodem  Sole,  ab  ejusdem  candelae  flamma,  ad  sensum  eadem  intensitate  :     immanis 
velocitas,  nam  semidiametrorum  terrestrium  20  millia,  quanta  est  circiter  Solis  a  Terra 

(r)  Cum  htec  scriberem,  nondum  •prodierant  Opera   Taurinensis  Academies  ;   nee  vero    hue  usque,   dum  hoc  Opus 
reimprimitur,  adhuc  videre  potui,  quie  Geometra  maximus  La  Grange  hoc  in  genere  protulit, 


A  THEORY  OF  NATURAL  PHILOSOPHY  331 

greater  effect  can  be  obtained,  than  is  the  case  for  gravity ;  for,  this  remains  the  same, 
the  velocity  of  descent  being  only  increased  by  fresh  accelerations. 

469.  But  if  fire  is  excited  only  by  the  fermentation  of  sulphurous  matter  ;  then,  when  Substances,    that 
none  of  this  matter  is  present,  there  will  be  no  danger  from  fire.     We  see  indeed,  the  less  sulphurous  mattel* 
of  this  substance  the  bodies  have,  the  less  liable  they  are  to  be  injured  by  fire  ;  thus,  a  material  are  not  necessarily 
is  woven  from  asbestos,  &  this  is  only  purified,  but  not  burned,  by  moderate  fire.     Further,  henc^perhaps^lii 
I  consider  that  all  our  earthy  substances  are  broken  up  by  fire,  provided  it  is  sufficiently  the  Sun  itself  there 
intense,  &  are  set  on  fire,  just  because  all  substances   of  this  kind  have  something  mixed 

with  them,  which  connects  a  large  number  of  inert  particles  together.  However,  if  there 
were  any  bodies  which  had  nothing  at  all  of  such  a  substance  mixed  with  them,  these  would 
be  unaltered  in  the  heart  of  the  most  vigorous  fire,  &  would  not  acquire  any  motion,  that 
is  to  say,  such  motion  as  the  bodies  about  us  acquire  from  fire,  not  through  the  entrance 
of  fiery  particles,  but  through  fermentation  excited  by  internal  forces.  Hence,  in  the  Sun 
itself,  &  in  the  stars,  in  which  our  terrestrial  bodies  would  burn  up  in  an  instant  of  time 
&  go  off  into  the  thinnest  of  vapours,  there  may  exist  bodies  altogether  lacking  in  such 
a  substance  ;  &  these  may  grow  &  live  without  the  slightest  injury  of  any  kind  to  their 
organic  structure.  Indeed  we  see  spots  very  close  to  the  Sun  lasting  sometimes  for  several 
months  even  ;  whereas  our  clouds,  to  which  these  spots  seem  to  bear  a  considerable  analogy, 
would  be  dissipated  in  a  very  short  time. 

470.  Now  this  will  appear  wonderful  to  a  man  who  is  obsessed  by  prejudices ;    nor  Example,  in  the 
will  he  be  able  to  understand  why  it  is  that   anything  can  live  in  the  Sun,  in  which  ^   which^some 
there  is  bound  to  be  ever  so  much  greater  burning  force,  while  on  earth  an  exceedingly  earths    have  with 
small  number  of  solar  rays,  collected  by  fairly  large  concave  mirrors  or  by  lenses,  will  break  areeunaffectcdthers 
up  all  substances.     However,  in  order  to  make  plain  how  such  a  prejudice  arises,  let  us 

suppose  that  our  substances  are  formed  from  those  earths,  which  are  termed  boluses,  such 
as  are  deposited  by  certain  minerals  of  different  kinds  &  ferment  with  acids ;  &  that  all 
bodies  around  us  either  are  formed  out  of  this  earth  or  are  largely  impregnated  with  it. 
Let  vinegar  be  taken  to  represent  fire  ;  then  if  any  of  these  bodies  fell  into  the  vinegar, 
they  would  be  very  quickly  broken  up  by  the  huge  motion  induced ;  &  if  we  placed  our 
hands  in  the  vinegar,  they  too  being  lost  by  the  fermentation  produced,  we  should  be 
forthwith  struck  with  horror  at  the  mere  vicinity  of  vinegar.  It  would  seem  to  us  that 
it  was  something  ridiculous  if  we  were  told  that  there  were  substances  which  were  in  no 
fear  of  vinegar,  but  could  last  in  it  for  a  long  time  without  slightest  motion  or  injury  to 
their  structure  ;  in  exactly  the  same  way  as  an  ordinary  man  would  think  it  ridiculous, 
if  he  were  told  that  in  the  heart  of  fire,  or  in  the  Sun  itself,  there  might  exist  bodies  which 
received  no  injury  from  it,  but  remained  at  rest  in  the  most  calm  fashion,  &  grew  &  lived. 

471.  So  much  on  the  subject  of  fire;    now  I  will  make  a  few  remarks  about  light,  Light;  the  theory 
which  is  given  off  by  fire,  &  which,  when  present  in  sufficient  quantity,  excites  fire.      It  is  to^bT^referred 
possible  that  light   may  be  a  sort  of  very  tenuous  effluvium,  or  a  kind  of  vapour  forced  altogether  before 
out  by  the  vigorous  igneous  fermentation.     Indeed,  in  my  judgment,  there  are  very  strong  elastic*  fluid68  m  *" 
arguments  in  favour  of  this  hypothesis,  as  opposed  to  all  other  hypotheses,  such  as  that 

of  waves.  On  the  hypothesis  of  waves,  Huygens  once  tried  to  explain  all  the  phenomena 
of  light ;  &  the  most  noted  of  the  geometers  of  our  age  have  tried  to  revive  this  theory, 
which  had  been  buried  with  Huygens;  but,  as  I  think,  unsuccessfully  (r).  For,  they  have 
explained,  &  even  then  poorly  enough,  a  very  few  of  the  properties  of  light,  leaving  the 
rest  untouched  ;  &  indeed  I  consider  that  such  properties  can  not  be  explained  in  any  way 
by  this  hypothesis  of  waves,  &  my  opinion  is  that  some  of  them  are  altogether  contrary  to  it. 
But  this  is  not  the  right  place  to  impugn  this  theory ;  indeed  I  have  already,  more  than 
once,  presented  my  view  in  other  places.  It  is  really  marvellous  how  excellently,  on  the 
hypothesis  of  emanating  effluvia,  all  the  different  properties  of  light  are  derived  from  my 
Theory  in  a  straightforward  way.  I  gave  a  very  full  explanation  of  this  in  the  second  part  of 
my  dissertation,  De  Lumine  ;  &  the  principal  points  of  this  work  I  will  touch  upon  here. 
Meanwhile,  I  will  just  mention  that  the  idea  of  effluent  matter  seems  to  be  altogether 
reasonable;  more  especially  from  the  fact  that,  in  a  very  great  agitation  amongst  particles, 
such  as  there  is  in  the  case  of  fire,  there  is  always  bound  to  be,  in  accordance  with  what  we 
have  seen  in  Art.  195,  an  abundance  of  particles  flying  off,  just  as  we  have  evaporations 
in  ebullition,  effervescence  &  fermentation. 

472.  The  principal  properties  of  light  are  : — its  constant  emission,  &  the  fact  that  Those  properties  of 

.1        •    '          •        •  r   1  ,r  f  .  ,      V,  ,  ,        light   for  which  we 

the  intensity  is  always  the  same  from  the  same  mass,  such  as  from  the  Sun,  or  from  the  have   to  find   the 
flame  of  the  same  candle ;    its  huge  velocity,  for  it  traverses  a  distance  equal  to  twenty  reason, 
thousand  times  the  semidiameter  of  the  Earth,  which  is  about  the  distance  of  the  Sun 

(r)  When  I  wrote  this,  the  Transactions  of  the  Academy  of  Turin  had  not  been  published  ;  and  even  now,  at  the  time 
of  this  reprint  of  my  work,  I  have  so  far  been  unable  to  see  what  that  excellent  geometer  La  Grange  has  published  on 
the  subject. 


332  PHILOSOPHIC  NATURALIS  THEORIA 

distantia,  percurrit  semiquadrante  horae ;  velocitatum  discrimcn  cxiguum  in  diversis 
radiis,  nam  celeritatis  discrimen  in  radiis  homogeneis  vix  ullum  esse,  si  quod  est,  colligitur 
pluribus  indiciis  :  propagatio  rectilinea  per  medium  diaphanum  ejusdem  densitatis  ubique 
cum  impedimento  progressus  per  media  opaca,  sine  ullo  impedimento  sensibili  ex  impactu 
in  se  invicem  radiorum  tot  diversas  directiones  habentium,  aut  in  partes  internas  diaphan- 
orum  corporum  utcunque  densorum  :  reflexio  partis  luminis  ad  angulos  asquales  in 
mutatione  medii,  parte,  quae  reflectitur,  eo  majore  respectu  luminis,  quo  obliquitas 
incidentiae  est  major  ;  refractio  alterius  partis  eadem  mutatione  cum  lege  constantis  rationis 
inter  sinum  incidentiae,  &  sinum  anguli  refracti ;  quae  ratio  [218]  in  diversis  coloratis 
radiis  diversa  est,  in  quo  stat  diversa  diversorum  coloratorum  radiorum  refrangibilitas  : 
dispersio  &  in  reflexione,  &  in  refractione  exiguse  partis  luminis  cum  directionibus  quibus- 
cunque  quaquaversus  :  alternatio  binarum  dispositionum  in  quovis  radio,  in  quarum 
altera  facilius  reflectatur,  &  in  altera  facilius  transmittatur  lux  delata  ad  superficiem 
dirimentem  duo  media  heterogenea,  quas  Newtonus  vocat  vices  facilioris  reflexionis,  & 
facilioris  transmissus,  cum  intervallis  vicium,  post  quae  nimirum  dispositiones  maxime 
faventes  reflexioni,  vel  refraction!  redeunt,  aequabilis  in  eodem  radio  ingresso  in  idem 
medium,  &  diversis  coloratis  radiis,  in  diversis  mediorum  densitatibus,  &  in  diversis 
inclinationibus,  in  quibus  radius  ingreditur,  ex  quibus  vicibus,  &  earum  intervallis  diversis 
in  diversis  coloratis  radiis  pendent  omnia  phenomena  laminarum  tenuium,  &  naturalium 
colorum  tarn  permanentium,  quam  variabilium,  uti  &  crassarum  laminarum  colores,  quae 
omnia  satis  luculenter  exposuit  in  celebri  dissertatione  De  Lumine  P.  Carolus  Benvenuti  e 
Soc.  nostra  Scriptor  accuratissimus  :  ac  demum  ilia,  quam  vocant  diffractionem,  qua 
radii  in  transitu  prope  corporum  acies  inflectuntur,  &  qui  diversum  colorem,  ac  diversam 
refrangibilitatem  habent,  in  angulis  diversis. 


Emissio    quomodo          473.  Quod  pertinet  ad  emissionem   jam  est  expositum  num.  199,  &  num   461  ;    ubi 
si"mui  etiam  ostensum  est  illud,  manente  eadem  massa  quae  emittit  effluvia,  ipsorum  multitudinem 


citissime    d  is  sol-  dato  tempore  esse  ad  sensum  eandem.     Porro  fieri  potest,  ut  massa,  quae  lumen  emittit, 

emittunt,UIut  "ignis  Penitus  dissolvatur,  ut  in  ignibus  subitis  accidit,  &  fieri  potest,  ut  perseveret  diutissime, 

subitus,    quaedam,  Id  potissimum  pendet  a  magnitudine  intervalli,  in  quo  fit  oscillatio  fermentationis,  &  a 

persVste'nt^slne  natura  arcus  attractivi  terminantis  id  intervallum  juxta  num.  195.      Quin  immo  si  Auctor 

sensibili  jactura.       Naturae  voluit  massam  vehementissima  etiam  fermentatione  agitatam  prorsus  indissolubilem 

quacunque  finita  velocitate,  potuit  facile  id  praestare  juxta  num.  460  per  alios  asymptoticos 

arcus  cum  areis  infinitis,  intra  quorum  limites  sit  massa   fermentescens  ;  quorum   ope  ea 

colligari  potest  ita,  ut  dissolvi  omnino  nequeat,  ponendo  deinde  materiam  luminis  emittendi, 

ultra  intervallum  earum  asymptotorum  respectu  particularum  ejus  massae,  &  citra  arcum 

attractivum  ingentis  areae,  sed  non  infinitae,  ex  quo  aliae  lucidae  particulae  evolare  possint 

post  alias.     Nee  illud,  quod  vulgo  objici  solet,  tanta  luminis  effusione  debere  multum 

imminui  massam  Solis,  habet  ullam  difficultatem,  posita  ilia  componibilitate  in  infinitum 

&  ilia  solutione  problematis  quae  habetur  num.  395.     Potest  enim  in  spatiolo  utcunque 

exiguo  haberi  numerus  utcunque  ingens  punctorum,  &  omnis  massa  luminis,  quas  diffusa 

tarn  immanem  molem  occupat,  potest  in  Sole,  vel  prope  Solem  occupavisse  spatiolum, 

quantum  libuerit,  parvum,  ut  idcirco  Sol  post  quotcunque  sae-[2i9]-culorum   millia  ne 

latum  quidem  unguem  decrescat.     Id  pendet  a  ratione  densitatis  luminis  ad  densitatem 

Solis,  quae  ratio  potest  esse  utcunque  parva  ;    &  quidem  pro  immensa  luminis  tenuitate 

sunt  argumenta  admodum  valida,  quorum  aliqua  proferam  infra. 


Unde  tanta  veloci-          474.  Celeritas  utcunque  magna  haberi  potest  ab  arcubus  repulsivis  satis  validis,  qui 

tfsS  'discrimln0<exi-  occurrant  post  extremum  limitem  oscillationis  terminatae  ab  arcu  ingenti  attractive  juxta 

guum,  &  in  radiis  num.   194  :    nam  si  inde  evadat  particula  cum  velocitate  nulla  ;    quadratum  velocitatis 

homogeneis    multo  totjus  definitur  ab  excessu  arearum  omnium  repulsivarum  supra  omnes  attractivas  juxta 

num.  178,  qui  excessus  cum  possit  esse  utcunque  magnus  ;   ejusmodi  celeritas  potest  itidem 

esse  utcunque  magna.     Verum  celeritatis  discrimen  in  particulis  homogeneis  erit  prorsus 

insensibile,  qui  a  particulae  luminis  ejusdem  generis  ad  finem  oscillationis  advenient  cum 

velocitatibus  fere  nullis  :    nam  eae,  quae  juxta  Theoriam  expositam  num.  195,  paullatim 

augent  oscillationem  suam,  demum  adveniunt  ad  limitem  cohibentem  massam,  &  avolant ; 


A  THEORY   OF  NATURAL  PHILOSOPHY  333 

from  the  Earth,  in  an  eighth  of  an  hour  ;  the  slight  differences  of  velocity  that  exist  in 
different  rays,  for  it  is  proved  from  several  indications  that  there  is  scarcely  any  difference 
for  homogeneous  light,  if  there  is  any  at  all ;  its  rectilinear  propagation  through  a  transparent 
medium  everywhere  equally  dense,  along  with  hindrance  to  progression  through  opaque 
media  ;  &  this  without  any  sensible  hindrance  due  to  impact  with  one  another  of  rays 
having  so  many  different  directions,  or  any  that  prevents  passage  into  the  inner  parts  of 
transparent  bodies,  no  matter  how  dense  they  may  be  ;  reflection  of  part  of  the  light  at 
equal  angles"  at  the  surface  of  separation  of  two  media,  the  part  that  is  reflected  being  greater 
with  regard  to  the  whole  amount  of  light,  according  as  the  obliquity  of  incidence  is  greater  ; 
refraction  of  the  other  part  at  the  same  surface  of  separation,  with  the  law  of  a  constant 
ratio  between  the  sines  of  the  angle  of  incidence  &  the  angle  of  refraction,  the  ratio  being 
different  for  differently  coloured  rays,  upon  which  depends  the  different  refrangibility  of 
the  differently  coloured  rays  ;  dispersion,  both  in  reflection  &  in  refraction,  of  a  very  small 
part  of  the  light  in  directions  of  every  description  whatever ;  the  alternation  of  propensity 
in  any  one  ray,  in  one  of  which  the  light  falling  upon  the  surface  of  separation  between 
two  media  of  different  nature  is  the  more  easily  reflected  &  in  the  other  is  the  more  easily 
transmitted,  which  Newton  calls  '  fits '  of  easier  reflection  &  easier  transmission,  with 
intervals  between  these  fits,  after  which  the  propensities  mostly  favouring  reflection  or 
refraction  return,  these  intervals  being  equal  in  the  same  ray  entering  the  same  medium, 
&  different  for  differently  coloured  rays,  for  different  densities  of  the  medium,  £  for 
the  different  inclinations  at  which  the  ray  enters  the  medium  ;  upon  these  fits  &  the 
different  intervals  between  them  for  differently  coloured  rays  depend  all  the  phenomena 
of  thin  plates,  &  of  natural  colours,  both  variable  &  permanent,  as  well  as  the  colours 
of  thick  plates,  all  of  which  have  been  discussed  with  considerable  clearness  by  Fr.  C. 
Benvenuti,  a  most  careful  writer  of  our  Society,  in  his  well-known  dissertation,  De 
Lumine.  Last  of  all,  we  have  that  property,  which  is  called  diffraction,  in  which  rays, 
passing  near  the  edge  of  a  body,  are  bent  inwards,  having  a  different  colour  &  different 
refrangibility  for  different  angles. 

473.  What  pertains  to  emission  has  been  already  explained  in  Art.  199  &  Art.  461  ;  How  emission  takes 
there  also  it  was  shown  that,  if  the  mass  emitting  the  effluvia  remained  the  same,  then  p^nT'that*  *lome 
the  amount    emitted    is    practically    the    same    in    any    given  time.     Further,    it   may  bodies    are    very 
happen  that    the    mass    emitting  the   light    is    completely    broken  up,  as  takes  place  in  ^"the  tkne^they 
sudden  flashes  of  fire  ;   or  it  may  be  that  this   mass  persists  for  a  very  long  time.     This  emit  light,  like  a 
to  a  very  great  extent  depends  on  the  size  of  the  interval  in  which  the  oscillation  due  to  wmfe^hers?*  Rke 
fermentation  takes  place,  &  on  the  nature  of  the  attractive  arc  at  the  end  of  that  interval,  the  Sun  persist  for 
by  Art.  195.     Nay,  if  the  Author  of  Nature  had  wished  that  a  mass,  agitated  by  the  most  ^^t  Jnylppar- 
vigorous  fermentation  even,  should  be  quite  irreducible  by  any  finite  force  whatever,  he  ent  loss. 
could  easily  have  accomplished  this,  as  shown  in  Art.  460,  by  other  asymptotic  arcs  with 
infinite  areas,  between  the  confines  of  which  the  fermenting  mass  would  be  situated.     By 
the  aid  of  these  arcs  the  mass  could  be  so  bound  together,  that  it  would  not  admit  of  the 
slightest  dissolution  ;    &  then  by   placing  the   material  for   emitting  light  further  from 
the  particles  of  the  mass  than  the  interval  between  those  asymptotes,  &  within  the  distance 
corresponding  to  an  attractive  arc  of  huge  but  finite  area ;  from  which  we  should  have 
particles,  one  after  the  other,  of  light  flying  off.      Nor  is  there  any  difficulty  from  the 
usual  argument  that  is  raised  in  objection  to  this,  that  the  mass  of  the  Sun  must  be  much 
diminished  by  such  a  large  emission  of  light ;  if  we  suppose  indefinitely  great  componibility, 
&  the  solution  of  the  problem,  given  in  Art.  395.     For  in  any  exceedingly  small  space 
there  may  be  any  huge  number  of  points  whatever  ;   &  the  whole  mass  of  the  light,  which 
is  diffused  throughout  &  occupies  such  an  immense  volume,  may,  in  the  Sun  or  near  the 
Sun,  have  occupied  a  space  as  small  as  ever  one  likes  to  assign  ;   so  that  the  Sun,  after  the 
lapse  of  any  number  of  thousands  of  centuries,  will  not  therefore  have  decreased  by  even 
a  finger's  breadth.     It  all  depends  on  the  ratio  of  the  density  of  light  to  the  density  of  the 
Sun,  &  this  ratio  can  be  any  small  ratio  whatever.     Indeed  there  are  perfectly  valid  arguments 
for  the  immense  tenuity  of  light,  some  of  which  I  will  give  below. 

474.  Any  velocity,  no  matter  how  great,  can  be  obtained  from  sufficiently  powerful  Whence  comes  the 
repulsive  arcs,  if  these  occur  after  the  last  limit  of  oscillation  within  the  confines  of  a  very  withstanding7'  the 
great  attractive  arc,  as  shown  in  Art.  104.  For  if  a  particle  goes  off  from  here  with  no  slight  differences  in 

3   ,      .  r     ,  i      •        •      i    r        i  i  j-     11     i  i  •  velocity,  &  the  still 

velocity,  the  square  of  the  whole  velocity  is  defined  by  the  excess  of  all  the  repulsive  areas  iess  differences   in 

over  all  the  attractive,  as  was  shown  in  Art.  178  ;   &,  as  this  excess  can  be  of  any  amount  homogeneous  rays. 

whatever,  the  velocity  can  also  be  of  any  magnitude  whatever.     Again,  the  difference  of 

velocity  for  homogeneous  particles  is  quite  insensible,  because  particles  of  light  of  the  same 

kind  come  to  the  end  of  their  oscillation  with  velocities  that  are  almost  zero  ;    for  those 

which,  according  to  the  Theory  set  forth  in  Art.  195,  increase  their  oscillation  gradually, 

arrive  at  the  boundary  limiting  the  mass  at  last,  &  then  fly  off.     Now,  if,  at  the  time  they 


334  PHILOSOPHIC  NATURALIS  THEORIA 

quo  si  turn,  cum  avolant,  advenirent  cum  ingenti  velocitate,  advenissent  utique  eodem,  & 
effugissent  in  oscillatione  praecedenti.  Demonstravimus  autem  ibidem,  exiguum  discrimen 
velocitatis  in  ingressu  spatii,  in  quo  datae  vires  perpetuo  accelerant  motum,  &  generant 
velocitatem  ingentem,  inducere  discrimen  velocitatis  genitae  perquam  exiguum  etiam 
respectu  illius  exigui  discriminis  velocitatis  initialis,  quod  demonstravimus  ibi  ratione  petita  a 
natura  quadrati  quantitatis  ingentis  conjunct!  cum  quadrate  quantitatis  multo  minoris,  quod 
quantitatem  exhibet  a  priore  ilia  differentem  multo  minus,  quam  sit  quantitas  ilia  parva, 
cujus  quadratum  conjungitur.  Discrimen  aliquod  sensibile  haberi  poterit ;  siqua  effugiunt, 
non  sint  puncta  simplicia,  sed  particulae  non  nihil  inter  se  diversse  :  nam  curva  virium, 
qua  massa  tota  agit  in  ejusmodi  particulas,  potest  esse  nonnihil  diversa  pro  illis  diversis 
particulis,  adeoque  excessus  summae  arearum  repulsivarum  supra  summam  attractivarum 
potest  esse  nonnihil  diversus  &  quadratum  velocitatis  ipsi  respondens  nonnihil  itidem 
diversum.  Hoc  pacto  particulae  luminis  homogeneas  habebunt  velocitatem  ad  sensum 
prorsus  asqualem ;  particulas  heterogeneae  poterunt  habere  nonnihil  diversa m,  uti  ex 
observatione  phsenomenorum  videtur  omnino  colligi.  Illud  unum  hac  in  re  notandum 
superest,  quod  curva  virium,  qua  massa  tota  agit  in  particulam  positam  jam  ultra  terminum 
oscillationum,  mutatis  per  oscillationem  ipsam  punctis  massae,  mutabitur  nonnihil :  sed 
quoniam  in  fortuita  ingenti  agitatione  massae  totius  celerrime  succedunt  omnes  diversse 
positiones  punctorum  ;  summa  omnium  erit  ad  sensum  eadem,  potissimum  pro  particula 
diutius  hasrente  in  illo  initio  suae  fugae,  ad  quod  advenit,  uti  diximus,  cum  velocitate 
perquam  exigua,  ut  idcirco  homogenearum  velocitas,  [220]  ubi  jam  deventum  fuerit  ad 
arcum  gravitatis,  &  vires  exiguas,  debeat  esse  ad  sensum  eadem,  &  discrimen  aliquod  haberi 
possit  tantummodo  in  heterogeneis  particulis  a  diverse  earum  textu.  Patet  igitur,  unde 
celeritas  ingens  provenire  possit,  &  si  quod  est  celeritatis  discrimen  exiguum. 


Unde  propagatio          475.  Quod  pertmet  ad  propagationem  rectilmeam  per  medium  homogeneum  diaph- 

rectilinea   :     incur-  a       j  TL  •  11  j-  v     •      •        i        •    •  i        V 

sum   immediatum  anum,  &  ad.  motum  nberum  sine  ullo  impedimento  a  particulis  ipsius  luminis,  vel  medii 
punctorum  lucis,  diaphani,  id  in  mea  Theoria  admodum  facile  exponitur,  quod  in  aliis  ingentem  difficultatem 

in     puncta     medii  in.          -j  j  j    •  j-  •  •  •  IT          i     i 

nullum    haberi:  pant.     Et  quidem  quod  pertmet  ad  impedimenta,  si  curva  virium  nullum  habeat  arcum 
virium    in   medio  asymptoticum  perpendicularem  axi  praeter  primum  :    ostensum  est  num.  362.  sola  satis 

homogeneo    exi-       '  i      •  t^j        •  •  j  i  • 

guam    inaequaiita-  ™agna  velocitate  obtinen  posse  apparentem  compenetrationem  duarum  substantiarum, 
tem  eludi  a  tenui-  quam  tenuitas,  &  homogeneitas  spatii,  per   quod   transitur,  plurimum    iuvat.     Quoniam 

tate,     &    celeritate    *  •  '.r   ,.?.,.,.  .      .  '  </,...-. 

luminis.  respectu  punctorum   matense   prorsus   mdivisibihum,   &  mextensorum  infinities   mnnita 

sunt  puncta  spatii  existentia  in  eodem  piano  ;  infinities  infinite  est  improbabilis  pro  quovis 
momento  temporis  directio  motus  puncti  materiae  cujus  vis  accurate  versus  aliud  punctum 
materiae,  ac  improbabilitas  pro  summa  momentorum  omnium  contentorum  dato  quovis 
tempore  utcunque  longo  evadit  adhuc  infinita.  Ingens  quidem  est  numerus  punctorum 
lucis,  &  propemodum  immensus,  sed  in  mea  Theoria  utique  finitus.  Ea  puncta  quovis 
momento  temporis  directiones  motuum  habent  numero  propemodum  immenso,  sed  in 
mea  Theoria  finite.  Verum  quidem  est,  ubicunque  oculus  collocetur  in  immensa 
propemodum  superficie  sphserse  circa  unam  fixam  remotissimam  descripta,  immo  intra 
ipsam  sphaeram,  videri  fixam,  &  proinde  aliquam  luminis  particulam  afncere  nostrum 
oculum  :  sed  id  fit  in  mea  Theoria  non  quia  accurate  in  omnibus  absolute  infinitis 
directionibus  adveniant  radii,  sed  quod  pupilla,  &  fibrae  oculorum  non  unicum  punctum 
£unt,  &  vires  punctorum  particulae  luminis  agunt  ad  aliquod  intervallum.  Hinc  quovis 
utcunque  longo  tempore  nullus  debet  accidere  casus  in  mea  Theoria,  in  quo  punctum 
aliquod  luminis  directe  tendat  contra  aliquod  aliud  punctum  vel  luminis,  vel  substantiae 
cujusvis,  ut  in  ipsum  debeat  incurrere.  Quamobrem  per  incursum,  &  immediatum 
impactum  nullum  punctum  luminis  aut  sistet  motum  suum,  aut  deflectet. 


si    satis    magnam          476.  Id  quidem  commune  est  omnibus  corporibus,  quae  corpora  inter  se  congrediuntur. 

antTq^evL,  soiida  Ea  nullum  habent  in  mea  Theoria  punctum  immediatum  incurrens  in  aliud  punctum  ; 

etiam,  transitura  quam  ob  causam  &  illud  ibidem  dixi,  si  nullae  vires  mutuae  adessent,  debere  utique  haberi 

sine3  umT  motuum  apparentem  quandam  compenetrationem  omnium  massarum  ;    sed  adhuc  vel  ex  hoc  solo 

perturbatione.          capite  veram  compenetrationem  haberi    nunquam    omnino  posse.     Vires  igitur  quae  ad 

aliquam  distantiam  protenduntur,  im-  [221]  -pediunt  progressum.    Eae  vires  si  circumquaque 

essent  semper  aequales ;    nullum  impedimcntum  haberet  motus,  qui  vi  inertiae  deberet 


A  THEORY  OF  NATURAL  PHILOSOPHY  335 

fly  off,  they  should  reach  this  boundary  with  a  very  great  velocity,  then  it  is  certain  that 
they  would  have  reached  it  &  flown  off  in  a  previous  oscillation.  Further,  in  the  same 
article,  we  have  proved  that  a  slight  difference  of  velocity  on  entering  a  space,  in  which 
given  forces  continually  accelerate  the  motion  &  generate  a  huge  velocity,  also  induces  a 
difference  in  the  velocity  generated  that  is  very  small  even  when  compared  with  the  slight 
difference  in  the  initial  velocity.  This  we  there  prove  from  an  argument  derived  from 
the  nature  of  the  square  of  a  very  large  quantity  compounded  with  the  square  of  a  quantity 
much  less  than  it ;  this  gives  a  quantity  differing  from  the  first  quantity  by  something 
much  less  than  the  small  quantity  of  which  the  square  was  added.  A  sensible  difference 
may  be  obtained,  if  what  fly  off  are  not  simple  points,  but  particles  somewhat  different 
from  one  another.  For  the  curve  of  forces,  with  which  the  mass  acts  upon  such  particles, 
can  be  somewhat  different  for  those  different  particles ;  &  thus,  the  excess  of  the  sum  of 
the  repulsive  areas  over  the  sum  of  the  attractive  may  be  somewhat  different,  &  therefore 
the  square  of  the  velocity  corresponding  to  this  excess  may  be  somewhat  different.  In 
this  way  particles  of  homogeneous  light  will  have  velocities  that  are  practically  equal ; 
but  particles  of  heterogeneous  light  may  have  velocities  that  are  somewhat  different ;  as 
seems  to  be  conclusively  shown  from  observations  of  phenomena.  One  thing  remains  to 
be  noted  in  this  connection,  namely,  that  the  curve  of  forces,  with  which  the  whole  mass 
acts  upon  a  particle  placed  already  beyond  the  limit  of  the  oscillation,  when  the  points 
of  the  mass  are  changed  on  account  of  the  oscillation,  will  be  somewhat  altered.  But  since 
in  a  very  large  irregular  agitation  of  the  entire  mass  all  the  different  positions  of  the  points 
follow  on  after  one  another  very  quickly,  the  sum  of  all  the  forces  will  be  practically  the 
same,  especially  in  the  case  of  a  particle  stopping  for  some  time  at  the  beginning  of  its  , 
flight ;  which  point  it  has  reached,  as  we  have  said,  with  a  velocity  that  is  exceedingly 
small.  Thus,  the  velocity  of  homogeneous  particles  must  on  that  account  be  practically 
the  same,  when  they  have  reached  the  arc  representing  gravitation  ;  &  a  difference  can 
only  be  obtained  in  heterogeneous  particles  owing  to  their  structure.  It  is  therefore 
clear  from  what  source  the  very  great  velocity  can  come,  &  also  the  slight  differences,  if 
there  are  any. 

475.  That    which    relates    to    the    rectilinear    propagation    through    a    transparent  The  reason  for 
homogeneous  medium,  &  the  free  motion,  without  hindrance,  by  particles  either  of  the  light  eation^there^Tan 
or  of  the  transparent  medium,  is  quite  easily  explained  in  my  Theory,  whereas  in  other  be   no'  immediate 
theories  it  begets  a  very  great  difficulty.     Also  as  regards  hindrance  to  this  motion,  so  ^'pcTnt^oV^ight 
long  as  the  curve  of  forces  has  no  asymptotic  arc  perpendicular  to  the  axis  besides  the  first,  &  the  points  of  the 
it  has  been  shown,  in  Art.  362,  that  merely  with  a  sufficiently  great  velocity  there  can  be  ^equ'aiit'    of1  the 
obtained  an  apparent  compenetration  of  two  substances ;    &  tenuity  &  homogeneity  of  forces  in  a  homo- 
space  traversed  will  assist  this  to  a  very  great  extent.     Now,  since,  compared  with  perfectly  f re ^"uded  "tf^the 
indivisible  &  non-extended  points  of  matter,  there  are  an  infinitely  infinite  number  of  points  tenuity  &  the  veio- 
of  space  existing  in  the  same  plane,  there  is  an  infinitely  infinite  improbability  that,  for  "ty0*1^*- 

any  instant  of  time  chosen,  the  direction  of  motion  of  any  one  point  of  matter  should  be 
accurately  directed  towards  any  other  point  of  matter  ;  &  this  improbability,  when  we 
consider  the  sum  of  all  the  instants  contained  in  any  given  time,  however  long,  still  comes 
out  simply  infinite.  The  number  of  points  of  light  is  indeed  very  large,  not  to  say 
enormous,  but  in  my  Theory  it  is  at  least  finite.  These  points  at  any  chosen  instant  of 
time  have  an  almost  immeasurable  number  of  directions  of  motion,  but  this  number  is 
finite  in  my  Theory.  It  is  indeed  true  that,  no  matter  where  an  eye  is  situated  upon  the 
well-nigh  immeasurable  surface  of  a  sphere  described  about  one  of  the  remotest  stars  as 
centre,  nay,  or  within  that  sphere,  the  star  will  be  seen  ;  &  thus,  it  is  true  that  some  particle 
of  light  must  affect  our  eye.  But  in  my  Theory,  that  does  not  come  about  because  rays 
of  light  come  to  it  accurately  in  every  one  of  an  absolute  infinity  of  directions ;  but  because 
the  pupil  &  the  nerves  of  the  eye  do  not  form  a  single  point,  &  the  forces  due  to  the 
points  of  a  particle  of  light  act  at  some  distance  away.  Hence,  in  any  chosen  time,  no 
matter  how  long,  there  need  not  happen  in  my  Theory  any  case,  in  which  any  point  of 
light  is  directed  exactly  towards  any  other  point  either  of  light,  or  of  any  substance,  so 
that  it  is  bound  to  collide  with  it.  Hence,  no  point  of  light  stays  its  motion,  or  deflects 
it,  through  collision  or  immediate  impact. 

476.  This  is  indeed  a  common  property  of  all  bodies,  that  is,  of  bodies  that  approach  If  they  possess 
one  another.     In  my  Theory,  they  have  no  point  directly  colliding  with  any  other  point,  fity,  anybodies. 
For  this  reason  I  also  stated,  in  the  above-mentioned  article,  that,  if  no  mutual  forces  were  even  solids,  win 
present,  there  is  always  bound  to  be  an  apparent  compenetration  of  all  bodies.     Yet,  from  solids,  without°any 
this  article  alone,  it  is  utterly  impossible  that  there  ever  can  be  real  compenetration.     Hence,  disturbance  of  their 
forces  extending  over  some  distance  will  hinder  the  progressive  motion.     If  these  forces 

are  always  equal  in  all  directions,  there  would  be  no  impediment  to  the  motion,  &  it  would 
necessarily  be  rectilinear  owing  to  the  force  of  inertia.  Hence,  nothing  but  a  difference  in  the 


336  PHILOSOPHIC  NATURALIS  THEORIA 

esse  rectilineus.  Quare  sola  differentia  virium  agentium  in  punctum  mobile  obstare  potest. 
At  si  nulla  occurrat  infinita  vis  arcus  asymptotici  cujuspiam  post  primum  ;  vires  omnes 
finitae  sunt,  adeoque  &  differentia  virium  secundum  diversas  directiones  agentium  finita 
est  semper.  Igitur  utcunque  ea  sit  magna,  ipsarri  finita  quaedam  velocitas  elidere  potest, 
quin  permittat  ullam  retardationem,  accelerationem,  deviationem,  quas  ad  datam  quampiam 
utcunque  parvam  magnitudinem  assurgat  :  nam  vires  indigent  tempore  ad  producendam 
novam  velocitatem,  quae  semper  proportionalis  est  tempori,  &  vi.  Hinc  si  satis  magna 
velocitas  haberetur  ;  quaevis  substantia  trans  aliam  quanvis  libere  permearet  sine  ullo 
sensibili  obstaculo,  &  sine  ulla  sensibili  mutatione  dispositionis  propriorum  punctorum, 
&  sine  ulla  jactura  nexus  mutui  inter  ipsa  puncta,  &  cohaesionis,  quod  ibidem  illustravi 
exemplo  ferrei  globuli  inter  magnetes  disperses  cum  satis  magna  velocitate  libere  permeantis, 
ubi  etiam  illud  vidimus,  in  hoc  casu  virium  ubique  finitarum  impenetrabilitatis  ideam, 
quam  habemus,  nos  debere  soli  mediocritati  nostrarum  velocitatum,  &  virium,  quarum 
ope  non  possumus  imprimere  satis  magnam  velocitatem,  &  libere  trans  murorum  septa,  & 
trans  occlusas  portas  pervadere. 
si  per  asymptoticos  477.  Id  quidem  ita  se  habet,  si  nullas  praeter  primam  asymptoti  habeantur.  quae  vires 

arcus      particulae      i       i  •    r     •  •      i  •  •  f-  }  •    '  •      i        r 

essent  prorsus  im-  absolute  intimtas  mducant  :    nam  si  per  ejusmodi  asymptoticos  arcus  particulse  nant  & 
permeabiies,    tum  indissolubiles,  &  prorsus  impenetrabiles  iuxta  num.  362  :    turn  vero  nulla  utcunque  magna 

recurrendum        ad,.  •      i        i  i  „  •  i  • 

moiem  imminutam  velocitate  posset  una  particula  alteram  transvolare,  &  res  eodem  recideret,  quo  in  communi 
quantum  oportet.  sententia  de  continua  extensione  materiae.  Tum  nimirum  oporteret  lucis  particulas 
minuere,  non  quidem  in  infinitum  (quod  ego  absolute  impossibile  arbitror,  quemadmodum 
&  quantitates,  quas  revera  infinite  parvae  sint  in  se  ipsis  tales,  ac  independenter  ab  omni 
nostro  cogitandi  modo  determinatae  :  nee  vero  earum  usquam  habetur  necessitas  in  Natura) 
sed  ita,  ut  adhuc  incursus  unius  particulae  in  aliam  pro  quovis  finite  tempore  sit,  quantum 
libuerit,  improbabilis,  quod  per  finitas  utique  magnitudines  prsestari  potest.  Si  enim 
concipiatur  planum  per  lucis  particulam  quancunque  ductum,  &  cum  ea  progrediens ; 
eorum  planorum  numerus  dato  quovis  finito  tempore  utcunque  longo  erit  utique  finitus ; 
si  particulas  inter  se  distent  quovis  utcunque  exiguo  intervallo,  quarum  idcirco  finito  quovis 
tempore  non  nisi  finitum  numerum  emittet  massa  utcunque  lucida.  Porro  quodvis  ex 
ejusmodi  planis  ad  medias,  qua  latissimae  sunt,  alias  particulas  luminis  inter  se  distantes 
finito  numero  vicium  appellet  utique  intra  finitum  quodvis  tempus,  cum  id  per  intervalla 
finita  tantummodo  debeat  accidere,  [222]  £  summa  ejusmodi  accessuum  pertinentium 
ad  omnia  plana  particularum  numero  finitarum  finita  erit  itidem,  utcunque  magna. 
Licebit  autem  ita  particularum  diametros  maximas  imminuere,  ut  spatium  plani  ad  datam 
quamvis  distantiam  protensi  circunquaque  etiam  exiguam,  habeat  ad  sectionem  maximam 
particulae  rationem,  quantum  libuerit,  majorem  ilia,  quam  exprimit  ille  ingens,  sed  finitus 
accessuum  numerus  :  ac  idcirco  numerus  directionum,  per  quas  possint  transire  omnia 
ilia  plana  ad  omnes  particulas  pertinentia  sine  incursu  in  ullam  particulam,  erit  numero 
earum,  per  quas  fieri  possit  incursus,  major  in  ratione  ingenti,  quantum  libuerit ;  etiam 
si  cum  ea  lege  progredi  deberent,  ut  altera  non  deberet  transire  in  majore  distantia  ab 
altera,  quam  sit  intervallum  illud  determinans  exiguum  illud  spatium,  ad  quod  assumpta 
est  particularum  sectio  minor  in  ratione,  quantum  libuerit,  magna.  Infinite  nusquam 
opus  erit  in  Natura,  &  series  finitorum,  quae  in  infinitum  progreditur,  semper  aliquod 
finitum  nobis  offert  ita  magnum,  vel  parvum,  ut  ad  physicos  usus  quoscunque  sufficiat. 


Asymptoticis     iis          478.  Quod  de  particulis  inter  se  collatis  est  dictum,  idem  locum  habet  &  in  particulis 


ess"  opus  "ealu™  resPectu  corporum  quoruncunque,  potissimum  si  corpora  juxta  meam  Theoriam  constituta 

tius  exciudenda  :  sint  particulis  distantibus  a  se  invicem,  &  non  continue  nexu  colligatis,  sive  extensionis 

expUcen^ur2  Tin'e  verc  continu£e  ilHus  veli,  aut  muri  continuam  infinitam  objicientis  resistentiam,  de  quo 

ipsis.  egimus  num.   362,  &  363.     Verum    ejusmodi  asymptoticorum   arcuum   nulla    mini    est 

necessitas  in  mea  Theoria,  &  hie  itidem  per  nexus,  ac  vires  limitum  ingentis,  quantum 

libuerit,  quanquam  non  etiam  infiniti  valoris,  omnia  prasstari  possunt  in  Natura  :    &  si 

principio  inductionis  inhasrere  libeat  ;    debemus  potius  arbitrari,  nullos  esse  alios  ejusmodi 

asymptoticos  arcus  in  curva,  quam  Natura  adhibet  :    cum  in  ingenti  intervallo  a  fixis  ad 

particulas  minimas,  quas  intueri  per  microscopia  possumus,  nullus  ejusmodi  nexus  occurrat, 

quod  indicat  motus  continuus  particularum  luminis  per  omnes  ejusmodi  tractus  ;  nisi  forte 

primus  ille  repulsivus,  &  postremus  ejus  naturae  arcus,  ad  gravitatem  pertinens,  indicio  sint, 

esse  &  alios  alibi  in  distantiis,  quae  citra  microscopiorum,  vel  ultra  telescopiorum  potestatem 


A  THEORY  OF  NATURAL  PHILOSOPHY  337 

forces  acting  on  a  moving  point  can  hinder  it.  But  if  no  infinite  force  occurs  corresponding 
to  any  asymptotic  arc  after  the  first,  all  the  forces  are  finite  ;  &  so  also  the  difference  between 
the  forces  acting  in  different  directions  will  be  always  finite.  Therefore,  no  matter  how 
great  the  force  may  be,  there  is  some  finite  velocity  capable  of  overcoming  it,  without 
suffering  any  retardation,  acceleration,  or  deviation  amounting  to  any  given  magnitude, 
no  matter  how  small.  For,  the  forces  require  time  to  produce  a  new  velocity,  this  being 
always  proportional  to  the  force  &  the  time.  Hence,  if  there  were  a  sufficiently  great  velocity, 
any  substance  would  pass  freely  through  any  other  substance,  without  any  sensible  hindrance, 
&  without  any  sensible  change  in  the  situation  of  the  points  belonging  to  either  substance, 
&  without  any  destruction  of  the  mutual  connection  between  the  points,  or  of  cohesion. 
There  also  I  gave  an  illustration  of  an  iron  ball  making  its  way  freely  through  a  group  of 
magnets  with  a  sufficiently  great  velocity ;  &  here  also  we  saw  that  we  owe  what  idea 
we  have  of  impenetrability,  in  the  case  of  forces  that  are  everywhere  finite,  merely  to 
the  moderate  nature  of  our  velocities  &  forces ;  for  by  their  help  alone  we  cannot  impress 
a  sufficiently  great  velocity,  &  freely  pass  through  barrier- walls,  or  shut  doors. 

477.  Now,  this  is  the  case,  so  long  as  there  are  no  asymptotic  arcs  besides  the  first,  n.  owing  to   the 
to  induce  absolutely  infinite  forces ;  but  if,  owing  to  such  asymptotic  arcs,  the  particles  totfce"rcs?fpartSes 
become  incapable  both  of  dissolution  &  penetration,  as  in  Art.  362,  then  indeed  by  no  become   'imperme- 
velocity,  however  great,  could  one  particle  pass  through  another  ;   &  the  matter  would  be  Ihoaid.  ha^e^ofau 
reduced  to  the  same  idea,  as  is  held  generally  about  the  continuous  extension  of  matter,  back  upon  diminu- 
Thus,  in  that  case  it  would  be  necessary  to  diminish  the  size  of  the  particles  of  light ;   not  faraswas'necessary! 
indeed  infinitely — for  I  consider  that  that  would  be  altogether  impossible,  just  as  also  I 

think  that  there  are  no  quantities  infinitely  small  in  themselves,  and  so  determined  without 
reference  to  any  process  of  human  thought ;  nor  is  there  anywhere  in  Nature  any 
necessity  for  such  quantities.  But  they  must  be  so  diminished  that  the  direct  collision 
of  one  particle  with  another  in  any  chosen  finite  time  will  still  be  improbable,  to  any  extent 
desired ;  &  this  can  be  secured  in  every  case  by  finite  magnitudes.  For  suppose  a 
plane  area  circumscribing  each  particle  of  light,  &  that  this  plane  moves  with  the  particle  ; 
then  the  number  of  these  planes  in  any  given  finite  time,  however  long,  will  in  every  case 
be. finite,  so  long  as  the  particles  are  distant  from  one  another  by  any  interval  at  all,  no 
matter  how  small ;  &  thus,  in  any  given  finite  time  the  mass,  however  luminous,  can  only 
emit  a  finite  number  of  these  particles.  Further,  any  one  of  these  planes  will  impinge,  at  their 
broadest  parts,  upon  the  middle  of  other  particles  of  light  distant  from  one  another  by  a 
finite  number  of  fits,  in  every  case  in  a  finite  time  ;  for,  this  can  only  take  place  through 
a  finite  interval.  The  sum  of  such  approaches  pertaining  to  all  the  planes  of  the  particles, 
finite  in  number,  will  also  be  finite,  no  matter  how  great  the  number  may  be.  But 
we  may  so  diminish  the  greatest  diameters  of  the  particles  that  the  area  of  the  plane, 
extended  in  all  directions  round  to  any  given  distance,  however  small,  may  bear  to  the 
greatest  section  of  the  particle  a  ratio  greater,  to  any  arbitrary  extent,  than  that  which 
is  expressed  by  the  huge  but  finite  number  of  the  approaches.  Hence,  the  number  of 
directions,  by  which  all  the  planes  pertaining  to  all  the  particles  may  pass  without  colliding 
with  any  particle,  will  be  greater  than  the  number  of  directions  in  which  there  may  be 
collision,  the  ratio  being  one  that  is  as  immense  as  we  please.  And  this  will  even  be  the  case, 
if  they  should  have  to  move  in  accordance  with  the  law  that  one  must  not  pass  at  a  greater 
distance  from  the  other  than  that  interval  which  determines  the  very  small  space,  to 
which  it  is  supposed  that  the  section  of  the  particle  bears  a  ratio  of  less  inequality,  no  matter 
what  the  magnitude.  There  will  nowhere  be  any  need  of  the  infinite  in  Nature  ;  a  series 
of  finites,  extended  indefinitely,  will  always  give  us  something  finite,  which  is  large  enough 
or  small  enough  to  satisfy  any  physical  needs. 

478.  All  that  has  been  said  with  regard  to  particles  referred  to  one  another,  the  same  There   k  no  ne^ 
will  hold  good  for  particles  in  reference  to  any  bodies ;  &  especially  if  the  bodies  are  formed,  branch^s^^ather, 
in  accordance  with  my  Theory,  of  particles  distant  from  one  another,  &  not  bound  together  they  should  be  ex- 

J         .       J  .  .  .  ?.  eluded ;    how  well 

by  a  continuous  connection,  or  possessing  the  truly  continuous  extension  of  the  skin  or  an  things  can  be 
wall  offering  a  continuous  infinite  resistance,  with  which  we  dealt  in  Art.  362,  363.  But  explained  without 
really  there  is  no  necessity  for  such  asymptotic  arcs  in  my  Theory ;  in  it  also,  by  means  of 
connections  &  forces  of  limits  of  any  value  however  great,  though  not  actually  infinite, 
everything  in  Nature  can  be  accomplished.  If  we  are  to  adhere  to  the  principle  of  induction, 
we  are  bound  rather  to  think  that  there  are  no  other  asymptotic  arcs  in  the  curve  which 
Nature  follows.  For,  in  the  mighty  interval  between  the  stars  &  the  smallest  particles 
that  are  visible  under  the  microscope,  no  connections  of  this  kind  occur,  as  is  indicated 
by  the  continuous  motion  of  the  particles  of  light  throughout  the  whole  of  these  regions. 
Unless,  perhaps,  that  first  repulsive  branch,  &  that  last  arc  of  the  nature  that  pertains 
to  gravity,  are  to  be  taken  as  a  sign  that  there  are  also  somewhere  others  like  them,  at 
distances  which  are  less  than  microscopical,  or  greater  than  those  within  the  range  of  the 
*  z 


338  PHILOSOPHIC  NATURALIS  THEORIA 

contrahuntur,  vel  protenduntur.  Ceterum  si  vires  omnes  finitae  sint,  &  puncta  materiae 
juxta  meam  Theoriam  simplicia  penitus,  &  inextensa  :  multo  sane  facilius  concipitur, 
qui  fiat,  ut  habeatur  haec  apparens  compenetratio  sine  ullo  incursu,  &  sine  ulla  dissolutione 
particularum  cum  transitu  aliarum  per  alias. 

Quomodo  rem  con-          4.79.  Porro  duo  sunt,  quorum  singula  rem  praestare  possunt,  velocitas  satis  magna, 

satis11  ma^na"  &&  1U2E  nimirum  utcunque  magnam  virium  inaequalitatem  potest  eludere,  &  virium  circum- 

aequaiitas  sensibilis  quaque   positarum   aequalitas,   quae   differentiam   relinquat   omnino   nullam.     Differentia 

quaque!"  Quomodo  nunquam    sane    habebitur    omnino    nulla,  ubi  [223]  punctum  materias  praetervolet  per 

haec  in  homogeneo  quandam  punctorum  veluti  silvam,  quorum  alia  ab  aliis  distent  :  necessario  enim  mutabit 

;ur'       distantiam  ab  iis,  a  quibus  minimum  distat,  jam  accedens  nonnihil,  jam  recedens.     Verum 

ubi   distributio   particularum   ad   aequalitatem   quandam    multum   accesserit,    inaequalitas 

virium  erit  perquam  exigua  ;  si  omnium  virium  habeatur  ratio,  quas  exercent  omnia  puncta 

disposita    circa  id   punctum  ad    intervallum,  ad  quod  satis    sensibiles  meae  curvae  vires 

protenduntur.     Concipiamus    enim   sphaeram   quandam,    quae   habeat    pro   semidiametro 

illam  distantiam,  ad  quam  protenduntur  flexus  curvae  virium  primigenise,  sive  ad  quam 

vires  singulorum  punctorum  satis  sensibiles  pertingunt.     Si  medium  satis  ad  homogenei- 

tatem  accedat  ;  secta  ilia  sphaera  in  duas  partes  utcunque  per  centrum,  in  utraque  numerus 

punctorum  materiae  erit  quamproxime  idem,  &  summa  virium  quam  proxime  eadem,  se 

compensantibus  omnibus  exiguis  inaequalitatibus  in  tanta  multitudine,  quod  in  omnibus 

fit  satis  numerosis  fortuitis  combinationibus  :    adeoque  sine  ullo  sensibili  impedimento, 

sine  ingenti  flexione  progredietur  punctum  quodcunque  motu  vel  rectilineo,  vel  tremulo 

quidem  nonnihil,  sed  parum  admodum,  &  ad  sensum  aeque  in  omnem  plagam. 

reioctafs0  exi8uam          4^°'  Qu°d  s^  accedat   ingens   velocitas  ;     multo    adhuc    minor   erit    inaequalitatum 

inzquaiitatem  eiu-  effectus,  turn  quod  multo    minus    habebunt  temporis  vires  ut  agant,  turn  quod  in  ipso 

turbineXeiSneomnon  contmuato  progressu  inasqualitates  jam  in  unam  plagam  prasvalebunt,  jam  in  aliam,  quibus 

cadente.  sibi  mutuo  celerrime  succedentibus,  magis  adhuc  uniformis,  &  rectilineus  erit  progressus. 

Sic  ubi  turbo  ligneus  gyrat  celerrime  circa  verticalem  axem  cuspide  tenuissima  innixum 

solo,  stat  utique,  inaequalitate  ponderis,  quas  ad  casum  determinat,  jam  ad  aliam  plagam 

jacente,  &  totam  inclinante  molem,  jam  ad  aliam,  qui,  celeritate  motus  circularis  imminuta, 

decidit  inclinatus,  quo  exigit  prasponderantia. 


QUOCJ  autem  homogeneitas  medii,  &  velocitas  praestant  simul,  id  adhuc  auget 
quid    is  multo  magis  is  nexus,  qui  est  inter  materiae  puncta  particulam  componentia,  &  aequali  ad 
prastet.  sensum  velocitate    delata,  qui  mutuis  viribus  cum  accessum  ad  se  invicem  punctorum 

particulam  componentium,  &  recessum  impediat,  cogit  totam  particulam  simul  trepidare 
eo  solo  motu,  quern  inducit  summa  inaequalitatum  pertinentium  ad  puncta  omnia,  quae 
summa  adhuc  magis  ad  aequalitatem  accedit  :  nam  in  fortuitis,  &  temere  hac,  iliac  dispersis, 
vel  concurrentibus  casu  circumstantiis,  quo  major  numerus  accipitur,  eo  inaequalitatum 
irregularium  summa  decrescit  magis. 

Rantatem  plun-  ,g2_  Demum  raritas  medii  ad  id  ipsum  confert  adhuc  magis  :    quo  enim  major  est 

mum       prodesse  :          .    '  .  ....  ^  J     . 

omnes  eas  quatuor  raritas,  eo  minor  occurnt  punctorum  numerus  mtra  illam  sphaeram,  adeoque  eo  minor 
causas  habere  v}rium  componendarum  multitudo,  &   inaequalitas    adhuc    multo    mi-r224l-nor.      Porro 

locum     in     lumme  .  v       •  ,  .         .  ,..  L      7i     .  v 

non  turbato  a  omnes  hae  quatuor  causae  aequalitatis  concurrunt,  ubi  agitur  de  radus  collatis  cum  alns 
radns  alia  du-ectione  rac|iis  :  homogeneitas,  nam  lumen  a  dato  puncto  progrediens  suam  densitatem  imminuit 
sum:  priores  tres  in  ratione  reciproca  duplicata  distantiarum  a  puncto  radiante,  adeoque  in  tarn  exiguo 
'  circuncluaclue  circa,  quodvis  punctum  intervallo,  quantum  est  id,  ad  quod  virium  actio 
sensibilis  protenditur,  ad  homogeneitatem  accedit  in  immensum  :  celeritas,  quae  tanta 
est,  ut  singulis  arteriae  pulsibus  quaevis  luminis  particula  fere  bis  centum  millia  Romanorum 
milliariorum  percurrat  :  nexus  particularum  mutuus,  nam  ipsae  luminis  particulae  ad 
diversos  coloratos  radios  pertinentes  habent  perennes  proprietates  suas,  quas  constanter 
servant,  ut  certum  refrangibilitatis  gradum,  &  potentiam  certo  impulsu  agitandi  oculorum 
fibras,  per  quam  certain  certi  coloris  sensationem  eliciant  :  ac  demum  tenuitas  immanis, 
qua  opus  est  ad  tantam  diffusionem,  &  tarn  perennem  efHuxum  sine  ulla  sensibili  imminu- 
tione  Solaris  massas,  &  cujus  indicium  aliquod  proferam  paullo  inferius.  Ubi  vero  agitur 
de  lumine  comparato  cum  substantiis  pellucidis,  per  quas  pervadit,  priora  ilia  tria 
tantummodo  locum  habent  respectu  particularum  luminis,  &  omnia  quatuor  respectu 
particularum  pellucidi  corporis,  quarum  nexus  non  dissolvitur,  nee  positio  turbatur 
quidquam  ab  intervolantibus  radiorum  particulis.  Quamobrem  errat  qui  putat,  mea 


A  THEORY  OF  NATURAL  PHILOSOPHY  339 

telescope.  Besides,  if  all  the  forces  are  finite,  and  points  of  matter,  in  accordance  with,  my 
Theory,  are  perfectly  simple  &  non-extended,  it  is  far  more  easily  understood  why  there 
can  be  this  apparent  compenetration,  without  any  collision,  &  without  any  dissolution  of 
the  particles  as  they  pass  through  one  another. 

479.  Further,  there  are  two  things,  each  of  which  can  accomplish  the  matter  ;  namely,  How  the   matter 
a  sufficiently  great  velocity,  such  as   will  foil   the  inequality  of   the  forces,  however  great  uti&ed  b  ^suffi- 
that  may  be ;  &  an  equality  of  the  forces  in  all  directions,  such  as  will  leave  the  difference  ciently  large  veio- 
absolutely  zero.     Now  the  difference  can  never  really  be  altogether  zero,  when  a  point  Clty  ^-ta  ^"l1?6 
of  matter  passes  through,  so  to  speak,  a  forest  of  points,  which  are  separated  from  one  forces  in  all  direc- 
another.     For,  of  necessity,  it  will  change  its  distance  from  those  points,  from  which  it  is  ti°ns-    ?°w.  ^^ 
least  distant,  at  one  time  approaching  &  at  another  receding.     But  when  the  distribution  a  homogeneous 
of  the  particles  approaches  very  closely  to  an  equality,  the  inequality  of  the  forces  will  be  medium- 
exceedingly  small,  so  long  as  account  is  taken  of  all  the  forces  exerted  by  all  the  points 

situated  about  that  point  at  an  interval  equal  to  that  over  which  the  forces  of  my  curve 
extend  while  still  fairly  sensible.  For,  imagine  a  sphere,  that  has  for  its  semi-diameter 
the  distance  over  which  the  windings  of  the  primary  curve  extend,  that  is,  the  distance  up 
to  which  the  forces  of  each  of  the  points  are  fairly  sensible.  If  the  medium  approximates 
sufficiently  closely  to  homogeneity,  &  the  sphere  is  divided  into  any  two  parts  by  a  plane 
through  the  centre,  the  number  of  points  of  matter  in  each  part  will  be  nearly  the  same  ; 
&  the  sum  of  the  forces  will  be  very  approximately  the  same,  as  the  slight  differences  taken 
as  a  whole  compensate  one  another  in  so  great  a  multitude ;  for  this  is  always  the  case  in 
sufficiently  numerous  fortuitous  combinations.  Thus,  without  any  impediment,  without 
any  very  great  flexure,  any  point  will  proceed  with  a  motion  that  is  rectilinear,  or  maybe 
somewhat  but  very  slightly  wavy,  &  practically  uniformly  so  in  every  direction. 

480.  But  if  the  velocity  is  very  great,  the  effect  of  inequalities  will  be  still  less ;   both  How  a  very  great 
because  the  forces  will  have  much  less  time  in  which  to  act,  &  because  in  such  a  continued  slight  y  inequality* 
progress  the  inequalities  will  prevail  first  on  one  side  &  then  on  the  other  ;  &  as  these  follow  example,    from    a 
one  another  very  quickly,  the  progress  will  be  still  more  uniform  &  nearer  to  rectilinear  ^p  not  faUhig.1118 
motion.     Thus,  when  a  wooden  spinning-top  spins  very  quickly  about  a  vertical  axis  with 

a  very  fine  point  resting  on  the  ground,  it  stays  perfectly  upright ;  for,  the  inequality  of 
its  weight,  which  disposes  it  to  fall,  lies  first  on  one  side  &  inclines  the  whole  mass  that  way, 
&  then  on  the  other  side  ;  while,  as  soon  as  the  circular  motion  decreases,  it  becomes  inclined 
to  the  side  to  which  the  preponderance  forces  it. 

481.  Again  the  effect  produced  by  the  homogeneity  of  the  medium  &  the  great  velocity  ?n  addition  there 
together  is  still  further  increased  by  the  connection  that  exists  between  the  points  of  matter  between  the  point" 
forming;  the  particle  &  moving  together  with  practically  the  same  velocity.     This  connection,  °*  a  particle ;  the 

1  ,  •  i  i  i  •  r    i  •          effect  of  this. 

since,  through  the  mutual  forces,  it  prevents  the  mutual  approach  or  recession  ot  the  points 
forming  the  particle,  will  force  the  entire  particle  to  move  as  a-whole  with  the  single  motion 
that  is  induced  by  the  sum  of  the  inequalities  pertaining  to  all  its  points ;  &  this  sum  will 
still  further  approximate  to  equality.  For,  in  circumstances  that  are  fortuitous,  distributed 
here  &  there  at  random,  or  concurring  by  chance,  the  greater  the  number  taken,  the 
more  the  sum  of  the  irregular  inequalities  decreases. 

482.  Lastly,  rarity  of  the  medium  is  of  still  further  assistance  ;    for,  the  greater  the  Great  effect  of 
rarity,  the  smaller  the  number  of  points  that  occur  within  the  sphere  imagined  above,  aif'four  of^these 
&  therefore  the  smaller  the  number  of  forces  to  be  compounded,  &  much  smaller  still  the  causes    hold    for 
inequality.     Further,  all  four  of  these  causes  of  inequality  occur  together,  when  we  are  bj^rays^roceedSg 
dealing  with  rays  of  light  in  regard  to  other  rays.     Homogeneity  we  have,  because  light  in  any  other  direc- 
proceeding  from  a  given  point  diminishes  its  density  in  the  inverse  ratio  of  the  squares  of  threeoAhemin^he 
the  distances  from  the  radiant  point ;  &  thus,  in  the  exceedingly  small  interval  round  about  more  dense  trans- 
any  point,  whatever  the  distance  may  be  over  which  a  sensible  action  of  the  forces  extend,  P31611* media- 

the  approach  to  homogeneity  is  exceedingly  great.  Velocity  also  we  have,  so  great  that 
in  a  single  beat  of  the  pulse  a  particle  of  light  travels  a  distance  of  nearly  two  hundred 
thousand  Roman  miles.  Mutual  connection  of  the  particles  also,  for  the  particles  of  light 
pertaining  to  differently  coloured  rays  have  all  their  special  lasting  properties,  which  they 
keep  to  unaltered,  such  as  a  definite  refrangibility  &  the  power  of  affecting  the  nerves  of 
the  eye  with  a  definite  impulse,  through  which  they  give  it  a  definite  sensation  of  a  definite 
colour.  Lastly,  an  extremely  great  tenuity,  such  as  is  necessitated  by  the  greatness  of 
the  diffusion  &  the  endurance  of  the  efflux  without  sensible  diminution  of  mass  in  the  case 
of  the  Sun  ;  &  of  this  I  will  bring  forward  some  evidence  a  little  further  on.  But  when 
we  are  dealing  with  light  in  regard  to  transparent  substances,  through  which  the  light 
passes,  the  first  three  only  hold  good  with  regard  to  the  particles  of  light,  but  all  four  with 
regard  to  the  particles  of  the  transparent  body ;  the  connections  between  the  particles  of 
the  body  are  not  broken,  nor  is  their  relative  position  affected  to  any  extent  by  the  particles 
of  the  rays  of  light  passing  through  them.  Therefore  he  will  be  mistaken,  who  thinks 


340  PHILOSOPHIC  NATURALIS  THEORIA 

indivisibilia  puncta  prasdita  insuperabili  potentia  repulsiva  pertlngente  ad  finitam  distantiam 
esse  tarn  subjecta  collisionibus,  quam  sunt  particulae  finitae  magnitudinis,  &  idcirco  nulli 
adminiculo  esse  pro  comprehendenda  mutua  lucis  penetratione  ;  nam  sine  cruribus  illis 
asymptoticis  posterioribus  meae  vires  repulsivae  non  sunt  insuperabiles,  nisi  ubi  puncta 
congredi  debeant  in  recta,  quae  ilia  jungit,  qui  casus  in  Natura  nusquam  occurrit. 

Pelluciditatem  oriri          483.  Et  vero  sola  homogeneitas  pelluciditatem  parit,  uti  jam  olim  notavit  Newtonus, 

a    sola  homogenei-  •  ...  .  vioij- 

tate:soiam  hetero-  nec  opacitas  ontur  ab  impactu  in  partes  corporum  solidas,  &  a  defectu  pororum  jacentmm 
geneitatem     impe-  in  directum,  uti  alii  ante  ipsum  plures  censuerant,  sed  ab  inaequali  textu  particularum 

dire  posse  progres-  i  £         «.F        .  ,        .  ,        .  *.  .    r          ,.      .. 

sum  per  insquaii-  neterogenearum,  quarum  alise  alns  minus  densis,  vel  etiam  pemtus  vacuis  amphoribus 
tatem  virium.  spatiolis  intermixtae  satis  magnam  inducunt  inaequalitatem  virium,  qua  lumen  in  omnes 
partes  detorquent,  ac  distrahunt,  flexu  multiplici,  &  ambagibus  per  internes  meatus 
continuis,  quibus  fit,  ut  si  paullo  crassior  occurrat  massa  corporis  ex  heterogeneis  particulis 
coalescentis,  nullus  radius  rectilineo  motu  totam  pervadat  massam  ipsam,  quod  nimirum 
ad  pelluciditatem  requiritur.  Indicia  rei  habemus  quamplurima  prseter  ipsam  omnem 
superiorem  Theoriam,  quae  rem  sola  evinceret ;  cum  nimirum  sine  inasqualitate  virium 
nullum  haberi  possit  libero  rectilineo  progressui  impedimentum.  Id  sane  colligitur  ex 
eo,  quod  omnium  corporum  tenuiores  laminae  pellucidae  sunt,  uti  norunt,  qui  microscopiis 
tractandis  assueverunt  :  id  [225]  evincunt  illae  substantiae,  quae  aliarum  poris  injectae 
easdem  ex  opacis  pellucidas  reddunt,  ut  charta  oleo  imbuta  fit  pellucida,  supplente  aerem 
ipso  oleo,  cum  quo  multo  minus  inaequaliter  in  lumen  agunt  particulae  chartae,  quam 
agerent  soli  aeri,  vel  vacuo  spatio  intermixtae.  Rem  autem  oculis  subjicit  vitrum  contusum 
in  minores  particulas,  quod  sola  irregularitate  figuras  particularum  temere  ex  contusione 
nascentium,  &  aeris  intermixti  inaequalitate  fit  opacum  per  multiplicationem  reflexionum, 
&  refractionum  irregularium  :  nec  aliam  ob  causam  aqua  in  glaciem  bullis  continuis 
interruptam  abiens  pelluciditatem  amittit,  ut  &  alia  corpora  sane  multa,  quae,  dura 
concrescunt  vacuolis  interrupta,  illico  opaca  fiunt. 


Reflexionem    non  .$*    Quamobrem  nec  reflexio    inde  ortum  ducit,  sed  habetur  etiam  in  pellucidis 

onri    ab    impactu,  '  .7  ....  ,,  ...      ,-J    . 

sed  ab  inzequaiitate  corponbus  ex  inaequalitate  virium  seu  repellentium,  seu  attrahentium,  uti  in  Optica  sua 
vinum  m  mutatione  Newtonus  tarn  multis  notissimis  argumentis  demonstravit,  quorum  unum  est  illud  ipsum 

medu ;   ubi  pro  re-  .  -    .    .  .  .  i  .  .     .r 

fractions  expiica-  ex  aspentate  supernciei  cujuscunque  cujusvis  corporis,  utcunque  nobis,  nudo  potissimum 
pramissa  inspectantibus  oculo,  laevis  appareat,  &  perpolita,  quod  num.  299  exposuimus ;  &  ex  eadem 
causa  oritur  etiam  refractio.  Si  velocitas  luminis  esset  satis  magna  ;  impediret  etiam 
hujusce  inaequalitatis  effectum,  qui  provenit  a  diversa  mediorum  constitutione  :  sed  ex 
ipsis  reflexionibus,  &  refractionibus  in  mutatione  medii,  conjunctis  cum  propagatione 
rectilinea  per  medium  homogeneum,  patet,  celeritatem  illam  tantam  luminis  satis  esse 
magnam  ad  eludendam  illam  inaequalitatem  tanto  minorem,  quae  habetur  in  mediis  homo- 
geneis,  non  illam  tanto  majorem,  quae  oritur  a  mediorum  discrimine.  Quod  vero  ad 
refractionis  explicationem  ex  Mechanica  requiritur,  exposuimus  a  num.  302,  ubi  adhibuimus 
principium  illud  virium  inter  duo  plana  parallela  agentium  aeque  in  distantiis  aequalibus 
ab  eorum  utroque,  cujus  explicationem  ad  luminis  particulas  jam  expediemus. 

ad  "uam          485-  Concipiatur  (/)  ilia  sphaerula,  cujus    semidiameter^  [226]  aequatur  distantiae  illi, 
extenditur  vis  ad  quam  agunt  actione  satis  sensibili  particulae  corporum  in  lucis  particulam,  quae  cum 

sensibilis   agens    in         _~ ^ _____ 

lumen :      inde     vis 

inter    bina     plana  (f)  Refert  MN  in  fig.  70  superficiem  dirimentem  duo  media,  GE  viam  radii  advenientis,  H  particulam  luminis  ; 

parallela  superficiei  jjj?  celeritatem,  efus  absolutam,  HS  parallelam,  SE  perpendicularem,  quis  est  eo  minor,  quo  radius  incidit  magis  obliquus  : 

interfuse  vis^ari?  '  abc  est  sph<era,  intra  quam  habetur  actio  sensibilis  in  particulam  H,  quie  est  adhuc  tota  in  priore  media  :  X,  X',  X"  sunt 
loca  plura  particulis  progredientis  inter  plana  AB,  CD  parallela  superficiei  MN,  sita  ad  distantiam  ab  ea  aqualem 
semidiametro  sphtera  He.  Particula  sita  inter  ilia  plana  ubicunque,  ut  in  X,  ea  sphtera  habebit  suum  segmentum  FRL 
ultra  superficiem  MN  :  sit  efus  axis  RT,  ^  eodem  axe  segmentum  QTZ  priori  tequale,  ac  mn  planum  per  centrum  par- 
allelum  MN.  Segmenta  mFLn,  mQZn  ejusdem  medii  agent  izqualiter.  Segmenta  FRL,  QTZ  incequaliter,  sed  eorum 
vires  dirigentur  per  axem  TR  in  alteram  e  binis  plagis  oppositis  :  adeoque  i3  differentia  virium  dirigetur  per  eundem,  qui 
quidem  perpendicularis  est  utique  plants  AB,  CD.  Ea  actione  viaiticurva  radii  sinuatur  per  XX'X".  Prout  vis  dirigetur 
versus  CD,  vel  versus  AB,  curva  erit  cava  versus  easdem,  y  in  mutatione  directionis  vis  ipsius  mutabitur  ftexus  curvtr. 
Si  autem  curva  evaserit  alicubi  parallela  piano  AB  ;  flectet  cursum  retro  ;  nisi  id  accidat  accurate  in  situ  vis  =  o,  qui 


A  THEORY  OF  NATURAL  PHILOSOPHY 


34' 


FIG.  70. 


342 


PHILOSOPHISE   NATURALIS  THEORIA 


FIG.  70. 


A  THEORY  OF  NATURAL  PHILOSOPHY  343 

that  my  indivisible  points,  endowed  with  an  insuperable  repulsive  force  extending  to  a 
finite  distance,  are  just  as  subject  to  collisions  as  particles  of  finite  magnitude  ;  &  therefore 
that  there  is  no  assistance  to  be  derived  from  them  in  understanding  the  mutual  penetration 
of  light  ;  for,  unless  there  are  those  asymptotic  branches  after  the  first,  my  repulsive  forces 
are  not  insuperable,  except  when  points  are  bound  to  move  together  in  one  straight  line 
joining  them,  a  circumstance  which  never  occurs  in  Nature. 

483.  Indeed  homogeneity  by  itself  creates  transparence,  as  was  long  ago    stated  by 

Newton  ;   &  opacity  does  not  arise  from  impact  with  the  solid  parts  of  bodies,  or  through  alone  ;    &    hetero- 

a  lack  of  pores  lying  in  a  straight  line,  as  many  others  before  Newton  thought,  but  from  geneity    al°ne    1S 

the  unequal  structure  of  heterogeneous  particles  ;   of  which  some  are  interspersed  amongst  venting  progressive 

others    of    less    density,  or    even    in   perfectly    empty  little    spaces,  of  considerable  size,  motion  through  m- 

and  thus  induce  an  inequality  great  enough  to  distort  the  light  in  all  directions,  &  to 

harass  it  with  manifold  windings  &  continuous    meandering  through  internal  channels  ; 

from  which  it  comes   about  that,  if  a  somewhat  thick   mass   occurs  of  a  body  formed 

from    heterogeneous    particles,   no    ray   with  rectilinear    motion    will  pass    through  the 

whole  of  that  mass  ;    which  is  the  requirement  for  transparence.     We  have  very  many 

pieces  of  evidence  on  the  subject,  in  addition  to  the  whole  of  the  Theory  given  above, 

which  of  itself  is  sufficient  to  prove  it.     For,  indeed,  without  inequality  of  forces  there 

can  be  no  impediment  to  free  rectilinear  progressive  motion.     This  can  truly  be  deduced 

from  the  fact  that  fairly  thin  plates  of  all  bodies  are  transparent,  as  is  known  to  those  who 

have  been  accustomed  to  microscopical  work.     Evidence  is  also  afforded  by  such  substances 

as,  on  injection  into  the  pores  of  other  substances,  turn  the  latter  from  opaque  to  transparent  ; 

thus,  paper  soaked  with  oil  becomes  transparent,  the  oil  taking  the  place  of  the  air  ;    for, 

with  it  the  particles  of  paper  act  far  less  unequally  upon  the  light  than  they  would  act, 

if  merely  air,  or  an  empty  space  were  interspersed.     Moreover,  glass  broken  up  into  fine 

particles  brings  the  matter  right  before  our  eyes  ;    for,  from  the  mere  irregularity  of  the 

shape  of  the  particles   randomly  produced   by  the  powdering,   &  the  inequality  of  the 

interspersed  air,  it  becomes  opaque  on  account  of  the  multiplication  of    reflections  & 

refractions  occurring  irregularly.   From  no  other  cause  does  water,  turning  into  ice  interrupted 

by  continuous  bubbles,  lose  its  transparence  ;    it  is  just  the  same  also  with  many  other 

bodies,  which,  as  they  grow,  are  interspersed  with  little  empty  spaces,  &  from  this  cause 

alone  become  opaque. 

484.  Therefore  also  reflection  does  not  arise  from  impact  ;    but  it  is  even  found  in  Reflection  does  not 

,j.j  i       .  ......  ,.  .  ,-,.,.  .      take  place  through 

transparent  bodies  due  to  the  inequality  of  forces,  whether  repulsive  or  attractive.     This  impact,  but  owing 
was  proved  by  Newton  in  his  Optics  by  a  large  number  of  arguments  that  are  well  known  ;  *°    inequality   of 

r  f     ,          '.       ,  i        '  °       i    •        A  i      •        i    f  i  forces    on    the 

one  of  these  is  that  very  reason  that  was  stated  in  Art.  299,  derived  from  the  roughness  medium  being 
of  any  surface  of  any  body,  no  matter  how  smooth  &  polished  it  appears  to  us,  especially  changed  :  where 

i       •     •          »       •  i     *i  *i     j  T*    r         •  •          f  -rr      t       the     principles  for 

when  viewed  with  the  naked  eye.     Refraction  also  arises  from  the  same  cause.     If   the  the  explanation  of 
velocity  of  light  were  great  enough,  it  would  prevent  even  the  effect  of  this  inequality  refraction    have 

J.         f    '         i        T/r  °    '   .  •    i    *        11          T>         r  i       r  i  been  premised. 

that  arises  from  the  different  constitution  of  the  media,  .out,  from  the  fact  that  there  are 
these  reflections  &  refractions  on  a  change  of  medium,  taken  in  conjunction  with  the  fact 
of  rectilinear  propagation  through  a  homogeneous  medium,  it  is  clear  that  the  great  velocity 
of  light  is  enough  to  foil  the  comparatively  small  inequality  that  is  found  in  homogeneous 
media,  but  is  not  enough  for  the  comparatively  greater  inequality  that  arises  from  a  difference 
in  the  media  traversed.  But  that  which  is  necessary  for  the  mechanical  explanation  of 
refraction  has  been  stated  in  Art.  302  onwards  ;  where  we  employed  the  idea  of  forces 
acting  between  two  parallel  planes,  the  forces  being  equal  for  equal  distances  from  either 
of  the  planes  ;  we  will  now  apply  this  idea  to  particles  of  light. 

48;.  Imagine  (/)  a  sphere,  of  which  the  semidiameter  is  equal  to  the  distance  up  to  Consideration      of 

i'ii  •   i  r        i       i  •   i         /•  i-    i  •  i     -1    r  •  i  -11  •  »     the   sphere     whose 

which  the  particles  01  a  body  act  upon  a  particle  or  light  with  a  fairly  sensible  action  ;  &  radius  is  the  dis- 

_   tance  to  which  the 

(f)  In  Fig.  70,  MN  is  the  surface  of  separation  between  the  two  media,  GE  the  path  of  an  approaching  ray,  H  a  particle  light    extends- 

of  light,  HE  its  absolute  velocity,  HS  the  parallel,  SE  the  perpendicular  component,  which  latter  is  the  less,  the  more  thence    the    force 

oblique  the  incidence  of  the  ray.     abc  is  the  small  sphere,  within  which  there  is  sensible  action  on  the  particle  H,  which  between  two  planes 

it  as  yet  altogether  in  the  first  medium.     X,X',X"  are  positions  of  the  particle  as  it  passes  between  the  planes  AB,  CD,  parallel  to    the 


parallel  to  the  surface  MN,  and  situated  at  a  distance  from  it  equal  to  the  semidiameter  of  the  sphere  He.     //  the  particle  tio^of  the  media, 

is  situated  anywhere  between  the  two  planes,  as  at  X,  the  sphere  will  have  its  segment  FRL  on  the  far  side  of  the  surface  between  which  the 

MN.    Let  the  axis  of  the  segment  be  RT,  and  let  QTZ  be  a  segment  having  the  same  axis  and  equal  to  the  former  force  acts. 

segment,  and  let  mn  be  a  plane  through  the  centre  parallel  to  MN.     Then  the  segments  mFLn,  mQZn,  lying  in  the 

same  medium,  will  act  equally  ;    but  the  segments  FRL,  QTZ  will  act  unequally  ;    yet  their  forces  will  be  directed 

along  the  axis  TR  in  one  or  other  of  the  two  opposite  directions,  and  thus  also  the  difference  between  these  forces  will  act 

along  the  same  straight  line,  which  is  perpendicular  to  the  planes  AB,  CD  in  every  case.     Owing  to  this  action  the 

curved  path  of  the  ray  will  wind  along  through  X,X',X".     According  as  the  force  is  directed  towards  CD  or  towards 

AB,  the  curve  will  be  concave  with  respect  to  these  same  planes,  and  when  the  force  changes  its  direction  the  flexure  of 

the  curve  will  also  change.     Moreover,  if  the  curve  should  anywhere  happen  to  become  parallel  to  the  plane  AB,  the 

path  will  be  reflected  ;  unless  it  should  fall  out  that  exactly  in  that  position  the  force  was  zero,  a  case  that  is  infinitely 


344  PHILOSOPHIC  NATURALIS  THEORIA 

lucis  particula  progrediatur  simnl.  Donee  ipsa  sphaerula  est  in  aliquo  homogenco  medio 
tota,.  vires  in  particulam  circunquaque  sequales  erunt  ad  sensum,  &  cum  nullus  habeatur 
immediatus  incursus,  motus  inertiae  vi  factus  erit  ad  sensum  rectilineus,  &  uniformis. 
Ubi  ilia  sphaerula  aliquod  aliud  ingressa  fuerit  diversae  naturae  medium,  cujus  eadem  moles 
exerceat  in  particulas  luminis  vim  diversam  a  prioris  medii  vi ;  jam  ilia  pars  novi  medii, 
quae  intra  sphserulam  immersa  erit,  non  exercebit  in  ipsam  particulam  vim  aequalem  illi, 
quam  exeret  pars  sphaerulae  ipsi  rcspondens  ex  altera  centri-parte,  &  facile  patet,  differentiam 
virium  debere  dirigi  per  axem  perpendicularem  illis  segmentis  sphaarulae,  per  quern  singulae 
utriusque  segmenti  vires  diriguntur,  nimirum  perpendiculariter  ad  superficiem  dirimentem 
duo  media,  quae  illud  prius  segmentum  terminal  :  &  quoniam  ubicunque  particula  sit  in 
aequali  distantia  a  superficie,  illud  segmentum  erit  magnitudinis  ejusdem  ;  vis  motum 
perturbans  in  iisdem  a  superficie  ilia  distantiis  eadem  erit.  Durabit  autem  ejusmodi  vis, 
donee  ipsa  sphserula  tota  intra  novum  medium  immergatur.  Incipiet  autem  immergi 
ipsa  sphaerula  in  novum  medium,  ubi  particula  advenerit  ad  distantiam  ab  ipsius  superficie 
sequalem  radio  sphasrulas,  &  immergetur  tota,  ubi  ipsa  particula  jam  immersa  fuerit,  ac 
ad  distantiam  eandem  processerit.  Quare  si  concipiantur  duo  plana  parallela  ipsi  superficiei 
dirimenti  media,  quae  superficies  in  exiguo  tractu  habetur  pro  plana,  ad  distantias  citra, 
&  ultra  ipsam  sequales  radio  illius  sphaerulae,  sive  intervallo  actionis  sensibilis ;  particula 
constituta  inter  ilia  plana  habebit  vim  secundum  directionem  perpendicularem  ipsis  planis, 
quae  in  data  distantia  ab  eorum  altero  utrovis  aequalis  erit. 


Tres  casus,  qui  ex-  486.  Porro  id  ipsum  est  id,  quod  assumpsimus  num.  302,  &  unde  derivavimus  reflexionis, 

vel  refractio°neIm  ac  refractionis  legem  :    nimirum  si  concipiatur  ejusmodi  vis  resoluta  in    duas,  alteram 

cum  recessu  a  per-  parallelam  iis  planis,  alteram  perpendicularem  :    ilia    vis    pot-[227]-est    perpendicularem 

sam  1Crefractionem  velocitatem  vel  extinguere  totam  ante,  quam  deveniatur  ad  planum  ulterius,  vel  imminuere, 

cum  accessu.  vel  augere.     In  primo  casu  debet  particula  retro  regredi,  &  describere  curvam  similem  illi, 

quam  descripsit  usque  ad  ejusmodi  extinctionem,  recuperando  iisdem  viribus  in  regressu, 

quod  amiserat  in  progressu,  adeoque  debet  egredi  in  angulo  reflexionis  aequali  angulo 

incidentiae  :    in    secundo  casu  habetur  refractio    cum  recessu  a  perpendiculo,  in    tertio 

refractio  cum  accessu  ad  ipsum,  &  in  utrolibet  casu,  quaecunque  fuerit  inclinatio  in  ingressu, 

debet   differentia   quadratorum   velocitatis   perpendicularis   in    ingressu,   &    egressu   esse 

constantis  cujusdam  magnitudinis  ex  principio  mechanico  demonstrato  num.   176  in  adn. 

&  inde  num.  305  est  erutum  illud,  sinum  anguli  incidentiae  ad  sinum  anguli  refracti  debere 

esse  in  constanti  ratione,  quae  est  celeberrima  lucis  proprietas,  cui  tota  innititur  Dioptrica 

&  prasterea  illud  num.  306  velocitatem  in  medio  prascedente  ad  velocitatem  in  medio 

sequente  esse  in  ratione  reciproca  sinuum  eorundem. 


Lumen    debere  in  487.  Hoc  pacto  ex  uniformi    Theoria   deductae    sunt  notissimae,  ac  vulgares  leges 

corpora  reagere  reflexionis,  ac  refractionis,  ex  quibus  plura  consectaria  deduci possunt.     Imprimis  quoniam 

sequahter   :     hinc     ,   ,  .'  *     ,  ,  r.  a 

immensa  lucis  ten-  debet  actio  semper  esse  mutua,  dum  corpora  agunt  in  lumen  ipsum  renectenuo,  oc  reirm- 
uitas :  qui  effectus  gendo  ;    debet  ipsum  lumen  agere  in  corpora,  ac  debet  esse  velocitas  amissa  a  lumine  ad 

ipsi  f also  tribuantur  °  .      .  '  r.  .  .r  .     , 

a  nonnuiiis.  velocitatem  acquisitam  a  centre  gravitatis  corporis  sistentis  lumen,  ut  est  massa  corporis 

ad  massam  luminis.  Inde  deducitur  immensa  luminis  tenuitas  :  nam  massa  tenuissima 
levissimae  plumulae  suspensae  filo  tenui,  si  impetatur  a  radio  repente  immisso,  nullum 
progressivum  acquirit  motum,  qui  sensu  percipi  possit.  Cum  tarn  immanis  sit  velocitas 
amissa  a  lumine ;  facile  patet,  quam  immensa  sit  tenuitas  luminis.  Newtonus  etiam 
radiorum  impulsioni  tribuit  progressum  vaporum  cometicorum  in  caudam ;  sed  earn 
ego  sententiam  satis  valido,  ut  arbitror,  argumento  rejeci  in  mea  dissertatione  De  Cometis. 
Sunt,  qui  auroras  boreales  tribuant  halitibus  tenuissimis  impulsis  a  radiis  solaribus,  quod 
miror  fieri  etiam  ab  aliquo,  qui  radios  putat  esse  undas  tantummodo,  nam  undae  progressivum 


Cams  est  in  infinitum  improbabilis.  Id  accidet  in  aliis  radiis  citius,  in  aliis  radiis  serins,  •pro  diversa  absoluta  celeritate 
radii,  pro  diversa  indinatione  incidentitz,  W  pro  diversa  natura,  vel  constitutione  particula,  abeuntibus  aliis  particulis 
per  QXIK,  aliis  per  QXX'I'K',  aliis  per  QXX'X'T'K".  Porro  perquam  exiguum  discrimen  in  vi,  vel  celeritate,  potest 
curvam  uno  aliquo  in  loco  a  positione  proxima  parallelismo  ad  ipsum  parallelismum  traducere,  quo  loco  superato  adhuc  summa 
actionum  usque  ad  O  potest  esse  ad  sensum  eadem.  Reliqua  sunt  Me,  ut  num.  306. 


ex- 


A  THEORY  OF  NATURAL  PHILOSOPHY  345 

suppose  that  this  sphere  moves  along  with  the  light  particle.  So  long  as  the  little  sphere 
is  altogether  In  a  homogeneous  medium,  the  forces  on  the  particle  all  round  it  are  practically 
equal ;  &,  since  no  immediate  impact  can  take  place,  the  motion  will  be  kept  practically 
rectilinear  &  uniform  by  the  force  of  inertia.  When  the  little  sphere  enters  some  other 
medium  of  a  different  nature,  the  same  volume  of  which  exerts  on  the  particles  of  light 
a  force  different  from  the  force  due  to  the  first  medium,  then,  that  part  of  the  new  medium 
which  is  intercepted  within  the  little  sphere  will  not  exert  on  the  particle  a  force  equal 
to  that  which  the  corresponding  part  on  the  other  side  of  the  centre  exerts ;  &  it  is  easily 
seen  that  the  difference  of  the  forces  must  be  directed  along  the  axis  perpendicular  to  these 
segments  of  the  sphere,  for  the  forces  due  to  each  segment  separately  are  so  directed  ;  that 
is  to  say,  perpendicular  to  the  surface  of  separation  between  the  two  media,  which  is  the 
bounding  surface  of  the  first  of  the  two  segments.  Now,  since  that  segment  will  be  of 
the  same  magnitude  whenever  the  distance  of  the  particle  from  the  surface  of  separation 
is  the  same,  the  force  determining  the  change  of  motion  will  be  the  same  at  equal  distances 
from  that  surface.  Further,  such  force  will  continue  unchanged  so  long  as  the  little  sphere 
is  altogether  immersed  in  the  new  medium.  Now,  the  little  sphere  will  commence  to  be 
immersed  in  the  new  medium  as  soon  as  the  particle  reaches  a  distance  from  the  surface 
of  separation  equal  to  the  radius  of  the  little  sphere  ;  &  it  will  become  altogether  immersed 
in  it  as  soon  as  the  particle  itself,  after  entering  it,  has  gone  forward  a  further  distance 
equal  to  the  radius.  Hence,  if  two  planes  are  imagined  to  be  drawn  parallel  to  the  surface 
of  separation  of  the  media,  &  this  surface  is  supposed  to  be  plane,  for  the  very  small  region 
extending  on  every  side  to  a  distance  equal  to  the  radius  of  the  little  sphere,  or  the  interval 
corresponding  to  sensible  action  ;  then,  a  particle  situated  between  those  planes  will  be 
under  the  influence  of  a  force  in  the  direction  perpendicular  to  the  planes,  which  will  be 
the  same  for  equal  distances  from  either  of  them. 

486.  Now,  this  reduces  to  that  very  same  supposition  that  we  made  in  Art.  302,  from  J.1?1??    cases< 
which  we  derived  the  laws  of  reflection  &  refraction.     Thus,  if  such  a  force  is  supposed  iy  reflection,  refrac- 
to  be  resolved  into  two  parts,  one  parallel  &  the  other  perpendicular  to  the  planes,  the  *lon  with  recession 

•  i          i  t          i     i        r     ^  T      i  i      •        i     r  i       r       i         from    the    normal, 

latter  force  may  either  destroy  the  whole  ot  the  perpendicular  velocity  before  the  further  &  refraction  with 

plane  is  reached,  or  it  may  reduce  it,  or  it  may  increase  it.     In  the  first  case  the  particle  approach  to    the 

must  turn  back  in  its  path  &  describe  a  curve  similar  to  that  which  it  has  already  described 

up  to  the  point  at  which  its  perpendicular  velocity  was  described  ;   &  on  its  return  it  will 

recover  the  velocity  it  lost  during  its  advance,  with  the  same  forces  ;  &  thus,  it  must  leave 

the  second  medium  with  an  angle  of  reflection  equal  to  its  angle  of  incidence.     In  the 

second  case  there  will  be  refraction  with  recession  from  the  normal ;   &  in  the  third  case, 

refraction  with  approach  to  the  normal.     In  either  of  these  cases,  whatever  the  inclination 

was  on  entering  the  second  medium,  the  difference  between  the  squares  of  the  velocities 

on  entering  &  leaving  must  be  of  some  constant  magnitude,  from  the  mechanical  principle 

demonstrated  in  the  note  to  Art.  176.     From  which,  in  Art.  305, 1  have  deduced  that  the  sine 

of  the  angle  of  incidence  must  bear  a  constant  ratio  to  the  sine  of  the  angle  of  refraction  ; 

&  this  is  the  very  well  known  property  of  light,  upon  which  is  established  the  whole  theory  of 

dioptrics.    Also,  in  addition,  in  Art.  306,  I  deduced  that  the  velocity  in  the  first  medium 

is  to  the  velocity  in  the  second  in  the  inverse  ratio  of  the  sines  of  these  angles. 

487.  In  this  way,  from  a  uniform  theory,  all  the  principal  well-known  laws  of  reflection  Lisht   ^"^J^™ 
&  refraction  have  been  derived  ;  &  from  these  a  large  number  of  corollaries  can  be  deduced.  On  the  bodies; 
First  of  all,  because  the  action  must  always   be  mutual,  so  long  as  bodies  act  upon  light,  hence  the  extreme 

n        •  f         •        •        i  -  i«   i  '  ill-  »      i  I'lii        tenuity    of     light ; 

reflecting  or  refracting  it,  the  light  must  react  on  the  bodies ;    &  the  velocity  lost  by  the  these    effects    are 
light  must  bear  a  ratio  to  the  velocity  gained  by  the  centre  of  gravity  of  the  body  resisting  *als??y .  attributed 

r    i      !•   i  i  •   i    •  '  i  1  •        r    i  r    i      i      i  i  e   to   light   itself    by 

the  motion  of  the  light,  which  is  equal  to  the  ratio  of  the  mass  of  the  body  to  the  mass  of  some  people, 
the  light.  From  this  we  deduce  the  extreme  tenuity  of  light.  For,  the  tiniest  mass  of 
the  lightest  feather  suspended  by  the  finest  of  strings,  if  it  should  be  struck  by  a  ray  of  light 
suddenly  falling  upon  it,  still  would  acquire  no  progressive  motion,  such  as  could  be  perceived. 
Since  the  velocity  lost  by  the  light  is  so  huge,  it  can  be  clearly  seen  how  exceedingly  small  must 
be  the  density  of  light.  Newton  even  attributed  to  the  impact  of  light  rays  the  progressive 
motion  tail  first  of  the  vapours  of  comets ;  but  I  overthrew  this  idea,  by  an  argument 
which  I  consider  to  be  perfectly  sound,  in  my  dissertation  De  Cometis.  Some  people 
attribute  the  aurora  borealis  to  exhalations  of  extremely  small  density  impelled  by  solar 
light-rays ;  &  I  am  astonished  that  this  should  be  put  forward  by  anyone  who  considers 

improbable.  This  reflection  will  take  place  sooner  in  some  rays  than  in  others,  according  to  different  velocities  of  the  rays, 
different  angles  of  incidence,  different  natures  and  constitutions  of  the  particle  ;  some  of  the  particles  will  pass  along  a 
path  QXIK,  others  along  QXX'I'K',  and  others  again  along  QXX'X'T'K".  Further,  a  very  slight  difference  in  the 
force  or  velocity  will  be  enough  to  turn  the  curve  in  some  one  position  of  the  particle  from  being  very  nearly  parallel 
to  being  exactly  parallel ;  if  this  position  is  once  passed,  the  sum  of  the  actions  thereafter  as  far  as  O  may  be  practically 
the  same.  The  rest  is  now  similar  to  that  which  has  been  stated  in  Art.  306. 


346  PHILOSOPHIC  NATURALIS  THEORIA 

motum  per  se  se  non  imprimunt  :  qui  autem  censent,  &  fluvios  retardari  orient!  Soli 
contraries,  &  Terrae  motus  fieri  ex  impulsu  radiorum  Solis,  ii  sane  nunquam  per  legitima 
Mechanics  principia  inquisiverunt  in  luminis  tenuitatem. 

4 

Tenuissimum    mo-  488.  Solis  particulis  tenuissimis  corporum  imprimunt  motum  radii,  ex  quo  per  internas 

iuminelprartic1ulis  Y"cs  aucto  oritur  calor,  &  quidem  in  opacis    corporibus   multo  facilius,  ubi  tantse  sunt 

corporum:   cal-  reflexionum,  &  refractionum  internae  vicissitudines  :  exiguo  motu  impresso  paucis  particulis, 

provenire  ab^arum  re^^lua  internae  mutuae  vires  agunt  juxta  ea,  quae  diximus  num.  467.     Sic  ubi  radiis  solaribus 

viribus   internis,  speculo  collectis  comburuntur  aliqua,  alia  calcinantur   [228]  etiam  ;    omnes  illi  motus  ab 

t'u°hicPSUm  proba"  internis   utique  viribus  oriuntur,   non  ab  impulsione  radiorum.     Regulus  antimonii  ita 

calcinatus  auget  aliquando  pondus  decima  sui  parte.     Sunt,  qui  id  tribuant  massae  radiorum 

ibi  collects.     Si  ad  ita  esset ;    debuisset  citissime  abire  ilia  substantia  cum  parte  decima 

velocitatis  amissse  a  lumine,  sive  citius,  quam  binis  arteriae  pulsibus  ultra  Lunam  fugere. 

Quamobrem  alia  debet  esse  ejus  phaenomeni  causa,  qua  de  re  fusius  egi  in  mea  dissertatione 

De  Luminis  Tenuitate. 


Densiora  agere  in          489.  Quoniam  lumen  in  sulphuris  particulas   agit  validissime,   nam  sulphurosae,   & 
l?,^.tn,,foftlu»:  Sied  oleosae  substantias  facillime  accenduntur  :    eae  contra  in  lumen  validissime  agunt.     Sub- 

Suipnurosu.,    cc    OIG~  .•  _ ,  .  ._  _.  .    * 

osa  pan  densitate  stantise  gencraliter  eo  magis  agunt  in  lumen,  quo  densiores  sunt,  &  attractionum  summa 
plus :  cur  id  ipsum.  pr3evalet,  ubi  radius  utrumque  illud  planum  transgressus  refringitur  :  &  idcirco  generaliter 
ubi  sit  transitus  a  medio  rariore  ad  densius,  refractio  fit  per  accessum  ad  perpendiculum, 
&  ubi  a  medio  densiore  ad  rarius,  per  recessum.  Sed  sulphurosa,  &  oleosa  corpora  multo 
plus  agunt  in  lucem,  quam  pro  ratione  suae  densitatis.  Ego  sane  arbitror,  uti  monui  num. 
467,  ipsum  ignem  nihil  esse  aliud,  nisi  fermentationem  ingentem  lucis  cum  sulphurea 
substantia. 

Lumen  in  progressu          490.  Lumen  per  media  homogenea  progredi  motu  liberrimo,  &  sine  ulla  resistentia 

entiam  'po's it f ve  me^">  Per  quod  propagetur,  eruitur  etiam  ex  illo,  quod  velocitas  parallela  maneat  constans, 

probatiir.  uti  assumpsimus  num.  302,  quod  assumptum  si  non  sit  verum,  manentibus  ceteris ;    ratio 

sinus  incidentiae  ad  sinum  anguli  refracti  non  esset  constans  :    sed  idem  eruitur  etiam  ex 

eo,  quod  ubi  radius  ex  acre  abivit  in  vitrum,  turn  e  vitro  in  aerem  progressus  est,  si  iterum 

ad  vitrum  deveniat ;  eandem  habeat  refractionem,  quam  habuit  prima  vice.      Porro  si 

resistentiam  aliquam  pateretur,  ubi  secundo  advenit  ad  vitrum  ;    haberet  refractionem 

major  em  :    nam  velocitatem  haberet   minorem,  quas  semel  amissa  non  recuperatur  per 

hoc,  quod   resistentia  minuatur,  &  eadem  vis  mobile    minori   velocitate    motum    magis 

detorquet  a  directione  sui  motus. 

Unde  lux  in  phos-  491.  Posteaquam  lux  intra  opaca  corpora  tarn  multis,  tarn  variis  erravit  ambagibus 
Lm'  aliqua  saltern  sui  parte  deveniet  iterum  ad  superficiales  particulas,  &  avolabit.  Inde  omnino 
ortum  habebit  lux  ilia  tarn  multorum  phosphororum,  quse  deprehendimus,  e  Sole  retracta 
in  tenebras  lucere  per  aliquot  secunda,  &  a  numero  secundorum  licet  conjicere  longitudinem 
itineris  confecti  per  tot  itus,  ac  reditus  intra  meatus  internos.  Sed  progrediamur  jam  ad 
reliqua,  quae  num.  472  proposuimus. 

Cur  in  majore  obii-          ^Q2.  Prime  quidem  illud  facile  perspicitur,  ex  Theoria,  quam  exposuimus,  cur,  ubi 

quitate   plus     lum-         v  T     •       •  •,•  •  .      ,.         .  r  .,    .  '.*..*  '  ,, 

inis  reflectatur.  raaius  inciait  cum  majore  inchnatione  ad  supernciem,  major  luminis  pars  renectatur. 
Et  quidem  In  dissertatione,  quam  superiore  anno  die  12  Novembris  legit  [229]  Bouguerius 
in  Academiae  Parisiensis  conventu  publico,  uti  habetur  in  Mercurio  Gallico  hujus  anni  ad 
mensem  Januarii,  profitetur,  se  invenisse  in  aqua  in  inclinatione  admodum  ingenti  reflex- 
ionem  esse  aeque  fortem,  ac  in  Mercurio  ut  nimirum  reflectantur  duo  trientes,  dum  in 
incidentia  perpendicular!  vix  quinquagesima  quinta  pars  reflectatur.  Porro  ratio  in 
promptu  est.  Quo  magis  inclinatur  radius  incidens  ad  superficiem  novi  medii,  eo  minor 
est  perpendicularis  velocitas,  uti  patet  :  quare  vires,  quae  agunt  intra  ilia  duo  plana,  eo 
facilius,  &  in  pluribus  particulis  totam  velocitatem  perpendicularem  elident,  &  reflex- 
ionem  determinabunt. 

Diversam   refrangi-  .,.,.  .  .  ..  .. 

biiitatem  non  pen-  493.  Verum  id  quidem  jam  suppomt,  non  in  omnes  lucis  particulas  eandem  exercen 

ceterftatea  articu*  v*m'  sec^  *n  "s  ^iscrimen  haberi  aliquod.  Ejusmodi  discrimina  diligenter  evolvam. 
larum  luminis,  sed  Inprimis  discrimen  aliquod  haberi  debet  ex  ipso  textu  particularum  luminis,  ex  quo  pendeat 
earum  textu'Vnduc"  constans  discrimen  proprietatum  quarundam,  ut  illud  imprimis  diversae  radiorum  refran- 
ente  vim  diversam.  gibilitatis.  Quod  idem  radius  refringatur  ab  una  substantia  magis,  ab  alia  minus  in  eadem 


A  THEORY  OF  NATURAL  PHILOSOPHY  347 

that  light-rays  are  only  waves ;  for,  waves  do  not  give  any  progressive  motion  of  themselves. 
Further  there  are  some  who  consider  that  rivers  running  in  a  direction  opposite  to  the  rising 
Sun  are  retarded,  &  that  the  motion  of  the  Earth  is  due  to  impulse  of  solar  rays ;  but  really 
such  people  can  never  have  investigated  the  tenuity  of  light  by  means  of  legitimate  mechanical 
principles. 

488.  The  rays  of  the  Sun  impress  a  motion  on  the  exceedingly  small  particles  of  bodies ;  There  is  a  very 
&  from  this,  when  increased  by  internal  forces,  arises  heat,  &  this  all  the  more  easily  in  the  to^the^artlcifs^'o'f 
case  of  opaque  bodies, where  there  are  such  a  number  of  internal  alternations  of  reflections  bodies    by  light; 
&  refractions.     If  a  slight  motion  is  impressed  on  but  a  few  particles,  the  internal  mutual  arise  &fro>mbUtheS 
forces  do  all  the  rest,  as  we  stated  in  Art.  467.     Thus,  when  some  substances  are  set  on  internal  forces,  as 
fire  by  solar  rays  collected  by  a  mirror,  while  some  are  even  reduced  to  powder,  all  the  B  here  proved- 
motions  arise  in  every  case  from  internal  forces,  &  not  from  the  impulse  of  the  light-rays. 

Regulus  of  antimony  (stibnite),  thus  calcined,  will  sometimes  increase  its  weight  by  a  tenth 
part  of  it ;  &  there  are  some  who  attribute  this  fact  to  the  mass  of  the  rays  so  collected. 
But  if  this  were  the  case,  the  substance  would  have  to  fly  off  very  quickly  with  a  velocity 
equal  to  a  tenth  part  of  the  velocity  lost  by  the  light,  or  more  quickly  than  would  be 
necessary  to  get  beyond  the  Moon  in  two  beats  of  the  pulse.  Hence  there  must  be  other 
causes  to  account  for  this  phenomenon,  with  which  I  have  dealt  fairly  fully  in  my 
dissertation  De  Luminis  Tenuitate. 

489.  Since  light  acts  very  strongly  on  the  particles  of  sulphur,  for  sulphurous  &  oily  Denser   substances 
substances  are  very  easily  set  on  fire,  these  on  the  other  hand  act  very  strongly  on  light.  ^  j^f!  bS'"^ 
In  general,  substances  have  the  greater  action  on  light,  the  denser  they  are;    &  the  sum  phurous  &  oily 
of  the  attractions  will  be  stronger  when  the  ray  is  refracted  as  it  passes  through  each  of  substances    more 

„..  .o,  1  1*1  °  so   than  others    of 

the  planes.     Tor  this  reason,  in  general,  when  a  ray  passes  from  a  less  dense  to  a  more  dense  equal  density;  the 
medium,  refraction  takes  place  with  approach  to  the  normal,  &  when  from  a  more  dense  reason  for  thls- 
to  a  less  dense,  with  recession  from  the  normal.     But  sulphurous  &  oily  bodies  act  much 
more  vigorously  upon  light  than  in  proportion  to  their  density.     I  am  firmly  convinced 
that  fire  is    nothing    else   but   an    exceedingly  great   fermentation  of    light   with   some 
sulphurous  substance,  as  I  stated  in  Art.  467. 

490.  That  light  progresses  through  homogeneous  media  with  a  perfectly  free  motion,  Positive   demon- 
without  suffering  any  resistance  from  the  medium  through  which  it  is  propagated,  is  proved  ^oTs'not  ^er 'an* 
by  the  fact  that  the  parallel  component  of  the  velocity  remains  unaltered.     We  made  this  resistance  in  its  pro- 
assumption  in  Art.  302;    &  if  the  assumption  is  not  true,  other  things  being  unaltered,  gressive  motion.  ^ 
the  ratio  of  the  sine  of  incidence  to  the  sine  of  refraction  cannot  be  constant.     Now  the 

same  thing  is  also  proved  by  the  fact  that  when  a  light-ray  goes  from  air  into  glass,  &  then 
proceeds  from  the  glass  into  air,  then,  if  once  more  it  should  come  to  glass,  it  will  have 
the  same  refraction  as  it  had  in  the  first  instance.  Moreover,  if  it  suffered  any  resistance, 
when  for  the  second  time  it  came  to  glass,  it  would  have  a  greater  refraction  ;  for,  the 
velocity  would  be  less,  &  once  having  lost  this  velocity,  the  particle  could  not  regain  it 
simply  because  the  resistance  was  diminished  ;  &  the  same  force  will  cause  a  body  moving 
with  a  smaller  velocity  to  deviate  from  the  direction  of  its  motion  to  a  greater  degree. 

491.  After  light  has  wandered  through  so  many  &  various  paths  within  opaque  bodies,  The  source  of  the 
at  some  part  at  least  it  will  once  more  arrive  at  the  superficial  particles  of  the  bodies  &  fly  hf>ht     ln    ff1*1111 

n-       rm  >      i  -11     •          •  i      i«  i       i  •  •  i    r  i  IT         phosphorous  bodies. 

otr.  i  his  alone  will  give  rise  to  the  light  that  we  perceive  with  so  many  phosphorous  bodies, 
which  on  being  withdrawn  from  the  Sun  into  the  shade  shine  for  some  seconds ;  &  from 
the  number  of  seconds  one  may  conjecture  the  length  of  the  path  described  by  so  many 
backward  &  forward  journeys  within  the  internal  channels.  But  let  us  now  go  on  to  the  rest 
of  those  things  that  we  set  forth  in  Art.  472. 

492.  In  the  first  place,  then,  it  is  easily  seen  from  the  Theory  which  I  have  expounded,  why    at  greater 
why  the  proportion  of  light  reflected  is  greater,  when  the  ray  falls  on  the  surface  with  greater  Jnore^oT  thought 
inclination  to  it.     Indeed,  in  a  dissertation,  read  on  November  I2th  of  last  year  by  Bouguer  reflected, 
before  a  public  convention  of  the  Paris  Academy,  as  is  reported  in  the  French  Mercury 

for  January  of  this  year  the  author  professed  to  have  found  for  water  at  a  very  great  inclination 

a  reflection  equal  to  that  with  mercury  ;  that  is  to  say,  two-thirds  of  the  light  was  reflected, 

while  at  perpendicular  incidence  barely  a  fifty-fifth  part  is  reflected.     Now,  the  reason 

for  this  is  not  far  to  seek.     The  more  inclined  the  incident  ray  is  to  the  surface  of  the 

new  medium,  the  less  is  its  perpendicular  velocity,  as  is  quite  clear  ;  hence,  the  forces  that 

act  between  the  two  planes  will  the  more  easily,  &  for  a  larger  number  of  particles,  destroy  Different   refrangi- 

the  whole  of  the  perpendicular  velocity,  &  thus  determine  reflection.  biiity   does    not 

T>          i  •  i  i  <•  •  i  11  -1          r   v    i_       L    j.    depend  on  different 

493.  Jout  this  supposes  that  the  same  force  is  not  exerted  on  all  particles  or  light,  but  velocities    of   the 
that  even  for  them  there  is  some  difference.     I  will  carefully  discuss  these  differences.     First  particles  of     light 
of  all,  there  is  bound  to  be  some  difference  owing  to  the  structure  of  the  particles  of  light  ;  their  different 
&  upon  this  will  depend  a  constant  difference  in  some  of  its  properties,  such  as  that  of  the  structure     which 
different  refrangibilities  of  rays,  in  particular.     The  fact  that  the  same  ray  is  refracted  by  3 


348  PHILOSOPHIC  NATURALIS  THEORIA 

etiam  inclinatione  incidentise,  id  quidem  provenit  a  diversa  natura  substantiae  refringentis, 
uti  vidimus  :  ac  eodem  pacto  e  contrario,  quod  e  diversis  radiis  ab  eodem  medio,  &  cum 
eadem  inclinatione,  alius  refringatur  magis,  alius  minus,  id  provenire  debet  a  diversa 
constitutione  particularum  pertinentium  ad  illos  radios.  Debet  autem  id  provenire  vel 
a  diversa  celeritate  in  particulis  radiorum,  vel  a  diversa  vi.  Porro  demonstrari  potest,  a 
sola  diversitate  celeritatis  non  pro,venire,  atque  id  prasstiti  in  secunda  parte  meae  dissertationis 
De  Lumine :  quanquam  etiam  radii  diversae  refrangibilitatis  debeant  habere  omnino 
diversam  quoque  celeritatem  ;  nam  si  ante  ingressum  in  medium  refringens  habuissent 
aequalem ;  jam  in  illo  inasqualem  haberent,  cum  velocitas  praecedens  ad  velocitatem 
sequentem  sit  in  ratione  reciproca  sinus  incidentiae  ad  sinum  anguli  refracti  :  &  hsec  ratio 
in  radiis  diverse  refrangibilitatis  sit  omnino  diversa.  Quare  provenit  etiam  a  vi  diversa, 
quae  cum  constanter  diversa  sit,  ob  constantem  in  eodem  radio,  utcunque  reflexo,  & 
refracto,  refrangibilitatis  gradum,  debet  oriri  a  diversa  constitutione  particularum,  ex 
qua  sola  potest  provenire  diversa  summa  virium  pertinentium  ad  omnia  puncta.  Cum  vero 
diversa  constanter  sit  harum  particularum  constitutio :  nihil  mirum,  si  diversam  in 
oculo  impressionem  faciant,  &  diversam  ideam  excitent. 

EX  eadem  refrao  494.  At  quoniam  experiments  constat,  radios  ejusdem  colons  eandem  refractionem 
denf  Icrforismem!s-  Pat*  eodem  corpore,  sive  a  stellis  fixis  provenerint,  sive  a  Sole,  sive  a  nostris  ignibus,  sive 
sorum  ab  omnibus  etiam  a  naturalibus,  vel  artificialibus  phosphoris,  nam  ea  omnia  eodem  telescopio  aeque 
evhfcf  eandemlbin  distincta  videntur  :  manifesto  patet,  omnes  radios  ejusdem  coloris  pertinentes  ad  omnia 
iis  celeritatem,  &  ejusmodi  lucida  corpora  eadem  velocitate  esse  praeditos,  &  eadem  [230]  dispositione  punc- 
textum.  torum  :  neque  enim  probabile  est,  (&  fortasse  nee  fieri  id  potest),  celeritatem  diversam 

a  diversa  vi  compensari  ubique  accurate  ita,  ut    semper    eadem  habeatur    refractio  per 

ejusmodi  compensationem. 

vices  facilioris  re-  495.  Sed  oportet  invenire  aliud  discrimen  inter  diversas  constitutiones  particularum 
flexionis  &c.,  oriri  pertinentium  ad  radios  eiusdem  refrangibilitatis  ad  explicandas  vices  faciliores  reflexionis,  & 

a    contractione,    &   f  .  .  J  .  °  .  .    r  ,  ,. 

expansione  particu-  facilions  transmissus ;  ac  mde  mini  prodibit  etiam  ratio  phsenomem  radiorum,  qui  in  rerlex- 
larum  in  progressu  jone  &  refractione  irregularitur  disperguntur,  &  ratio  discriminis  inter  eos,  qui  reflectuntur 

inducente     dis-  r    .°       ,,  .          .      ,.        .  i* 

crimen.  potius,  quam  refnngantur,  ex  quo  etiam  fit,  ut  in  majore  inclinatione  renectantur  plures. 

Newtonus  plures  innuit  in  Optica  sua  hypotheses  ad  rem  utcunque  adumbrandam,  quarum 
tamen  nullam  absolute  amplectitur  :  ego  utar  hie  causa,  quam  adhibui  in  ilia  dissertatione 
De  Lumine  parte  secunda,  quae  causa  &  existit  &  rei  explicandae  est  idonea  :  quamobrem 
admitti  debet  juxta  legem  communem  philosophandi.  Ubi  particula  luminis  a  corpore 
lucido  excutitur  fieri  utique  non  potest,  ut  omnia  ejus  puncta  eandem  acquisierint  veloci- 
tatem, cum  a  punctis  repellentibus  diversas  distantias  habuerint.  Debuerunt  igitur  aliqua 
celerius  progredi,  quae  sociis  relictis  processissent,  nisi  mutuae  vires,  acceleratis  lentioribus, 
ea  retardassent,  unde  necessario  oriri  debuit  particulae  progredientis  oscillatio  quaedam, 
in  qua  oscillatione  particula  ipsa  debuit  jam  produci  non  nihil,  jam  contrahi  :  &  quoniam 
dum  per  medium  homogeneum  particula  progreditur,  inaequalitas  summae  actionum  in 
punctis  singulis  debet  esse  ad  sensum  nulla  ;  durabit  eadem  per  ipsum  medium  homogeneum 
reciprocatio  contractionis,  ac  productionis  particulae,  quae  quidem  productio,  &  contractio 
poterit  esse  satis  exigua  ;  si  nimirum  nexus  punctorum  sit  satis  validus  :  sed  semper  erit 
aliqua,  &  potest  itidem  esse  non  ita  parva,  nee  vero  debet  esse  eadem  in  particulis  diversi 
textus. 


in  Hmitibus    ejus          496.  Porro  in  ea  reciprocatione  figure  habebuntur  limites  quidam  productionis  maxi- 

dfutiu°SC^>erstare  m3e>  &  maximae  contractionis,  in  quibus  juxta  communem  admodum  indolem  maximorum, 

formam :  in  diver-  &  minimorum  diutissime  perdurabitur,   motu  reliquo,   ubi  jam  inde  discessum  fuerit  ad 

vtriunTs^  m  m^m  distantiam  sensibilem  cum  ingenti  celeritate  peracto,  uti  in  pendulorum  oscillationibus 

esse  diversam.         videmus,  pondus  in  extremis  oscillationum  Hmitibus  quasi  haerere  diutius,  in  reliquis  vero 

locis  celerrime  praetervolare  :    ac  in  alio  virium  genere  diverse  a  gravitate  constanti,  ilia 

mora  in  extremis  limitibus  potest  esse  adhuc  multo  diuturnior,  &  excursus  in  distantiis 

sensibilibus  ab  utrovis  maximo  multo  magis  celer.     Deveniet  autem  particula  ad  medium 

extremarum  illarum  duarum  dispositionum  diutius  perseverantium  post  aequalia  temporum 

intervalla,  ut  aequales  pendulorum    oscillationes  sunt  aeque    diuturnae,  ac  idcirco    dum 

particula  progreditur  per  medium  homogeneum,  recurrent  illae  ipsae  binae  dispositiones 

post  aequa-[23l]-lia    intervalla  spatiorum  pendentia  a  constanti  velocitate  particulae,  & 


A  THEORY  OF  NATURAL  PHILOSOPHY  349 

one  substance  more,  &  by  another  substance  less,  even  for  the  same  inclination  of  incidence, 
is  due  to  the  different  nature  of  the  refracting  substance,  as  we  have  seen  ;  &  in  the  same 
way,  on  the  other  hand,  the  fact  that,  of  different  rays,  &  with  the  same  inclination,  one  ray 
is  refracted  &  another  less,  by  the  same  medium,  is  due  to  the  different  constitution  of 
the  particles  pertaining  to  those  rays.  Further,  it  is  bound  to  be  due  either  to  a  different 
velocity  in  the  particles  of  the  rays,  or  to  a  different  force.  Lastly,  it  can  be  proved  that 
it  is  not  due  to  the  difference  of  velocity  alone  ;  &  this  I  showed  in  the  second  part  of  my 
dissertation  De  Lumine  ;  although  indeed  rays  of  different  refrangibilities  are  bound  to 
have  altogether  different  velocities  also.  For,  if  before  entering  the  refracting  substance 
they  had  equal  velocities,  then  after  entering  they  would  have  unequal  velocities ;  since 
the  first  velocity  is  to  the  second  in  the  inverse  ratio  of  the  sines  of  the  angles  of  incidence 
&  refraction  ;  &  this  ratio  for  rays  of  different  refrangibilities  is  altogether  different.  Hence 
it  must  also  be  due  to  a  difference  of  force  ;  &  since  this  must  be  constantly  different,  on 
account  of  the  constant  degree  of  refrangibility  in  the  same  ray,  however  it  may  be  reflected 
or  refracted,  it  must  be  due  to  a  difference  in  the  constitution  of  the  particles,  from  which 
alone  there  can  arise  a  difference  in  the  sum  of  the  forces  pertaining  to  all  points  forming 
them.  Now,  since  the  constitution  of  these  particles  is  constantly  different,  it  is  no  wonder 
that  they  make  a  different  impression  on  the  eye,  &  incite  a  different  sensation. 

494.  Now,  since  it  is  proved  by  experiment  that  rays  of  the  same  colour  suffer  the  same  From  the  equality 
refraction  by   the   same   body,   whether   they  come   from   the  fixed  stars,    or   from  the  pL"  of  ^he 'colour 
Sun,  or  from  our  fires,  or  even  from  natural  or  artificial  phosphorous  substances,  for  they  coming  from    ail 
all  appear  equally  distinct  when  viewed  with  the  same  telescope ;   it  is  clearly  evident  that  {j^r^i^'roved 
all  rays  of  the  same  colour  pertaining  to  such  light -giving  bodies  are  endowed  with  the  that  for  such  rays 
same  velocities,  &  the  same  distribution  of  their  points.     For,  it  is  very  improbable,  not  to  Velocity  *&e  struck 
say  impossible,  that  a  difference  in  velocity  should  be  everywhere  exactly  balanced  by  a  ture. 
difference  in  force  to  such  a  degree  that  by  means  of  such  a  balance  there  should  always 

be  the  same  refraction  obtained. 

495.  But  another  difference  must  be  found  amongst  the  different  constitutions  of  the  Fits.  of  easier  re- 
particles  belonging  to  rays  of  the  same  refrangibility,  to  account  for  the  fits  of  easier  reflection  duetto'  contraction 
&  easier  transmission.     From  it  I  shall  obtain  also  the  reason  for  the  phenomenon  of  rays  that  &  expansion  of  the 
are  irregularly  scattered  in  reflection  &  refraction  ;  &  the  reason  for  the  difference  between  mduce^a^itoence 
those  that  are  reflected  in  preference  to  being  refracted,  from  which  also  it  comes  about  in  the  progressive 
that  the  greater  the  angle  the  more  numerous  the  rays  reflected.     Newton  suggests  several 
hypotheses,  in  his  Optics,  to  give  a  rough  idea  of  the  matter  ;    but  he  does  not  adhere 

absolutely  to  any  one  of  them.  I  will  use  in  this  connection  the  reason  that  I  employed  in 
the  dissertation  De  Lumine,  in  the  second  part  ;  this  reason  both  really  exists  &  is  fitted  for 
explaining  the  matter  ;  &  therefore,  according  to  the  usual  rule  in  philosophizing,  this 
reason  should  be  admitted.  When  a  particle  of  light  is  driven  off  from  a  light-giving  body, 
it  cannot  in  any  case  happen  that  all  the  points  forming  it  have  acquired  the  same  velocity  ; 
for,  they  will  have  been  at  different  distances  from  the  repelling  points  of  the  body.  Therefore 
some  of  them  are  bound  to  progress  more  quickly  than  others,  &  the  former  would  have  left 
their  fellows  behind  in  their  advance,  unless  the  mutual  forces  had  retarded  them,  while  the 
slower  ones  were  accelerated.  Owing  to  this,  there  must  necessarily  have  arisen  a  certain 
oscillation  of  the  particle  as  it  goes  along,  &  due  to  this  oscillation  the  particle  itself  must  have 
been  alternately  extended  &  contracted  to  some  extent.  Now,  since  during  the  progress 
of  a  particle  through  a  homogeneous  medium  inequality  of  the  sum  of  the  actions  at  all 
points  of  it  must  be  practically  zero,  the  same  alternation  of  extension  &  contraction  of 
the  particle  will  continue  right  through  the  homogeneous  medium,  although  the 
contraction  &  expansion  will  indeed  be  but  slight,  if  the  connections  between  the  points 
are  fairly  strong.  But  there  will  always  be  some  oscillation,  &  it  may  also  not  be  so  very 
small,  nor  need  it  be  the  same  for  particles  of  different  structure. 

496.  Further,  in  this  alternation  of    figure  there  will  be  certain  bounding  forms,  At  the  boundaries 
corresponding  to  maximum  extension  &  maximum  contraction ;    &  in  these,  according  to  ^  ^  Sp^atl°he 
a  universal  property  of  all  maxima  &  minima,  there  will  be  quite  a  long  pause  ;   whereas,  particle   will'  pre- 
the  rest  of  the  motion,  after  a  departure  from  them  has  taken  place  to  a  sensible  distance,  longer  •**&  the  sum 
is  accomplished  with  a  great  velocity.     Thus,  we  see  in  the  oscillations  of  pendulums  that  of  the  forces    at 
the  weight  at  the  extreme  ends  of  the  oscillations  seems  to  pause  for  a  considerable  time, 

whereas  in  other  positions  it  flies  past  very  quickly.  In  another  kind  of  forces  different 
from  constant  gravitation,  this  delay  at  the  extreme  ends  may  be  still  more  prolonged,  & 
the  motion  at  sensible  distances  from  either  maximum  much  more  swift.  Moreover  the 
particle  will  reach  the  mean,  between  the  two  extreme  dispositions  that  last  for  some 
considerable  time,  after  equal  intervals  of  time  ;  just  as  equal  oscillations  of  pendulums 
are  of  equal  duration.  Hence,  as  a  particle  proceeds  through  a  homogeneous  medium, 
those  two  dispositions  recur  after  equal  intervals  of  space,  depending  on  the  constant  velocity 


350  PHILOSOPHIC  NATURALIS  THEORIA 

a  constant!  tempore,  quo  particular  cujusvis  oscillatio  durat.  Demum  summa  virium, 
quam  novum  medium,  ad  quod  accedit  particula,  exercet  in  omnia  particulae  puncta, 
non  erit  sane  eadem  in  diversis  illis  oscillantis  particulae  dispositionibus. 

inde  binae  disposi-  497.  Hisce  omnibus  rite  consideratis,  concipiatur  jam  ille  fere  continuus  affluxus 
vkium  in°maxima  particularum  etiam  homogenearum  ad  superficiem  duo  heterogenea  media  dirimentem. 
particuiarum  parte  Multo  maximus  numerus  adveniet  in  altera  ex  binis  illis  oppositis  dispositionibus,  non 
uSltibust  kTpart'e  quidem  in  medio  ipsius,  sed  prope  ipsam,  &  admodum  exiguus  erit  numerus  earum.  quse 
exigua  appeiiente  adveniunt  cum  dispositione  satis  remota  ab  illis  extremis.  Quae  in  hisce  intermediis 
inter  eos  dispersio.  adveniunt,  mutabunt  utique  dispositiones  suas  in  progressu  inter  ilia  duo  plana,  inter 
quas  agit  vis  motum  particulae  perturbans,  ita,  ut  in  datis  ab  utrovis  piano  distantiis  vires 
ad  diversas  particulas  pertinentes,  sint  admodum  diversae  inter  se.  Quare  illse,  quae  retro 
regredientur,  non  eandem  ad  sensum  recuperabunt  in  regressu  velocitatem  perpendicu- 
larem,  quam  habuerunt  in  accessu,  adeoque  non  reflectentur  in  angulo  reflexionis  aequali 
ad  sensum  angulo  incidentise,  &  illae,  quae  superabunt  intervallum  illud  omne,  in  appulsu 
ad  planum  ulterius,  aliae  aliam  summam  virium  expertae,  habebunt  admodum  diversa 
inter  se  incrementa,  vel  decrementa  velocitatum  perpendicularium,  &  proinde  in  admodum 
diversis  angulis  egredientur  disperses.  At  quae  advenient  cum  binis  illis  dispositionibus 
contrariis,  habebunt  duo  genera  virium,  quarum  singula  pertinebunt  constanter  ad  classes 
singulas,  cum  quarum  uno  idcirco  facilius  in  illo  continue  curvaturae  flexu  devenietur 
ad  positionem  illis  planis  parallelam,  sive  ad  extinctionem  velocitatis  perpendicularis 
cum  altero  difficilius  :  adeoque  habebuntur  in  binis  illis  dispositionibus  oppositis  binae 
vices,  altera  facilioris,  altera  difficilioris  reflexionis,  adeoque  facilioris  transitus,  quae  quidem 
regredientur  post  aequalia  spatiorum  intervalla,  quanquam  ita,  ut  summa  facilitas  in  media 
dispositione  sita  sit,  a  qua  quae  minus,  vel  magis  in  appulsu  discedunt,  magis  e  contrario, 
vel  minus  de  ilia  facilitate  participent.  Is  ipse  accessus  major,  vel  minor  ad  summam 
illam  facilitatem  in  media  dispositione  sitam  in  Benvenutiana  dissertatione  superius 
memorata  exhibetur  per  curvam  quandam  continuam  hinc,  &  inde  aeque  inflexam  circa 
suum  axem,  &  inde  reliqua  omnia,  quas  ad  vices,  &  earum  consectaria  pertinent,  luculen- 
tissime  explicantur. 


Unde  discrimen  498.  Porro  hinc  &  illud  patet,  qui  fieri  possit,  ut  e  radiis  homogeneis  ad  eandem 
reVex^'ad  ^trans-  superficiem  advenientibus  alii  transmittantur,  &  alii  reflectantur,  prout  nimirum  advenerint 
missum.  in  altera  e  binis  dispositionibus  :  &  quoniam  non  omnes,  qui  cum  altera  ex  extremis  illis 

dispositionibus  adveniunt,  adve-[232J-niunt  prorsus  in  media  dispositione,  fieri  utique 
poterit,  ut  ratio  reflexorum  ad  transmissos  sit  admodum  diversa  in  diversis  circumstantiis, 
nimirum  diversi  mediorum  discriminis,  vel  diversas  inclinationis  in  accessu  :  ubi  enim 
inaequalitas  virium  est  minor,  vel  major  perpendicularis  velocitas  per  illam  extinguenda 
ad  habendam  reflexionem,  non  reflectentur,  nisi  illae  particulas,  quse  advenerint  in  dispositione 
illi  medias  quamproxima,  adeoque  multo  pauciores  quam  ubi  vel  insequalitas  virium  est 
major,  vel  velocitas  perpendicularis  est  minor,  unde  fiet,  ut  quemadmodum  experimur, 
quo  minus  est  mediorum  discrimen,  vel  major  incidentiae  angulus,  eo  minor  radiorum 
copia  reflectetur  :  ubi  &  illud  notandum  maxime,  quod  ubi  in  continue  flexo  curvaturae 
viae  particulae  cujusvis,  quae  via  jam  in  alteram  plagam  est  cava,  jam  in  alteram,  prout 
prasvalent  attractiones  densioris  medii,  vel  repulsiones,  devenitur  identidem  ad  positionem 
fere  parallelam  superficiei  dirimenti  media,  velocitate  perpendiculari  fere  extincta,  exiguum 
discrimen  virium  potest  determinare  parallelismum  ipsum,  sive  illius  perpendicularis 
velocitatis  extinctionem  totalem  :  quanquam  eo  veluti  anfractu  superato,  ubi  demum 
reditur  ad  planum  citerius  in  reflexione,  vel  ulterius  in  refractione,  summa  omnium  actionum 
quae  determinat  velocitatem  perpendicularem  totalem,  debeat  esse  ad  sensum  eadem, 
nimirum  nihil  mutata  ad  sensum  ab  exigua  ilia  differentia  virium,  quam  peperit  exiguum 
dispositionis  discrimen  a  media  dispositione. 


Unde  discrimen  in          40.9.  Atque  hoc  pacto  satis  luculenter  jam  explicatum  est  discrimen  inter  binas  vices, 

mtervalhs  viaum.     se(j  SUperest  exponendum,  unde  discrimen  intervalli  vicium,  quod  proposuimus  nurn.  472. 

Quod  diversi  colorati  radii  diversa  habeant  intervalla,  nil  mirum  est  :    nam  &    diversas 


A  THEORY  OF  NATURAL  PHILOSOPHY  351 

of  the  particle,  &  on  the  constant  time  for  which  any  oscillation  of  the  particle  lasts.  Lastly, 
the  sum  of  the  forces,  which  the  new  medium,  approached  by  the  particle,  exerts  upon 
all  the  points  of  the  particle,  will  not  really  be  the  same  for  the  different  dispositions  of 
the  oscillating  particle. 

497.  All  such  things  being  duly  considered,  a  conception  can  be  now  formed  of  the  almost  Hence,  we  have  the 
continuous  flow  of  even  homogeneous  particles  towards  the  surface  of  separation  of  two  positions5  fielding 
unlike  media.     By  far  the  greater  number  of  them  will  arrive  at  the  surface  in  one  or  other  fits,  with  the  greater 
of  those  two  opposite  dispositions ;   not  indeed  exactly  so,  but  very  nearly  so.     A  very  few  tic°es°   which  Pare 
of  them  will  reach  the  surface  with  a  disposition  considerably  removed  from  those  extremes,  striking   in   those 
Those  that  do  arrive  in  these  intermediate  states,  will  in  all  cases  change  their  dispositions  f™1  ^  SfewSthat 
in  their  passage  between  the  two  planes,  between  which  the  force  disturbing  the  motion  strike    in     states 
of  the  particle  acts ;   &  in  such  a  manner  that  at  any  given  distance  from  either  plane  the  twee^^them   bwe 
forces  pertaining  to  different  particles  will  be  altogether  different.     Therefore,  those  which  have  dispersion, 
return  on  their  path,  will  not  recover  a  velocity  on  the  return,  that  is  practically  equal 

to  that  perpendicular  velocity  that  it  had  on  approach  ;  &  thus,  it  will  not  be  reflected 
at  an  angle  of  reflection  practically  equal  to  the  angle  of  incidence.  Those,  which  manage 
to  pass  over  the  whole  of  the  interval  between  the  two  planes,  on  moving  away  from  the 
further  plane,  will,  under  the  influence  of  different  sums  of  forces  for  different  particles, 
have  quite  different  increments  or  decrements  of  the  perpendicular  velocities ;  &  they 
will  emerge  at  quite  different  angles  from  one  another,  in  all  directions.  But,  those  that 
reach  the  surface  with  either  of  those  two  opposite  dispositions  will  have  but  two  kinds 
of  forces ;  &  each  of  these  will  remain  constant  for  its  corresponding  class  of  particles. 
Hence,  with  one  of  these  classes  there  will  be  more  easy  approach  in  its  continually  curving 
path  to  a  position  parallel  to  the  planes,  corresponding  to  the  extinction  of  the  perpendicular 
velocity ;  &  with  the  other,  this  will  be  more  difficult.  Therefore  there  will  be  produced, 
in  consequence  of  the  two  opposite  dispositions,  two  fits,  the  one  of  more  easy,  &  the  other 
of  more  difficult  reflection,  or  more  easy  transmission  ;  these  fits  recur  at  equal  intervals 
of  space.  However,  these  will  take  place  in  such  a  manner  that  the  greatest  facility  of 
reflection  will  correspond  to  the  mean  disposition  ;  &  the  less  or  more  the  particles  depart 
from  this  mean  on  striking  the  surface,  the  more  or  the  less,  respectively,  will  they  participate 
in  that  facility.  This  greater  or  less  approach  to  the  maximum  facility,  corresponding 
to  the  mean  disposition,  has  been  represented  in  the  dissertation  by  Benvenuti  mentioned 
above  by  a  continuous  curve,  which  is  equally  inflected  on  each  side  of  its  axis ;  &  from 
this  curve  all  the  other  points  that  relate  to  fits  &  their  consequences  are  explained  in  a 
most  excellent  manner. 

498.  Further,  from  this  also  it  is  clear  how  it  comes  about  that,  out  of  a  number  of  The  cause  of  the 
homogeneous  rays  reaching  the  same  surface,  some  are  transmitted  &  others  are  reflected,  dlfference    in  tne 


he  amount 
according  as  they  reach  it  in  one  or  other  of  two  dispositions.     Since,  of  those  particles  of  light  reflected  to 

which  do  [not]  reach  the  surface  with  one  of  the  two  extreme  dispositions,  not  all  reach  it  ^*e^hlch  1S  trans" 

in  the  mean  disposition  exactly  ;  it  may  happen  that  the  ratio  of  reflections  to  transmissions 

will  be  altogether  different  in  different  circumstances  of,  say,  various  differences  between 

the  media,  or  different  inclinations  of  approach.     For  when  the  inequality  of  the  forces 

is  less  or  the  perpendicular  velocity,  which  has  to  be  destroyed  by  the  inequality  to  produce 

reflection,  is  greater,  only  those  particles  are  reflected  which  reach  the  surface  in  dispositions 

very  near  to  that  mean  disposition  ;   &  so,  much  fewer  are  reflected  than  is  the  case  when 

the  inequality  of  forces  is  greater  or  the  perpendicular  velocity  is  less.     Hence,  it  comes 

about  that  the  less  the  difference  between  the  media,  or  the  greater  the  angle  of  incidence, 

the  smaller  the  proportion  of  rays  reflected  ;    which  is  in  agreement  with  experience.     In 

this  connection  also  it  is  especially  to  be  observed  that  when  in  the  continuous  winding 

of  the  curved  path  of  any  particle,  the  path  being  at  one  time  concave  on  one  side  &  at 

another  time  on  the  other,  according  as  the  attractions  or  the  repulsions  of  the  denser 

medium  are  more  powerful,  a  position  nearly  parallel  to  the  surface  of  separation  between 

the  media  is  attained  several  times  in  succession,  as  the  perpendicular  velocity  is  nearly 

destroyed,   a   very  slight   difference   of   the  forces   will   be   sufficient   to   produce   exact 

parallelism,  or  the  total  extinction  of  that  perpendicular  velocity.     Although,  when  these, 

so  to  speak,  tortuosities  are  ended  as  the  particle  at  length  reaches  the  nearer  plane  in  reflection 

&  the  further  plane  in  refraction,  the  sum  of  all  the  actions,  which  determines  the  total 

perpendicular  velocity,  must  be  practically  the  same  ;    that  is  to  say,  in  no  wise  changed 

to  any  sensible  extent  by  the  slight  difference  of  forces,  such  as  produced  the  slight  difference 

of  disposition  from  the  mean  disposition. 

499.  In  this  way  we  have  a  sufficient  explanation  of  the  difference  between  the  two  J*Le  cause  of  the 

-         ~? '  <,,  .   ,  i-     •        i-rr  i  it  i       difference    in     the 

fits ;   but  we  have  still  to  explain  the  source  of  the  difference  in  the  intervals  between  the  intervals    between 
fits,  which  we  propounded  in  Art.  472.      There  is  nothing  wonderful  in  the  fact  that  successive  fits, 
differently  coloured  rays  should  have  different  intervals.     For,  different  velocities  require 


352  PHILOSOPHIC  NATURALIS  THEORIA 

velocitates  diversa  requirunt  intervalla  spatii  inter  vices  oppositas,  quando  etiam  eas  vices 
redeant  aequalibiw  temporis  intervallis,  &  diversus  particularum  heterogenearum  textus 
requirit  diversa  oscillationum  tempora.  Quod  in  diversis  mediis  particulae  ejusdem  generis 
habeant  diversa  intervalla,  itidem  facile  colligitur  ex  diversa  velocitate,  quam  in  iis  haberi 
post  refractionem  ostendimus  num.  493  ;  sed  praaterea  in  ipsa  mediorum  mutatione 
inaequalis  actio  inter  puncta  particulam  componentia  potest  utique,  &  vero  videtur  etiam 
debere  oscillationis  magnitudinem,  &  fortasse  etiam  ordinem  mutare,  adeoque  celeritatem 
oscillationis  ipsius.  Demum  ejusmodi  mutatio  pro  diversa  inclinatione  vias  particular 
advenientis  ad  superficiem,  diversa  utique  esse  debet,  ob  diversam  positionem  motuum 
punctorum  ad  superficiem  ipsam,  &  ad  massam  agentem  in  ipsa  puncta.  Quamobrem 
patet,  eas  omnes  tres  causas  debere  discrimen  aliquod  exhibere  inter  diversa  intervalla, 
uti  reapse  ex  observatione  colligitur. 

Discnmen  id  non  coo.  Si    possemus    nosse  peculiares    constitutiones    particula-tessl-rum    ad    diversos 

posse   definin,    nisi        ,3  r  .  .  L    JOJ  _ 

per  observationes :  coloratos  radios  pertmentium,  ordinem,  &  numerum,  ac  vires,  &  velocitates  punctorum 
vef  ^ntdere  a  sola  singulorum  ;  turn  mediorum  constitutionem  suam  in  singulis,  ac  satis  Geometrias,  satis 
imaginationis  haberemus,  &  mentis  ad  omnia  ejusmodi  solvenda  problemata  ;  liceret  a 
priori  determinare  intervallorum  longitudines  varias,  &  eorundem  mutationes  pro  tribus 
illis  diversis  circumstantiis  exhibere.  Sed  quoniam  longe  citra  eum  locum  consistimus 
debemus  illas  tantummodo  colligere  per  observationes,  quod  summa  dexteritate  Newtonus, 
praestitit,  qui  determinatis  per  observationem  singulis,  mira  inde  consectaria  deduxit, 
&  Naturae  phenomena  explicavit,  uti  multo  luculentius  videre  est  in  ilia  ipsa  Benvenutiana 
dissertatione.  Illud  unum  ex  proportionibus  a  Newtono  inventis  haud  difficulter  colligitur, 
ea  discrimina  non  pendere  a  sola  particularum  celeritate,  nam  celeritatum  proportiones, 
novimus  per  sinuum  rationem  :  &  facile  itidem  deducitur  ex  Theoria,  quod  etiam  multo 
facilius  infertur  partim  ex  Theoria,  &  partim  ex  observatione,  radium,  qui  post  quotcunque 
vel  reflexiones,  vel  refractiones  regulares  devenit  ad  idem  medium,  eandem  in  eo  velocitatem 
habere  semper  ;  nam  velocitates  in  reflexione  manent,  &  in  mutatione  mediorum  sunt  in 
ratione  reciproca  sinus  incidentiae  ad  sinum  anguli  refracti  :  ac  tarn  Theoria,  quam  observatio 
facile  ostendit,  ubi  planis  parallelis  dirimantur  media  quotcunque,  &  radius  in  data 
inclinatione  ingressus  e  primo  abeat  ad  ultimum,  eundem  fore  refractionis  angulum  in 
ultimo  medio,  qui  esset,  si  a  primo  immediate  in  ultimum  transivisset.  Sed  haec  innuisse 
sit  satis. 

Quod  de  crystalio  501.  Illud   etiam   innuam   tantummodo,   quod   Newtonus   in   Opticis   Quaestionibus 

isiandica  Newtonus  exponit   esse  miram  quandam  crystalli  Islandicae  proprietatem,  quae  radium  quemvis,  dum 

prodidit,  id  in  nac        F ,      ,       ,,  .      .     *• ,  ..  •  .  i          r  •       •         !•          •         •  J  i  •    o 

Theoria  nuiiam  refrmgit,  discerpit  in  duos,  &  ahum  usitato  modo  retrmgit,  ahum  musitato  quodam,  ubi  & 
habere  difficuita-  certa2  qusedam  observantur  leges,  quarum  explicationes  ipse  ibidem  insinuat  haberi  posse 
per  vires  diversas  in  diversis  lateribus  particularum  luminis,  ac  solum  adnotabo  illud,  ex 
num.  423  patere,  in  mea  Theoria  nullam  esse  difficultatem  agnoscendi  in  diversis  lateribus 
ejusdem  particulae  diversas  dispositiones  punctorum,  &  vires,  qua  ipsa  diversitate  usi  sumus 
superius  ad  explicandam  solidorum  cohassionem,  &  organicam  formam,  ac  certas  figuras 
tot  corporum,  quse  illas  vel  affectant  constanter,  vel  etiam  acquirunt. 

piffractionem  esse          502.  Remanet  demum  diffractio  luminis  explicanda,  quam  itidem  num.  472  proposue- 
inchoatam    reflexi-  ramus<     j?a  est  qusedam  velut  inchoata  reflexio,  &  refractio.     Dum  radius  advenit  ad  earn 

oncm,  vci  rciitiCtiQ"  '        -  »•  i  i  i  *  •«•  -t  • 

nem.  distantiam  a  corpore  diversas  naturae  ab  eo,  per  quod  progreditur,  quae  vinum  maequahtatem 

inducit,  incurvat  viam  vel  accedendo,  vel  recedendo,  &  directionem  mutat.  Si  corporis 
superficies  ibi  esset  satis  ampla,  vel  reflecteretur  ad  angulos  asquales,  vel  immergeretur 
intra  novum  illud  medium,  &  refrin-[234"|-geretur  ;  at  quoniam  acies  ibidem  progressum 
superficiei  interrumpit ;  progreditur  quidem  radius  aciem  ipsam  evitans  &  circa  illam 
praetervolat  ;  sed  egressus  ex  ilia  distantia  directionem  conservat  postremo  loco  acquisitam, 
&  cum  ea,  diversa  utique  a  priore,  moveri  pergit  :  ut  adeo  tota  luminis  Theoria  sibi  ubique 
admodum  conformis  sit,  &  cum  generali  Theoria  mea  apprime  consentiens,  cujus  rami 
quidam  sunt  bina  Newtoni  praeclarissima  comperta  virium,  quibus  caslestia  corpora  motus 
peragunt  suos  &  quibus  particulae  luminis  reflectuntur,  refringuntur,  diffringuntur.  Sed 
de  luce,  &  coloribus  jam  satis. 


De sapore,  & odore :  503.  Post  ipsam  lucem,  quae  oculos  percellit,  &  visionem  parit,  ac  ideam  colorum 

ratione^densrtat'is  excitat,  pronum  est  delabi  ad  sensus  ceteros,  in  quibus  multo  minus  immorabimur,  cum 

odoris  propagati.      circa  eos  multo  minora  habeamus  comperta,  quae  determinatam  physicam  explicationem 

ferant.     Saporis  sensus  excitatur  in  palato  a  salibus.     De  angulosa  illorum  forma  jam 


°n 


A  THEORY  OF  NATURAL  PHILOSOPHY  353 

different  intervals  of  space  between  opposite  fits,  when  those  fits  recur  also  at  equal  intervals 
of  time  ;  &  a  difference  in  the  structure  of  heterogeneous  particles  requires  a  difference 
in  the  periods  of  oscillation.  It  is  also  easily  seen  that  particles  of  the  same  kind  have 
different  intervals  in  different  media,  owing  to  that  difference  in  velocity,  which,  in 
Art.  493,  was  proved  to  exist  after  refraction.  But,  in  addition,  on  changing  the 
medium,  an  unequal  action  between  the  points  composing  the  particle  certainly  can 
and,  apparently  indeed,  is  bound  to  alter  the  magnitude  of  the  oscillation  also,  &  perhaps 
even  the  order  ;  &  thus  the  velocity  of  that  oscillation  must  alter.  Further,  such  a  change, 
for  a  difference  in  the  inclination  of  the  path  of  the  particle  approaching  the  surface, 
is  in  every  case  bound  to  be  different,  on  account  of  the  difference  in  situation  of 
the  motions  of  the  points  with  respect  to  the  surface  &  the  mass  acting  upon  the  points. 
Hence,  it  is  clear  that  all  three  of  these  causes  must  stand  for  some  difference  between 
diverse  intervals  ;  &  indeed  we  can  deduce  as  much  from  observation. 

500.  If  we  could  know  the  particular  constitutions  of  particles  for  differently  coloured  This  difference  can- 
rays,  the  order,  number,  forces  &  velocities  of  each  point,  &  the  constitution  of  each  medium  n°ven  ^  unless  ^b7 
for  each  ray,  and  if  we  had  a  sufficiency  of  geometry,  imagination  &  intelligence  to  solve  observation;  it 
all  problems  of   this  kind,  we  could  determine  from  first  principles  the  various  lengths 

of  the  intervals,  &  could  give  the  changes  due  to  each  of  the  three  different  circumstances. 
But  since  this  is  far  beyond  us,  we  are  bound  to  deduce  them  from  observation  alone.  This 
Newton  accomplished  with  the  greatest  dexterity  ;  having  determined  each  by  observation, 
he  deduced  from  them  wonderful  consequences  ;  &  explained  the  phenomena  of  Nature  ; 
as  also  it  is  to  be  seen  much  better  in  the  dissertation  by  Benvenuti.  There  is  one  thing 
that  can  be  without  much  difficulty  derived  from  the  proportions  discovered  by  Newton, 
namely,  that  the  differences  do  not  solely  depend  upon  the  velocities  of  the  particles  ;  for 
we  know  the  proportions  of  the  velocities  by  the  ratio  of  the  sines.  It  can  also  easily  be 
deduced  from  the  Theory,  &  indeed  much  more  easily  can  it  be  inferred  partly  from  the 
Theory  &  partly  from  observation,  that  a  ray  which,  after  any  number  of  regular  reflections 
&  refractions,  comes  to  the  same  medium  will  always  have  the  same  velocity  in  it  as  at  first. 
For  the  velocities  remain  unaltered  in  reflection,  &  on  a  change  of  medium  they  are  in 
the  inverse  ratio  of  the  sine  of  the  angle  of  incidence  to  the  sine  of  the  angle  of  refraction. 
Both  the  Theory,  &  observation,  clearly  show  that,  when  any  number  of  media  are  separated 
by  parallel  planes,  &  a  ray,  entering  at  a  given  inclination,  leaves  the  first  &  reaches  the 
last,  there  will  be  the  same  angle  of  refraction  in  the  last  medium  as  there  would  have  been, 
if  it  had  passed  directly  from  the  first  medium  into  the  last.  But  a  mere  mention  of  these 
things  is  enough. 

501.  I  will  also  merely  mention  that,  as  was  stated  by  Newton  in  his  Questions  at  the  That  which  Newton 
end  of  his  Optics,  there  is  a  wonderful  property  of  Iceland  Spar  ;    namely,  that  when  it  inland  "spar™^ 
refracts  a  ray  of  light  it  divides  it  into  two,  refracting  one  part  according  to  the  normal  ?ents  no  difficulty 
manner,  &  the  other  in  an  unusual  way  ;  with  the  latter  also  definite  laws  are  observed. 

Newton  himself  suggested  that  the  explanation  of  these  laws  could  be  attributed  to  different 
forces  on  different  sides  of  the  particles  of  light  ;  &  I  will  only  remark  that,  according  to 
Art.  423,  it  is  evident  that  in  my  Theory  there  is  no  difficulty  over  admitting  for  different 
sides  of  the  same  particle  different  dispositions  of  the  points,  &  different  forces  ;  we  have 
already  employed  this  sort  of  difference  to  explain  cohesion  of  solids,  &  organic  form,  & 
all  those  shapes  of  bodies,  such  as  they  always  endeavour  to  acquire,  &  indeed  do  acquire. 

502.  Finally,  we  have  to  explain  diffraction,  which  we  also  enunciated  in  Art.  472.  Diffraction  is   in- 
This  is,  so  to  speak,  an  incomplete  reflection  or  refraction.     When  a  ray  of  light  attains  complete  reflection 
the  distance,  from  a  body  of  a  different  nature  from  one  through  which  it  passes,  which 

induces  an  inequality  of  forces,  its  path  becomes  curved,  either  by  approach  or  recession, 
&  the  direction  is  altered.  If  the  surface  of  the  body  at  the  point  in  question  is  sufficiently 
wide,  the  ray  will  either  be  reflected  at  equal  angles,  or  it  will  enter  the  new  medium  & 
be  reflected.  But  when  a  sharp  edge  terminates  the  run  of  the  surface,  the  ray  will  pass 
on,  slipping  by  the  edge,  &  flying  past  &  round  it.  But,  on  emergence  from  that  distance, 
the  ray  will  preserve  the  direction  acquired  in  the  last  position,  &  with  this  direction,  which 
will  be  altogether  different  from  that  which  it  had  originally,  it  will  continue  its  motion. 
Thus  the  whole  theory  of  light  will  be  quite  consistent,  &  in  close  agreement  with  my  Theory. 
Of  this  Theory,  the  two  most  noted  discoveries  of  Newton  with  respect  to  forces  are  just 
branches  ;  namely,  the  forces  with  which  the  heavenly  bodies  keep  up  their  motions,  & 
those  by  which  particles  of  light  are  reflected,  refracted  &  diffracted.  But  I  have  now 
said  sufficient  about  light  &  colour. 

503.  After  light,  which  affects  the  eyes,  begets  vision,  &  excites  the  idea  of  colours,  we  Concerning  taste  & 
naturally  come  to  the  other  senses  ;    over  these  I  will  spend  far  less  time,  since  we  have  ^ny'  p^opJiTwith 
far  less  knowledge  of  them,  such  as  will  help  us  to  give  a  definite  physical  explanation,  regard  to  the  ratio 
The   sense   of   taste   is   excited   in  the   palate   by  salts.     I   have  already  spoken  of  the 

A  A 


354  PHILOSOPHIC  NATURALIS   THEORIA 

diximus  num.  464,  quae  ad  diversum  excitandum  motum  in  papillis  palati  abunde  sufficit ; 
licet  etiam  dum  dissolvuntur,  vires  varias  pro  varia  punctorum  dispositione  exercere  debeant, 
quae  saporum  discrimen  inducant.  Odor  est  quidam  tenuis  vapor  ex  odoriferis  corporibus 
emissus,  cujus  rei  indicia  sunt  sane  multa,  nee  omnino  assentiri  possum  illi,  qui  odorem 
etiam,  ut  sonum,  in  tremore  medii  cujusdam  interpositi  censet  consistere.  Porro  quae 
evaporationum  sit  causa,  explicavimus  abunde  num.  462.  Illud  unum  hie  innuam,  errare 
illos,  uti  pluribus  ostendi  in  prima  parte  meae  dissertationis  De  Lumine,  qui  multi  sane 
sunt,  &  praestantes  Physici,  qui  odoribus  etiam  tribuunt  proprietatem  lumini  debitam, 
ut  nimirum  eorum  densitas  minuatur  in  ratione  reciproca  duplicata  distantiarum  a  corpore 
odorifero.  Ea  proprietas  non  convenit  omnibus  iis,  quae  a  dato  puncto  diffunduntur  in 
sphaeram,  sed  quae  diffunduntur  cum  uniformi  celeritate,  ut  lumen.  Si  enim  concipiantur 
orbes  concentrici  tenuissimi  datae  crassitudinis ;  ii  erunt  ut  superficies,  adeoque  ut  quadrata 
distantiarum  a  communi  centre,  ac  densitas  materiae  erit  in  ratione  ipsorum  reciproca  : 
si  massa  sit  eadem  :  ut  ea  in  ulterioribus  orbibus  sit  eadem,  ac  in  citerioribus ;  oportet 
sane,  tota  materia,  quae  erat  in  citerioribus  ipsis,  progrediatur  ad  ulteriores  orbes  motu 
uniformi,  quo  fiet,  ut,  appellente  ad  citeriorem  superficiem  orbis  ulterioris  particula,  quae 
ad  citeriorem  citerioris  appulerat,  appellat  simul  ad  ulteriorem  ulterioris  quae  appulerat 
simul  ad  ulteriorem  citerioris,  materia  tota  ex  orbe  citeriore  in  ulteriorem  accurate  translata  : 
quod  nisi  fiat,  vel  nisi  loco  uniformis  progressus  habeatur  accurata  compensatio  velocitatis 
imminutae,  &  impeditae  a  progressu  partis  vaporum,  quae  compensatio  accurata  est  admodum 
improbabilis ;  non  habebitur  densitas  reciproce  proportionalis  orbibus,  sive  eorum  super- 
ficiebus,  vel  distantiarum  quadratis. 


De  sono  difficult^  [235]  504.  Sonus  geometricas  determinationes  admittit  plures,  &  quod  pertinet  ad 
undis  Ixcitetis,11  in  vibrationes  chordae  elasticas,  vel  campani  aeris,  vel  motum  impressum  aeri  per  tibias,  & 
fluido  elastico.  tubas,  id  quidem  in  Mechanica  locum  habet,  &  mihi  commune  est  cum  communibus  theoriis. 
Quod  autem  pertinet  ad  progressum  soni  per  aerem  usque  ad  aures,  ubi  delatus  ad  tympanum 
excitat  eum  motum,  a  quo  ad  cerebrum  propagate  idea  soni  excitatur,  res  est  multo  opero- 
sior,  &  pendet  plurimum  ab  ipsa  medii  constitutione  :  ac  si  accurate  solvi  debeat  problema, 
quo  quaeratur  ex  data  medii  fluidi  elasticitate  propagatio  undarum,  &  ratio  inter  oscillationum 
celeritates,  a  qua  multipliciter  variata  pendent  omnes  toni,  &  consonantiae,  ac  dissonantiae, 
&  omnis  ars  musica,  ac  tempus,  quo  unda  ex  dato  loco  ad  datam  distantiam  propagatur  ; 
res  est  admodum  ardua  ;  si  sine  subsidiariis  principiis,  &  gratuitis  hypothesibus  tractari 
debeat,  &  determination!  resistentiae  fluidorum  est  admodum  affinis,  cum  qua  motum  in 
fluido  propagatum  communem  habet.  Exhibebo  hie  tantummodo  simplicissimi  casus 
undas,  ut  appareat,  qua  via  ineundam  censeam  in  mea  Theoria  ejusmodi  investigationem. 


QUO  pacto  onantur  cOr    Sit  in  recta  linea  disposita  series  punctorum  ad  data  intervalla  aequalia  a  se  invicem 

undae  in   serie  con-  J    •'  ,  .  ..  .  r.  ,,  .   ..  ... 

tinua  punctorum  se  distantium,  quorum  bma  quaeque  sibi  proxima  se  repellant  vinbus,  quae  crescant  immmutis 
invicem     repeiien-  distantiis,  &  dentur  ipsae.     Concipiatur  autem  ea  series  utraque  parte  in  infinitum  producta, 

tium.  .         '   .  .    .  ...  ^     . r  .  r. 

&  uni  ex  ejus  punctis  concipiatur  externa  vi  celernme  agente  in  ipsum  multo  magis,  quam 
agant  puncta  in  se  invicem,  brevissimo  tempusculo  impressa  velocitas  quaedam  finita  in 
ejusdem  rectae  directione  versus  alteram  plagam,  ut  dexteram,  ac  reliquorum  punctorum 
motus  consideretur.  Utcunque  exiguum  accipiatur  tempusculum  post  primam  systematis 
perturbationem,  debent  illo  tempusculo  habuisse  motum  omnia  puncta.  Nam  in  momento 
quovis  ejus  tempusculi  punctum  illud  debet  accessisse  ad  punctum  secundum  post  se 
dexterum,  &  recessisse  a  sinistro,  velocitate  nimirum  in  eo  genita  majore,  quam  generent 
vires  mutuae,  quae  statim  agent  in  utrumque  proximum  punctum,  aucta  distantia  a  sinistro,  & 
imminuta  a  dextero,  qua  fiet,  ut  sinistrum  urgeatur  minus  ab  ipso,  quam  a  sibi  proximo 
secundo  ex  ilia  par^e,  &  dexterum  ab  ipso  magis,  quam  a  posteriore  ipsi  proximo,  & 
differentia  virium  producet  illico  motum  aliquem,  qui  quidem  initio,  ob  differentiam 
virium  tempusculo  infinitesimo  infinitesimam,  erit  infinities  minor  motu  puncti  impulsi, 
sed  erit  aliquis :  eodem  pacto  tertium  punctum  utraque  ex  parte  debet  illo  tempusculo 
infinitesimo  habere  motum  aliquem,  qui  erit  infinitesimus  respectu  secundi,  &  ita  porro. 


A  THEORY  OF  NATURAL  PHILOSOPHY  355 

angular  forms  of  salts,  in  Art.  464  ;  these  are  quite  sufficient  for  the  excitement  of  different 
motions  in  the  papillae  of  the  palate  ;  although,  even  when  they  are  dissolved,  they  must 
exert  different  forces  for  different  dispositions  of  the  points,  which  induce  differences  in 
taste.  Smell  is  a  sort  of  tenuous  vapour  emitted  by  odoriferous  bodies  ;  of  this  there  are 
really  many  points  in  evidence.  I  cannot  agree  altogether  with  one  who  thinks  that  smell, 
like  sound,  consists  of  a  sort  of  vibration  of  some  intervening  medium.  Moreover,  I  have 
fully  explained,  in  Art.  462,  what  is  the  cause  of  evaporations.  I  will  but  mention  here  this 
one  thing,  namely,  that,  as  I  showed  in  several  places  in  the  first  part  of  my  dissertation 
De  Lumine,  those  many  and  distinguished  physicists  are  mistaken  who  attribute  to  smell 
the  same  property  as  that  proper  to  light,  namely,  that  the  density  diminishes  in  the  inverse 
ratio  of  the  squares  of  the  distances  from  the  odoriferous  body.  That  is  a  property  that 
does  not  apply  to  all  things  that  are  diffused  throughout  a  sphere  from  a  given  point ;  but 
only  with  those  that  are  thus  diffused  with  uniform  velocity,  as  light  is.  For  if  we  imagine 
a  set  of  concentric  spherical  shells  of  given  very  small  thickness,  they  will  be  like  surfaces. 
Hence,  they  will  be  in  the  same  ratio  as  the  squares  of  the  distances  from  the  common 
centre  ;  &,  the  density  of  matter  will  be  inversely  proportional  to  them,  if  the  mass  is  the 
same.  Now,  in  order  that  it  may  be  the  same  in  the  outer  shells  as  it  is  in  the  inner,  it  is 
necessary  that  the  whole  of  the  matter  which  was  in  the  inner  shells  should  proceed  to  the 
outer  shells  with  a  uniform  motion  ;  then,  it  would  come  about  that  two  particles,  which 
have  reached  simultaneously  the  inner  &  outer  surfaces  of  the  inner  shell  respectively, 
will  reach  simultaneously  the  inner  &  outer  surfaces  of  the  outer  shell ;  &  the  whole  of 
the  matter  will  be  transferred  accurately  from  the  inner  shell  to  the  outer.  If  this  is  not 
the  case,  or,  failing  uniform  progression,  if  instead  there  is  not  an  accurate  compensation  of 
the  velocity  thus  diminished  &  hindered  by  the  advance  of  part  of  the  vapours  (&  such  an 
accurate  compensation  is  in  the  highest  degree  improbable),  then  the  density  cannot  be 
inversely  proportional  to  the  shells,  i.e.,  to  their  surfaces,  or  the  squares  of  the  distances. 

504.  Sound  admits  of  several  geometrical  determinations ;    &  matters  pertaining  to  Sound ;     difficulty 
vibrations  of  an  elastic  cord  or  bell-metal,  or  the  motion  given  to  the  air  by  flutes  &  wavese^itedln^n 
trumpets,  all  belong  to  the  science  of  Mechanics  ;  &  for  them  my  Theory  is  in  agreement  elastic  fluid, 
with  the  ordinary  theories.     But,  with  respect  to  the  progression  of  sound  through  the  air 

to  the  ears,  where  it  is  carried  to  the  ear-drum  &  excites  the  motion  by  means  of  which, 
when  propagated  to  the  brain,  the  idea  of  sound  is  produced,  the  matter  is  much  more 
laborious,  &  depends  to  a  very  large  extent  on  the  constitution  of  the  medium  itself.  If 
it  is  necessary  to  solve  the  problem,  in  which  it  is  desired  to  find  the  propagation  of  waves 
from  a  given  elasticity  of  a  fluid  medium,  &  the  ratio  between  the  velocities  of  the  oscillations 
upon  which,  in  its  manifold  variations,  depend  all  musical  sounds,  harmonious  or  discordant, 
the  whole  art  of  music,  &  the  time  in  which  a  wave  is  propagated  from  a  given  point  to  a 
given  distance  ;  then,  the  matter  is  very  hard,  especially  if  it  has  to  be  treated  without 
the  help  of  subsidiary  principles  or  unfounded  hypotheses.  It  is  closely  allied  to  the 
determination  of  the  resistance  of  fluids,  with  which  subject  it  has  common  ground  in  the 
motion  propagated  in  a  fluid.  I  will  explain  here  merely  waves  of  the  very  simplest  kind  ; 
so  that  the  manner  in  which  I  consider  in  my  Theory  such  an  investigation  should  be 
undertaken  will  be  seen. 

505.  Suppose  we  have  a  series  of  points  situated  in  one  straight  line  at  given  equal  The   manner  in 
intervals  of  distance  from  one  another  ;  &  of  these  let  any  two  consecutive  points  repel  one  ^^  i^^continu- 
another  with  forces,  which  increase  as  the  distance  decreases,  &  suppose  that  the  magnitudes  ous  series  of  points 
of  these  forces  are  also  given.     Also  suppose  that  this  series  is  continued  on  either  side  to  ^her 

infinity ;  &  suppose  that,  by  means  of  an  external  force  acting  very  quickly  on  one  of  the 
points  of  the  series  much  more  than  the  points  act  upon  one  another,  there  is  impressed 
upon  it  in  a  very  short  time  a  certain  finite  velocity  in  the  direction  of  the  straight  line 
towards  either  end  of  it,  say  towards  the  right ;  then  we  have  to  consider  the  motion  of 
all  the  other  points.  No  matter  how  small  the  interval  of  time  taken,  after  the  initial 
disturbance  of  the  system,  in  that  interval  all  points  must  have  had  motion.  For,  in  any 
instant  of  that  interval  of  time,  that  point  must  have  approached  the  next  point  to  it  on 
the  right,  &  have  receded  from  the  one  on  the  left ;  a  velocity  being  generated  in  it  greater 
than  that  which  the  mutual  forces  would  give.  These  forces  immediately  act  on  the  points 
next  to  it  on  either  side,  the  distance  on  the  left  being  increased,  &  on  the  right  diminished. 
Thus,  the  point  on  the  left  will  be  impelled  by  that  point  less  than  by  the  next  one  to  it 
on  its  left,  &  the  one  on  the  right  more  than  by  the  next  one  to  the  right  of  it.  The  difference 
of  forces  will  immediately  produce  some  motion  ;  this  motion  indeed  at  first,  owing  to 
the  difference  of  forces  in  an  infinitesimal  time  being  itself  infinitesimal,  will  be  infinitely 
less  than  the  motion  of  the  point  under  the  action  of  the  external  force  ;  but  there  will 
be  some  motion.  In  the  same  way,  a  third  point  on  either  side  must  in  that  infinitesimally 
small  time  have  some  motion,  which  will  be  infinitesimal  with  respect  to  that  of  the  second  ; 


356  PHILOSOPHIC  NATURALIS  THEORIA 

Post  tempusculum  utcunque  exiguum  omnia  puncta  aequilibrium  amittent,  &  motum 
habebunt  aliquem.  Interea  cessante  actione  vis  impellentis  punctum  primum  incipiet 
ipsum  retar-[236]-dari  vi  repulsiva  secundi  dexteri  praevalente  supra  vim  secundi  sinistri, 
sed  adhuc  progredietur,  &  accedet  ad  secundum,  ac  ipsum  accelerabit  :  verum  post  aliquod 
tempus  retardatio  continua  puncti  impulsi,  &  acceleratio  secundi  reducent  ilia  ad  veloci- 
tatem  eandem  :  turn  vero  non  ultra  accedent  ad  se  invicem,  sed  recedent,  quo  recessu 
incipiet  retardari  etiam  punctum  primum  dexterum,  ac  paullo  post  extinguetur  tota 
velocitas  puncti  impulsi,  quod  incipiet  regredi  :  aliquanto  post  incipiet  regredi  &  punctum 
secundum  dexterum,  &  aliquanto  post  tertium,  ac  ita  porro  aliud.  Sed  interea  punctum 
impulsum,  dum  regreditur,  incipiet  urgeri  magis  a  primo  sinistro,  &  acceleratio  minuetur  : 
turn  habebitur  retardatio,  turn  motus  iterum  reflexus.  Dum  id  punctum  iterum  incipit 
regredi  versus  dexteram,  erit  aliquod  e  dexteris,  quod  tune  primo  incipiet  regredi  versus 
sinistram,  &  dum  per  easdem  vices  punctum  impulsum  iterum  reflexit  motum  versus 
sinistram,  aliud  dexterum  remotius  incipiet  regredi  versus  ipsam  sinistram,  ac  ita  porro 
motus  semper  progreditur  ad  dexteram  major,  &  incipient  regredi  nova  puncta  alia  post 
alia.  Undae  amplitudinem  determinabit  distantia  duorum  punctorum,  quae  simul  eunt 
&  simul  redeunt,  ac  celeritatem  propagationis  soni  tempus,  quod  requiritur  ad  unam 
oscillationem  puncti  impulsi,  &  distantia  a  se  invicem  punctorum,  quas  simul  cum  eo  eunt, 
&  redeunt ;  &  quod  ad  dexteram  accidit  ad  sinistram.  Sed  &  ea  perquisitio  est  longe 
altioris  indaginis,  quam  ut  hie  institui  debeat ;  &  ad  veras  soni  undas  elasticas  referendas 
non  sufficit  una  series  punctorum  jacentium  in  directum,  sed  congeries  punctorum,  vel 
particularum  circumquaque  dispersarum,  &  se  repellentium. 


Solutio  difficuitatis  506.  Interea  illud  unum  adjiciam,  in  mea  Theoria  admodum  facile  solvi  difficultatem, 

pa^atione'm  ^ectiii-  quam  Eulerus  objecit  Mairanio,  explicanti  propagationem  diversorum  sonorum,  a  quibus 
neam  diversorum  diversi  toni  pendent,  per  diversa  genera  particularum  elasticarum,  quae  habentur  in  acre, 
f a^cTus  "fn^h'ac  quorurn  singula  singulis  sonis  inserviant,  ut  diversi  sunt  colorati  radii  cum  diverse  constant! 
Theoria.  refrangibilitatis  gradu,  &  colore.  Eulerus  illud  objicit,  uti  tarn  multa  sunt  sonorum 

genera,  quae  ad  nostras,  &  aliorum  aures  simul  possint  deferri,  ita  debere  haberi  continuam 
seriem  particularum  omnium  generum  ad  ea  deferenda,  quod  haberi  omnino  non  possit, 
cum  circa  globum  quenvis  in  eodem  piano  non  nisi  sex  tantummodo  alii  globi  in  gyrum 
possint  consistere.  Difficultas  in  mea  Theoria  nulla  est,  cum  particulas  aliae  in  alias  non 
agant  per  immediatum  contactum,  sed  in  aliqua  distantia,  quae  diametro  globorum  potest 
esse  major  in  ratione  quacunque  utcunque  magna.  Cum  igitur  certi  globuli  in  iisdem 
distantiis  possint  esse  inertes  respectu  certorum,  &  activi  respectu  aliorum ;  patet,  posse 
multos  diversorum  generum  globulos  esse  permixtos  ita,  ut  actionem  aliorum  sentiant 
alii.  Quin  [237]  immo  licet  activi  sint  globuli,  fieri  debet,  ut  alii  habeant  motus  conformes 
turn  eos,  qui  pendent  a  viribus  mutuis  inter  duos  globulos,  a  quibus  proveniunt  undae, 
turn  eos  qui  pendent  ab  interna  distributione  punctorum,  a  qua  proveniunt  singularum 
particularum  interni  vibratorii  motus,  &  qui  itidem  ad  diversum  sonorum  genus  plurimum 
conferre  possint,  &  dissimilium  globorum  oscillationes  se  mutuo  turbent,  similium  perpetuo 
post  primas  actiones  actionibus  aliis  conformibus  augeantur,  quemadmodum  in  consonantibus 
instrumentorum  chordis  cernimus,  quarum  una  percussa  sonant  &  reliquae.  LJbique 
libertas  motuum,  &  dispositionis,  qua?  sublato  immediate  impulsu,  &  accurata  continuitate 
in  corporum  textu,  acquiritur  ad  explicandam  naturam,  est  perquam  idonea,  &  opportuna. 


De  caiore  &  frigore .  coy.  Quod  pertinet  ad  tactiles  propnetates,  quid  sit  solidum,  fluidum,  ngidum,  molle 

materiae     cientis      ,        •  n      -i        r        -i  i  v         •  -j    i        •  -j 

caiorem  expansio  elasticum,  flexile,  fragile,  grave,  abunde  explicavimus  :    quid  laevigatum,  quid  asperum, 

orta  abeiasticitate:  per  se  patet.     Caloris  causam  repono  in  motu  vehementi  intestine  particularum  igneae, 

vek>citasJU  ut    tor-  vel  sulphureae  substantias  fermentescentis  potissimum  cum  particulis  luminis,  &  qua  ratione 

rentis  cujusdam.       j^  fierj  possit,  exposuimus.     Frigus  haberi  potest  per  ipsum  defectum  ejusmodi  substantiae, 

vel  defectum  motus  in  ipsa.     Haberi  possunt  etiam  particulae,  quae  frigus  cieant  actione 

sua,  ut  nitrosas,  per  hoc,  quod  ejusmodi  particularum  motum  sistant,  &  eas,  attractione 


A  THEORY  OF  NATURAL  PHILOSOPHY  357 

&  so  on.  Thus,  after  the  lapse  of  any  short  interval  of  time,  however  small,  all  points  will 
lose  their  equilibrium  &  have  some  motion.  Further,  the  action  of  the  force  acting  upon 
the  first  point  will  itself  begin  to  be  retarded  by  the  repulsive  force  of  the  next  point  on 
the  right  prevailing  over  the  force  from  the  next  on  the  left ;  but  it  will  still  progress, 
approach  the  second  &  accelerate  it.  However,  after  some  time,  the  continuous  retardation 
of  the  first  point,  &  the  acceleration  of  the  second,  will  reduce  them  to  the  same  velocity  ; 
&  then  they  will  no  longer  approach  one  another,  but  will  recede  from  one  another.  When 
this  recession  starts,  the  first  point  on  the  right  will  also  begin  to  be  retarded,  &  a  little 
while  afterwards  the  whole  of  the  velocity  of  the  point  impelled  by  the  external  force  will 
be  destroyed,  &  it  will  commence  to  go  backwards ;  shortly  afterwards,  the  second  point 
on  the  right  will  also  commence  to  go  backwards  ;  shortly  after  that,  the  third  point ; 
&  so  on,  one  after  the  other.  But  meanwhile,  as  it  returns,  the  point,  that  was  impelled 
by  the  external  force,  will  be  more  under  the  action  of  the  first  point  on  the  left,  & 
its  acceleration  will  be  diminished ;  there  will  follow  first  a  retardation,  &  then  once  more 
a  reversal  of  motion.  When  the  point  once  more  begins  to  move  towards  the  right,  there 
will  be  some  one  of  the  points  on  the  right,  which  then  for  the  first  time  is  beginning  to 
move  backwards  to  the  left ;  &  when,  after  the  same  changes,  the  point  impelled  once 
more  reverses  its  motion  &  moves  towards  the  left,  there  will  be  another  point  on  the  right, 
further  off,  which  will  begin  to  move  backwards  towards  the  left.  In  this  way,  the  motion 
will  always  proceed  further  to  the  right,  &  fresh  points,  one  after  the  other,  will  begin  to 
reverse  their  motion.  The  distance  between  two  points,  which  go  forward  &  backward 
simultaneously,  will  determine  the  amplitude  of  the  wave  ;  the  velocity  of  propagation  of 
sound  will  be  found  from  the  time  that  is  required  for  one  oscillation  of  the  impelled  point, 
&  the  distance  between  points,  whose  motion  backwards  &  forwards  is  simultaneous ;  & 
what  happens  on  the  right  will  also  happen  on  the  left.  But  the  investigation  is  one  of 
far  too  great  difficulty  to  be  properly  treated  here  ;  to  render  an  account  of  the  true  elastic 
waves  of  sound,  one  series  of  points  lying  in  a  straight  line  is  insufficient ;  we  must  have 
groups  of  points  or  of  particles,  scattered  in  all  directions  round  about,  &  repelling  one 
another. 

506.  I  will  add  just  one  other  thing  ;  in  my  Theory,  it  is  quite  easy  to  give  a  solution  The  solution  of  the 
of  the  difficulty,  which  Euler  brought  forward  in  opposition  to  Mairan ;    the  latter  tried  spfct^thTrectifi- 
to  explain  the  propagation  of  the  different  sounds,  upon  which  different  musical  tones  near  propagation  of 
depend,  by  the  presence  of  different  kinds  of  elastic  particles  in  the  air  ;    each  kind   of  co^es "quhe0  easfiy 
particle  was  of  service  to  the  corresponding  sound,  just  as  there  are  differently  coloured  from  my  Theory, 
rays  of  light,  having  a  constant  different  degree  of  refrangibility,  &  a  different  colour.     Euler's 

objection  was  that  there  are  so  many  kinds  of  sounds,  which  can  be  borne  simultaneously 
to  our  ears  &  to  those  of  others,  that  there  must  be  a  continuous  series  of  particles  of  all 
the  different  kinds  to  carry  these  sounds  ;  &  that  this  was  quite  impossible,  since  only  six 
spheres  could  lie  in  a  circle  in  the  same  plane  round  a  sphere.  There  is  no  such  difficulty 
in  my  Theory,  since  particles  do  not  act  upon  one  another  by  immediate  contact,  but  at 
some  distance,  such  as  can  bear  to  the  diameter  of  the  spheres  any  ratio  whatever,  however 
large.  Since,  then,  certain  little  spheres  can  be  Inert,  when  placed  at  the  same  distances, 
with  regard  to  some  &  active  with  regard  to  others,  it  is  clear  that  a  large  number  of  little 
spheres  of  different  kinds  can  be  so  intermingled  that  some  of  them  feel  the  action  of  others. 
Nay  indeed,  even  if  the  little  spheres  are  active,  there  are  bound  to  be  some  that  have 
congruent  motions ;  not  only  those  motions  which  depend  upon  the  mutual  forces  between 
two  little  spheres  by  which  waves  are  produced,  but  also  those  which  depend  on  the  internal 
distribution  of  the  points  forming  them  from  which  arise  the  internal  vibratory  motions 
of  the  several  particles.  These,  too,  may  contribute  towards  a  different  class  of  sounds  to 
a  very  great  extent ;  &  they  will  disturb  the  mutual  oscillations  of  unlike  spheres,  &,  after 
the  first  actions,  the  oscillations  of  like  spheres  will  be  increased  by  congruent  actions ; 
just  as  in  the  consonant  strings  of  instruments  we  see  that,  when  one  of  them  is  struck, 
all  the  others  sound  as  well.  The  freedom  of  motion  everywhere,  &  of  arrangement,  which 
is  acquired  by  the  removal  of  the  ideas  of  immediate  impact  &  accurate  continuity  in  the 
structure  of  bodies,  is  most  suitable  &  convenient  for  the  purpose  of  explaining  the  nature 
of  sound. 

507.  With  respect  to  tactile  properties,  we  have  had  full  explanations  of  solid,  fluid.  Heat  &  cold ;  the 
rigid,  soft,  elastic,  flexible,  fragile  &  heavy  bodies ;    what  a  smooth,  or  a  rough,  body  is,  ^atter^producing 
is    self-evident.     I  consider  the  cause  of   heat  to  consist  of  a  vigorous    internal  motion  heat    arises    from 
of  the    particles    of  fire,  or  of  a  sulphurous  substance  fermenting   more  especially  with  of^th^Lmer3*0" 
particles  of  light ;   &  I  have  shown  the  mode  in  which  this  may  take  place.     Cold  may  velocity   as '  of  a 
be  produced  by  a  lack  of  this  substance,  or  by  a  lack  of  motion  in  it.     Also  there  may  be  torrent- 
particles  which  produce  cold  by  their  own  action,  such  as  nitrous  substances,  through 

something  which  stops  the  motion  of  such  particles,  &,  as  their  attraction  overcomes  their 


358  PHILOSOPHIC  NATURALIS  THEORIA 

mutuas  ipsarum  vires  vincente,  ad  se  rapiant,  ac  sibi  affundant  quodammodo,  veluti  alligatas. 
Potest  autem  generari  frigus  admodum  intensum  in  corpore  calido  per  solum  etiam  accessum 
corporis  frigefacti  ob  solum  ejusmodi  substantiae  defectum.  Ea  enim,  dum  fermentat, 
&  in  suo  naturali  volatilizationis  statu  permanet,  nititur  elasticitate  sua  ipsa  ad  expansionem, 
per  quam,  si  in  aliquo  medio  conclusa  sit,  utcunque  inerte  respectu  ipsius,  ad  aequalitatem 
per  ipsum  diffunditur,  unde  fit,  ut  si  uno  in  loco  dematur  aliqua  ejus  pars,  statim  illuc- 
ex  aliis  tantum  devolet,  quantum  ad  illam  aequalitatem  requiritur.  Hinc  nimirum,  si 
in  acre  libero  cesset  fermentantis  ejusmodi  substantias  quantitas,  vel  per  imminutam  con- 
tinuationem  impulsuum  ad  continuandum  motum,  ut  imminuta  radiorum  Solis  copia  per 
hyemem,  ac  in  locis  remotioribus  ab  yEquatore,  vel  per  accessum  ingentis  copise  particularum 
sistentium  ejusdem  substantias  motum,  unde  fit,  ut  in  climatis  etiam  non  multum  ab 
^Equatore  distantibus  ingentia  pluribus  in  locis  habeantur  frigora,  &  glacies  per  nitrosorum, 
efHuviorum  copiam  ;  e  corporibus  omnibus  expositis  aeri  perpetuo  crumpet  magna  copia 
ejusdem  fermentescentis  ibi  adhuc,  &  elasticae  materiae  igneas  ;  &  ea  corpora  remanebunt 
admodum  frigida  per  solam  imminutionem  ejus  materiae,  quibus  si  manum  admoveamus, 
ingens  illico  ex  ipsa  manu  particularum  earundem  multitude  avolabit  transfusa  illuc,  ut 
res  ad  aequalitatem  redu-[238]-catur,  &  tarn  ipsa  cessatio  illius  intestini  motus,  qua  immuta- 
bitur  status  fibrarum  organici  corporis,  quam  ipse  rapidus  ejus  substantiae  in  aliam 
irrumpentis  torrens,  earn  poterit,  quam  adeo  molestam  experimur,  frigoris  sensationem, 
excitare. 


tione,°&naffluxu.Xa"  5°8-  Torrentis  ejusmodi  ideamhabemus  in  ipso  velocissimo  aeris  motu,  qui  si  in  aliqua 

spatii  parte  repente  ad  fixitatem  reducatur  in  magna  copia,  ex  aliis  omnibus  advolat 
celerrime,  &  horrendos  aliquando  celeritate  sua  effectus  parit.  Sic  ubi  turbo  vorticosus, 
&  aerem  inferne  exsugens  prope  domum  conclusam  transeat,  aer  internus  expansiva  sua 
vi  omnia  evertit  :  avolant  tecta,  diffringuntur  fenestras,  &  tabulate,  ac  omnes  portae,  quae 
cubiculorum  mutuam  communicationem  impediunt,  repente  dissiliunt,  &  ipsi  parietes 
nonnunquam  evertuntur,  ac  corruunt,  quemadmodum  Romae  ante  aliquot  observavimus 
annos,  &  in  dissertatione  De  Turbine  superius  memorata,  quam  turn  edidi,  pluribus  exposui. 

Attractio,  quae  po-  509.  Verum  haec  sola  substantiae  hujusce  fermentantis  expansiva  vis  non  est  satis  ad 

nfotum  'sistere"1*  rem  explicandam,  sed  requiritur  etiam  certa  vis  mutua,  qua  ejusmodi  substantia  in 
fixare  :  communi-  alias  quasdam  attrahatur  magis,  in  alias  minus,  quod  qui  fieri  possit,  vidimus,  ubi  de 
»tu^tatenfqpao<st  dissolutione,  &  praecipitatione  egimus  :  &  ejusmodi  attractio  potest  esse  ita  valida,  ut 
partem  fixatam  :  motum  ipsum  intestinum  prorsus  impediat  appressione  ipsa,  ac  fixationem  ejus  substantiae 
varia  inducat,  quae  si  minor  sit,  permittet  quidem  motus  fermentatorii  continuationem,  sed  a 
se  totam  massam  divelli  non  permittet,  nisi  accedente  corpore,  quod  majorem  exerceat 
vim,  &  ipsam  sibi  rapiat.  Hie  autem  raptus  fieri  potest  ob  duplicem  causam  ;  primo 
quidem,  quod  alia  substantia  majorem  absolutam  vim  habeat  in  ejusmodi  substantiam 
igneam,  quam  alia,  pari  etiam  particularum  numero  :  deinde,  quod  licet  ea  aeque,  vel 
etiam  minus  trahat,  adhuc  tamen  cum  utraque  in  minoribus  distantiis  trahat  plus,  in 
majoribus  minus,  ilia  habeat  ejus  substantiae  multo  minus  etiam  pro  ratione  attractionis 
suae,  quam  altera  ;  nam  in  hoc  secundo  casu,  adhuc  ab  hac  posteriore  avellerentur  particulae 
affusae  ipsius  particulis  ad  distantias  aliquanto  majores,  &  affunderentur  particulis  prioris 
substantiae,  donee  in  utravis  substantia  haberetur  aequalis  saturitas,  si  ejus  partes  inter 
se  conferantur,  &  asqualis  itidem  attractiva  vis  particularum  substantiae  igneae  maxime 
remotarum  a  particulis  utriusque  substantiae,  quibus  ea  affunditur  :  sed  copia  ipsius 
substantiae  igneae  possit  adhuc  esse  in  iis  binis  substantiis  in  quacunque  ratione  diversa  inter 
se  ;  cum  possit  in  altera  ob  vim  longius  pertinentem  certa  vis  haberi  in  distantia  majore, 
quam  in  altera,  adeoque  altitude  ejusmodi  veluti  marium  in  altera  esse  major,  minor  in 
altera,  &  in  iisdem  distantiis  possit  in  altera  haberi  ob  vim  majorem  densitas  major  sub- 
stantias ipsius  igneae  affusae,  quam  in  altera.  Ex  hisce  quidem  principiis,  ac  diversis 
combinationibus,  mirum  sa-[239]-ne,  quam  multa  deduci  possint  ad  explicationem  Naturae 
per  quam  idoneis. 

Quae  a    diffuskme  510.  Sic  etiam  ex  hac  diffusione  ad  ejusmodi  aequalitatem  eandem  inter  diversas 

consequanturpo^  ejusdem  substantias  partes,  sed  admodum  diversam  inter  substantias  diversas,  facile  intelli- 

simum  respectu  re-  gitur,  qui  fiat,  ut  manus  in  hyeme  exposita  libero  aeri  minus  sentiat  frigoris,  quam  solido 

&rrongfcfciationis.S'  cuipiam  satis  denso  corpori,  quod  ante  ipsi  aeri  frigido  diu  fuerit  expositum,  ut  marmori, 

&  inter  ipsa  corpora  solida,  multo  majus  frigus  ab  altero  sentiat,  quam  ab  altero,  ac  ab 

acre  humido  multo  plus,  quam  a  sicco,  rapta  nimirum  in  diversis  ejusmodi  circumstantiis 


A  THEORY  OF  NATURAL  PHILOSOPHY  359 

mutual  forces,  these  substances  draw  these  particles  towards  themselves  &  surround  themselves 
with  them  as  if  the  particles  were  bound  to  them.  Moreover,  a  very  intense  cold  can  be 
produced  in  a  warm  body  merely  by  the  approach  of  a  body  made  cold  by  a  mere  defect 
of  such  a  substance.  For,  the  substance,  while  it  ferments,  &  remains  in  its  natural  state 
of  volatilization,  avails  itself  of  its  own  elasticity  to  expand ;  &  thereby,  if  it  is  enclosed  in  any 
medium,  however  inert  it  may  be  with  respect  to  the  medium,  the  substance  diffuses  through 
the  medium  equally.  Hence,  it  comes  about  that,  if  from  any  one  place  there  is  taken  away 
some  part  of  the  substance,  immediately  there  flies  to  it  from  other  places  just  that  quantity 
which  is  required  for  equality.  Thus,  for  instance,  if  in  the  open  air  a  quantity  of  such  fer- 
menting substance  is  lacking,  whether  through  a  diminution  in  the  continued  impulses  neces- 
sary for  the  continued  motion,  such  as  the  diminished  supply  of  rays  from  the  Sun  in  winter, 
or  in  places  more  remote  from  the  equator,  or  whether  through  the  presence  of  a  large 
supply  of  particles  that  stop  such  motion  of  the  substance,  due  to  which  there  is,  even  in 
regions  not  far  distant  from  the  equator,  great  coldness  in  several  places,  &  ice,  through 
an  abundance  of  nitrous  exhalations ;  then,  from  all  bodies  exposed  to  such  air  there  will 
rush  forth  a  great  abundance  of  the  substance  still  fermenting  in  them,  &  of  the  elastic 
matter  of  fire.  The  bodies  themselves  will  remain  quite  cold,  merely  by  the  diminution 
of  this  matter  ;  &  if  we  touch  them  with  the  hand,  immediately  a  large  number  of  these 
particles  will  fly  out  of  the  hand  &  be  transfused  into  the  bodies,  so  as  to  bring  about  equality  ; 
&  not  only  the  cessation  of  that  internal  motion  by  which  the  state  of  the  nerves  of  the 
organic  body  is  altered,  but  also  the  rapid  rush  of  the  substance  entering  into  the  other, 
will  give  rise  to  that  feeling  of  cold  which  we  experience  so  keenly. 

508.  We  have  an  idea  of  such  a  rush  in  the  very  swift  motion  of  the  air  ;  if  the  air  in  An  illustration 
some  part  of  space  is  suddenly  reduced  to  fixation  in  large  quantities,  air  will  rush  in  violently  ^°."^  thef  fi*atlon 
from  all  other  places,  &  sometimes  produces  dreadful  effects  by  its  velocity.     Thus,  when 

a  whirlwind,  sucking  out  the  air  below,  passes  near  to  a  house  that  is  shut  up,  the  air  inside 
the  house  overcomes  everything  by  its  expansive  force  ;  roofs  fly  off,  windows  are  broken, 
the  floors  &  all  the  doors  that  prevent  mutual  communication  between  the  rooms  are 
suddenly  burst  apart,  &  the  very  walls  are  sometimes  overthrown  &  fall  down  ;  just  as 
was  seen  at  Rome  some  years  ago,  &  as  I  fully  explained  in  the  dissertation  De  Turbine 
already  mentioned,  which  I  published  at  the  time. 

509.  But  the  mere  expansive  force  of  such  a  fermenting  substance  is  insufficient  to  The  at  fraction 

explain  thoroughly  what  happens  ;    we  require  also  a  certain  mutual  force,  due  to  which  yhl.cli  can.  st°2  & 
.°     '  ,  IT  .         ,  .   ,     fix  internal  motion ; 

the  substance  is  attracted  more  by  some  bodies  &  less  by  others ;   &  the  manner  in  which  motion   shared  so 

this  can  happen  was  explained  when  we  dealt  with  solution  &  precipitation.     Such  an  aj  ^t^tio^after 

attraction  may  be  so  powerful  as  to  prevent  that  internal  motion  altogether  by  its  pressure,  a   part   is    fixed ; 

&  lead  to  fixation  of  the  substance  ;    but  if  this  is  fairly  small,  it  will  indeed  allow  some  d^e2tnkillds  of 

fermentatory  motion  to  go  on,  but  will  not  allow  the  whole  mass  to  be  broken  up,  unless 

a  body  approaches  which  exerts  a  greater  force  &  draws  the  substance  to  itself.     Now 

this  attraction  can  take  place  in  two  ways.     In  the  first,  because  one  substance  has  a  greater 

absolute  force  on  this  fiery  substance  than  another,  for  the  same  number  of  particles ;   in 

the  second,  because  although  the  one  attracts  the  substance  equally  or  even  less  than  the 

other,  yet,  since  either  of  them  attracts  it  more  at  smaller  distances  &  less  at  greater  distances, 

the  one  has  much  less  of   the   substance   in  proportion  to  its  attraction  than  the  other. 

In  this  second  case,  particles  will  still  be  torn  away  from  the  latter  body,  intermingled 

with  particles  of  the  substance,  to  distances  somewhat  greater,  &  will  be  surrounded  with 

particles  of  the  former,  until  in  both  there  will  be  an  equal  saturation  when  parts  of  it  are 

compared  with  one  another  ;  &  also  an  equal  attractive  force  for  particles  of  the  fiery  substance 

that  are  remote  from  particles  of  either  of  the  substances  by  which  it  is  surrounded.     But 

there  still  may  be  an  abundance  of  the  fiery  substance  in  each  of  the  two  substances,  in 

any  ratio,  different  for  each.     For,  in  the  one,  due  to  a  more  extended  continuation  of 

the  force,  there  may  be  had  a  given  force  at  a  greater  distance  than  in  the  other  ;   &  thus 

the  depth,  so  to  speak,  of  the  oceans  surrounding  the  one  may  be  greater  than  for  the  other  ; 

&  for  the  same  distances,  for  the  one  there  may  be,  on  account  of  the  greater  force,  a  greater 

density  of  the  affused  fiery  substance,  than  for  the  other.     From  these  principles,  &  different 

combinations  of  them,  it  is  truly  wonderful  how  many  things  can  be  derived  extremely 

suitable  to  explain  the  phenomena  of  Nature. 

510.  Thus,  from  the  principle  of  such  diffusion  tending  to  establish  the  same  equality  The    consequences 

i  vrr  r   i  °  T          i  •         T<T  r        of  this    diffusion 

between  different  parts  of  the  same  substance,  but  an  equality  that  is  quite  different  for  ten(iing    to  estab- 
different  substances,  it  is  easily  seen  how  It  comes  about  that  in  winter  the  hand  when  exposed  Hsh    equality; 
to  the  open  air,  feels  the  cold  less  than  when  exposed  to  a  solid  body  of  sufficient  density,  Batter  of  refrigera- 
such  as  marble,  which  has  previously  been  exposed  to  the  same  cold  air  for  a  long  time;  tion  &  congelation. 
&  amongst  solids,  feels  far  more  cold  from  some  than  from  others,  from  damp  air  much 
more  than  from  dry.     For,  in  different  circumstances  of  the  same  kind,  in  the  same  time, 


360  PHILOSOPHIC  NATURALIS  THEORIA 

eodem  tempore  admodum  diversa  copia  igneas  substantial,  quae  calorem  in  manu  fovebat. 
Atque  hie  quidem  &  analogiae  sunt  quaedam  cum  iis,  quae  de  refractione  diximus  :  nam 
plerumque  corpora,  quae  plus  habent  materias,  nisi  oleosa,  &  sulphorosa  sint,  majorem 
habent  vim  refractivam,  pro  ratione  densitatis  suae,  &  corpora  itidem  communiter,  quo 
densiora  sunt,  eo  citius  manum  admotam  calore  spoliant,  quae  idcirco  si  lineam  telam 
libero  expositam  aeri  contingat  in  hyeme,  multo  minus  frigescit,  quam  si  lignum,  si  marmora 
si  metalla.  Fieri  itidem  potest,  ut  aliqua  substantia  ejusmodi  substantiam  igneam  repellat 
etiam,  sed  ob  aliam  substantiam  admixtam  sibi  magis  attrahentem,  adhuc  aliquid  surripiat 
magis,  vel  minus,  prout  ejus  admixtae  substantiae  plus  habet,  vel  minus.  Sic  fieri  posset, 
ut  aer  ejusmodi  substantiam  igneam  respueret,  sed  ob  heterogenea  corpora,  quae  sustinet,  inter 
quae  inprimis  est  aqua  in  vapores  elevata,  surripiat  nonnihil ;  ubi  autem  in  ipso  volitantes 
particulae,  quae  ad  fixitatem  adducunt,  vel  expellunt  ejusmodi  substantiam  igneam,  accedant 
ad  alias,  ut  aqueas,  fieri  potest,  ut  repente  habeantur  &  concretiones,  atque  congelationes, 
ac  inde  nives,  &  grandines.  A  diffusione  vero  ad  aequalitatem  intra  idem  corpus  fieri  utique 
debet,  ut  ubi  altius  infra  Terrae  superficiem  descensum  sit,  permanens  habeatur  caloris 
gradus,  ut  in  fodinis,  ad  exiguam  profunditatem  pertinente  effectu  vicissitudinum,  quas 
habemus  in  superficie  ex  tot  substantiarum  permixtionibus  continuis,  &  accessu,  ac  recessu 
solarium  radiorum,  quae  omnia  se  mutuo  compensant  saltern  intra  annum,  antequam 
sensibilis  differentia  haberi  possit  in  profundioribus  locis  :  ac  ex  diversa  vi,  quam  diversas 
substantiae  exercent  in  ejusmodi  substantiam  igneam,  provenire  debet  &  illud,  quod 
experimenta  evincunt,  ut  nimirum  nee  eodem  tempore  aeque  frigescant  diversae  substantiae 
aeri  libero  expositae,  nee  caloris  imminutio  certam  densitatum  rationem  sectetur,  sed 
varietur  admodum  independenter  ab  ipsa.  Eodem  autem  pacto  &  alia  innumera  ex  iisdem 
principiis,  ubique  sane  conformibus  admodum  facile  explicantur. 


Eodem  pacto  expli-  511.  Patet  autem  ex  iisdem  principiis  repeti  posse  explica-[24o]-tionem  etiam  praeci- 

tem  :&priencTpia  Puorum  omnium  ex  Electricitatis  phaenomenis,  quorum  Theoriam  a  Franklino  mira  sane 
Frank  linianae  sagacitate  inventam  in  America  &  exornavit  plurimum,  &  confirmavit,  ac  promovit  Taurini 
tads"*     EIectnci"  P.  Beccaria  vir  doctissimus  opere  egregio  ea  de  re  edito  ante  hos  aliquot  annos.     Juxta 
ejusmodi  Theoriam  hue  omnia  reducuntur  :    esse  quoddam  fluidum  electricum,  quod  in 
aliis  substantiis  &  per  superficiem,  &  per  interna  ipsarum  viscera  possit  pervadere,  per 
alias  motum  non  habeat,  licet  saltern  harum  aliquae  ingentem  contineant  ejusdem  substantiae 
copiam  sibi  firmissime  adhaerentem,  nee  sine  frictione,  &  motu  intestine  effundendam, 
quarum  priora  sint  per  communicationem  electrica,  posteriora  vero  electrica  natura  sua  : 
in  prioribus  illis  diffundi  statim  id  fluidum  ad  aequalitatem  in  singulis  ;  licet  alia  majorem, 
alia  minorem  ceteris  paribus  copiam  ejusdem  poscant  ad  quandam  sibi  veluti  connaturalem 
saturitatem  :    hinc  e  duobus  ejusmodi  corporibus,  quse  respectu  naturae  suse  non  eundem 
habeant  saturitatis  gradum,  esse  alterum  respectu  alterius  electricum  per  excessum,  & 
alterum  per  defectum,  quae  ubi  admoveantur  ad  earn  distantiam,  in  qua  particulae  circa 
ipsa  corpora  diffusae,  &  iis  utcunque  adhaerentes  ad  modum  atmosphaerarum  quarundam, 
possint  agere  aliae  in  alias,  e  corpore  electrico  per  excessum  fluere  illico  ejusmodi  fluidum 
in  corpus  electricum  per  defectum,  donee  ad  respectivam  aequalitatem  deventum  sit,  in 
quo  effluxu  &  substantiae  ipsae,  quae  fluidum  dant,  &  recipiunt,  simul  ad  se  invicem  accedant, 
si  satis  leves  sint,  vel  libere  pendeant,  &  si  motus  coacervatae  materiae  sit  vehemens,  explo- 
siones  habeantur,  &  scintillae,  &  vero  etiam  fulgurationes,  tonitrua,  &  fulmina.     Hinc 
nimirum  facile  repetuntur  omnia  consueta  electricitatis  phaenomena,  praeter  Batavicum 
experimentum  phialae,  quod  multo  generalius  est,  &  in  Frankliniano  piano  aeque  habet 
locum.     Id  enim  phasnomenum  ad  aliud  principium  reducitur  :    nimirum  ubi    corpora 
natura  sua  electrica  exiguam  habent  crassitudinem,  ut  tenuis  vitrea  lamella,  posse  in  altera 
superficie  congeri  multo  majorem  ejus  fluidi  copiam,  dummodo  ex  altera  ipsi  ex  adverse 
respondente  aequalis  copia  fluidi  ejusdem  extrahatur  recepta  in  alterum  corpus  per  com- 
municationem electricum,  quod  ut  per  satis  amplam  superficiei  partem  fieri  possit,  non 
excurrente  fluido  per  ejusmodi  superficies ;   aqua  affunditur  superficiei  alteri,  &  ad  alteram 
manus  tota  apprimitur,  vel  auro  inducitur  superficies  utraque,  quod  sit  tanquam  vehiculum, 
per  quod  ipsum  fluidum  possit  inferri,  &  efferri,  quod  tamen  non  debet  usque  ad  marginem 
deduci,  ut  citerior  inauratio  cum  ulteriore  conjungatur,  vel  ad  illam  satis  accedat  :   si  enim 
id  fiat,  transfuse  statim  fluido  ex  altera  superficie  in  alteram,  obtinetur  aequalitas,  &  omnia 
cessant  electrica  signa. 


A  THEORY  OF  NATURAL  PHILOSOPHY  361 

a  different  quantity  of  the  fiery  substance  is  seized,  &  this  originally  kept  the  hand  warm. 
Here,  too,  there  are  certain  analogies  with  what  we  have  said  about  refraction.  For,  very 
many  bodies  possessing  a  considerable  amount  of  matter,  unless  they  are  oily  or  sulphurous, 
have  a  greater  refractive  force  in  proportion  to  their  density  ;  &  commonly,  too,  the  denser 
they  are,  the  more  quickly  they  withdraw  heat  from  the  hand  that  touches  them  ;  &  thus, 
if  the  hand  touches  a  linen  cloth  exposed  to  the  open  air  in  winter,  it  is  made  cold  to  a  far 
less  degree  than  it  would  be  in  the  case  of  wood,  marble,  or  metal.  Further  it  may  be 
that  some  substance  of  this  sort  even  repels  the  fiery  substance  ;  but,  owing  to  the  fact 
that  another  substance  mixed  with  it  has  a  stronger  attraction,  it  will  still  carry  off  some 
of  the  fiery  substance,  more  or  less  in  amount  according  as  there  is  more  or  less  of  the  second 
substance  mixed  with  it.  Thus,  it  might  be  the  case  that  air  would  reject  a  fiery  substance 
of  this  sort ;  but,  owing  to  the  presence  of  heterogeneous  bodies  in  it,  amongst  which 
there  is  in  particular  water  uplifted  in  the  form  of  vapour,  it  seizes  some  portion  of  it.  Also, 
when  particles  hovering  in  it,  which  either  induce  fixity,  or  repel  such  fiery  substance,  approach 
others,  like  those  of  water -vapour,  it  may  happen  that  sudden  concretions  &  congelations  take 
place  ;  &  thus  cause  snow  &  hail.  But  from  a  diffusion  tending  to  produce  equality  within 
the  same  body  it  must  come  about  that,  when  one  goes  deeper  down  beneath  the  surface 
of  the  Earth,  there  is  a  permanent  degree  of  warmth.  Thus,  in  mines,  the  effect  of  the 
vicissitudes  which  take  place  on  the  surface  owing  to  the  continual  mingling  of  so  many 
substances,  &  the  accession  &  recession  of  the  solar  rays,  only  continues  for  a  very  small 
depth  ;  for  these  all  compensate  one  another  in  the  course  of  a  year  at  any  rate,  before 
any  sensible  difference  can  be  produced  in  places  of  fair  depth.  Because  of  this,  and  also 
on  account  of  the  different  force  exerted  by  different  substances  on  this  fiery  substance, 
it  must  come  about,  as  is  proved  experimentally,  that  different  bodies  are  not  cooled  equally 
in  the  same  time  when  exposed  to  the  open  air,  nor  is  the  diminution  of  heat  in  a  fixed 
ratio  to  the  density,  but  varies  altogether  independently  of  it.  In  the  same  way,  innumerable 
other  things  can  be  quite  readily  derived  from  these  same  principles,  which  agree  with  one 
another  perfectly. 

511.  Further,  it  is  clear  that  from  these  principles  there  can  be  derived  an  explanation  Electricity  can  also 
of  all  the  chief  phenomena  in  electricity ;  the  theory  of  these,  discovered  by  Franklin  in  ^m^wa'y^Frlnk- 
America  with  truly  marvellous  sagacity,  has  been  greatly  embellished  &  confirmed,  &  even  lin's  principles  of 
further  developed  at  Turin  by  Fr.  Beccaria,  a  most  learned  man,  in  his  excellent  work  tricky C0ry  °*  el& 
on  this  subject,  published  some  years  ago.  According  to  such  theory,  all  things  reduce 
to  this ;  there  is  a  certain  electric  fluid,  which  can  in  some  substances  move  along  the  surface 
&  also  through  their  inward  parts ;  but  has  no  motion  through  others,  although  some  of 
these  at  any  rate  hold  an  abundance  of  the  substance  very  firmly  adherent  to  themselves, 
&  not  to  be  loosened  without  friction  &  internal  motion.  Of  these,  the  former  are  electric 
by  communication,  the  latter  electric  by  nature.  In  the  former,  the  fluid  is  immediately 
diffused  to  produce  equality  on  each  of  them  ;  although  some  of  them  require  more,  others 
less,  of  the  fluid  to  produce,  so  to  speak,  an  intrinsic  saturation,  other  things  being  the  same. 
Thus,  of  two  of  these  bodies,  of  which  the  saturation  corresponding  to  their  natures  is 
not  the  same,  one  will  be  electric  by  excess,  &  the  other  by  defect,  with  respect  to  one 
another.  If  these  bodies  approach  one  another  to  within  that  distance,  for  which  the 
particles  surrounding  the  bodies,  &  adhering  to  them  like  atmospheres,  can  act  upon  one 
another  ;  then,  from  the  body  that  is  electric  by  excess  this  fluid  will  immediately  flow 
towards  the  one  that  is  electric  by  defect,  until  equality  is  reached.  During  this  flow, 
the  substances  which  respectively  yield  &  receive  the  fluid  will  simultaneously  approach 
one  another,  if  they  are  light  enough,  or  if  they  are  freely  suspended ;  &  if  the  motion  of 
the  concentrated  matter  is  vigorous,  there  will  be  explosions,  &  sparks,  &  even  lightning, 
thunder,  &  thunderbolts.  Hence,  forsooth,  can  be  derived  all  the  customary  phenomena 
of  electricity,  besides  the  experiment  of  the  Leyden  Jar,  which  is  much  more  general,  & 
the  same  holds  equally  good  for  Franklin's  plate.  For  this  phenomenon  reduces  to  another 
principle  ;  namely,  that  when  bodies  that  are  naturally  electric  have  a  very  small  thickness, 
such  as  a  thin  glass  plate,  there  can  be  collected  on  one  of  the  surfaces  a  much  greater  amount 
of  the  fluid,  &  at  the  same  time  from  the  other  surface  exactly  opposite  to  it  there  can  be 
withdrawn  an  equal  amount  of  the  fluid,  &  this  may  be  passed  into  another  body  by  electric 
communication.  In  order  that  this  can  take  place  over  a  sufficiently  ample  part  of  the 
surface,  as  the  fluid  does  not  run  away  from  such  surfaces,  water  is  brought  into  contact 
with  one  surface,  &  the  other  is  pressed  with  the  whole  hand ;  or  each  of  the  surfaces  is 
overlaid  with  gold,  which  forms  as  it  were  a  medium  through  which  the  fluid  can  be  borne 
either  in  or  out.  The  gold,  however,  must  not  be  brought  right  up  to  the  edge,  so  that 
the  inner  gilding  touches  the  outer,  or  even  approaches  it  too  closely ;  for  if  this  happens, 
the  fluid  is  immediately  transfused  from  one  surface  to  the  other,  equality  is  obtained, 
&  all  signs  of  electricity  cease. 


362  PHILOSOPHISE  NATURALIS  THEORIA 

Eorum    expiicatio  tjI2.  Hujusmodi  Theoriae  ea  pars,  quae  continet  respectivam  [241]  illam  saturitatem, 

conspirat  cum  iis,  quse  diximus  de  ignea  substantia,  ubi  ipsam  respectivam  saturitatem 
abunde  explicavimus.  Dum  autem  fluidum  vi  mutua  agente  abit  ex  altera  substantia 
in  alteram  :  facile  patet,  debere  ipsa  etiam  ea  corpora,  quorum  particulse  ipsum  fluidum, 
quanquam  viribus  inaequalibus,  ad  se  trahunt,  ad  se  invicem  accedere,  ac  facile  itidem 
patet,  cur  aer  humidus,  in  quo  ob  admixtas  aquae  particulas  vidimus  citius  manum  frigescere, 
electricis  phasnomenis  contrarius  sit,  vaporibus  abripientibus  illico,  quod  in  catena  a  globi 
sibi  proximi  frictione  in  ipso  excitatum,  &  avulsum  congeritur.  Secunda  pars,  ex  qua 
Batavicum  experimentum  pendet,  &  successus  plani  Frankliniani,  aliquanto  difricilior, 
explicatione  tamen  sua  non  caret.  Fieri  utique  potest,  ut  in  certis  corporibus  ingens  sit 
ejus  substantise  copia  ob  attractionem  ingentem,  &  ad  exiguas  distantias  pertinentem, 
congesta,  quae  in  aliquanto  majore  distantia  in  repulsionem  transeat,  sed  attraction!  non 
praevalentem.  Haec  repulsio  cum  ilia  copia  materiae  potest  esse  in  causa,  ne  per  ejusmodi 
substantias  transire  possit  is  vapor,  &  ne  per  ipsam  superficiem  excurrat,  nee  vero  ad  earn 
accedat  satis ;  nisi  alterius  substantiae  adjunctae  actio  simul  superveniat,  &  adjuvet.  Turn 
vero  ubi  lamina  sit  tenuis,  potest  repulsio,  quam  exercent  particulae  fluidi  prope  alteram 
superficiem  siti,  agere  in  particulas  sitas  circa  superficiem  alteram  :  sed  adhuc  fieri  potest, 
ut  ea  non  possit  satis  ad  vincendam  attractionem,  qua  haerent  particulis  sibi  proximis  : 
verum  si  ea  adjuvetur  ex  una  parte  ab  attractione  corporis  admoti  per  communicationem 
electrici,  &  ex  altera  crescat  accessu  novi  fluidi  advecti  ad  superficiem  oppositam,  quod 
vim  ipsam  repulsivam  intendat  :  turn  vero  ipsa  praevaleat.  Ipsa  autem  praevalente, 
effluet  ex  ulteriore  superficie  ejus  fluidi  pars  novum  illud  corpus  admotum  ingressa,  ac 
ex  ejus  partis  remotione,  cessante  parte  vis  repulsivae,  quam  nimirum  id,  quod  effluit, 
exercebat  in  particulas  citerioris  superficiei,  ipsi  citeriori  superficiei  adhaereat  jam  idcirco 
major  copia  fluidi  electrici  admota  per  aquam,  vel  aurum,  donee  tamen,  communicatione 
extrorsum  restituta  per  seriem  corporum  sola  communicatione  electricorum,  defluxus  ex 
altera  superficie  pateat  ad  alteram.  Porro  explicationem  hujusmodi  &  illud  confirmat, 
quod  experimentum  in  lamina  nimis  crassa  non  succedit.  Quod  autem  per  substantiam 
natura  sua  electricam  non  permeet,  ut  aequalitatem  acquirat,  id  ipsum  provenire  posset 
ab  exigua  distantia,  ad  quam  extendatur  ingens  ejus  attractiva  vis  in  illam  substantiam 
fluidam,  &  aliquanto  majore  distantia  suarum  particularum  a  se  invicem  :  nam  in  eo  casu 
altera  particula  substantiae  per  se  electricae,  utut  spoliata  magna  parte  sui  fluidi,  non  poterit 
rapere  partem  satis  magnam  fluidi  alteri  parti  affusi,  &  appressi. 


Quod  videatur  esse  513.  Haec  quidem  an  eo  modo  se  habeant,  defmire  non  licet  [242]  nisi  &  illud  ostendatur 

discrimen     inter  sjmui  rem  aliter  se  habere  non  posse.    Sed  illud  jam  patet,  Theoriam  meam,  servato  semper 

materiam      elec-  .  *   .  .  J.      •*,... 

tricam,  &  igneam.  eodem  agendi  modo,  suggerere  ideam  earum  etiam  dispositionum  materiae,  quae  possint 
maxime  omnium  ardua,  &  composita  explicare  Naturae  phaenomena,  ac  corporum  discrimina. 
Illud  unum  hie  addam  ;  quoniam  &  ingens  inter  igneam  substantiam,  &  electricum  fluidum 
analogia  deprehenditur,  &  habetur  itidem  discrimen  aliquod  ;  fieri  etiam  posse,  ut  inter 
se  in  eo  tantummodo  discrepent,  quod  altera  sit  cum  actuali  fermentatione,  &  intestine 
motu,  quamobrem  etiam  comburat,  &  calefaciat,  &  dilatet,  ac  rarefaciat  substantias,  altera 
ad  fermentescendum  apta  sit,  sed  sine  ulla,  saltern  tanta  agitatione,  quantam  fermentatio, 
inducit  orta  ex  collisione  ingenti  mutua,  vel  ex  aliarum  admixtione  substantiarum,  quse 
sint  ad  fermentandum  idoneae. 


r»e  magnetica  yi  :  51^.  Quod  ad  magneticam  vim  pertinet,  adnotabo  illud  tantummodo,  ejus  phaenomena 

variationem  "Ven-  omnia  reduci  ad  solam  attractionem  certarum  substantiarum  ad  se  invicem.     Nam  directio, 

dere  ab  attractione,  ad  quam  &  inclinatio,  &  declinatio  reducitur,  repeti  utique  potest  ab  attractione  ipsa  sola. 

ar^m^ingeSfum  Videmus  acum  magneticam  inclinari  statim  prope  fodinas  ferri,  intra  quas  idcirco  nullus 

attrahentium.          est  pyxidis  magneticae  usus.     Si  ingens  adesset  in  ipsis  polis,  &  in  iis  solis,  massa  ferrea  ; 

omnes  acus  magneticae  dirigerentur  ad  polos  ipsos  :   sed  quoniam  ubique  terrarum  fodinae 

ferreae  habentur,  si  circa  polos  eaedem  sint  in  multo  majore  copia,  quam  alibi  ;    dirigentur 

utique  acus  polos  versus,  sed  cum  aliqua  deviatione  in  reliquas  massas  per  totam  Tellurem 

dispersas,  quae  nunquam  poterit  certum  superare  graduum  numerum  ;    nisi  plus  sequo  ad 

fodinam  aliquam  accedatur.     Declinatio  ejusmodi  diversa  erit  in  diversis  locis,  ob  diversam 


A  THEORY  OF  NATURAL  PHILOSOPHY  363 

512.  That  part  of  this  theory,  which  deals  with  the  relative  saturation,  agrees  with  Explanation  of 
what  we  have  said  with  respect  to  the  fiery  substance,  when  we  gave  a  full  explanation  of  n^Ttoory. 

its  relative  saturation.  Moreover,  when  the  fluid,  under  the  action  of  a  mutual  force, 
passes  from  one  substance  to  another,  it  is  readily  seen  that  those  bodies,  of  which  the 
particles  attract  the  fluids  to  themselves  although  with  unequal  forces,  must  also  attract 
one  another.  It  is  also  quite  clear  why  moist  air,  in  which,  on  account  of  the  admixture 
of  water  particles,  we  see  that  the  hand  is  cooled  more  rapidly,  works  in  an  exactly  opposite 
manner  with  electric  phenomena,  the  vapour  immediately  carrying  off  the  fluid,  that  is 
accumulated  in  a  chain,  after  it  has  been  excited  in  a  sphere  very  close  to  it  by  friction  & 
expelled  from  it  into  the  chain.  The  second  part,  upon  which  the  Leyden  jar  experiment 
depends,  as  also  the  Franklin  plate,  is  somewhat  more  difficult,  yet  does  not  altogether 
lack  an  explanation.  For,  it  may  indeed  be  the  case  that  in  certain  bodies  there  may  be 
concentrated  a  huge  amount  of  the  substance,  due  to  a  huge  attraction,  which  however 
only  lasts  for  exceedingly  small  distances ;  &  this  attraction  for  a  somewhat  greater  distance 
may  pass  into  a  repulsion,  without  however  overcoming  the  attraction.  This  repulsion 
taken  in  conjunction  with  the  large  amount  of  matter  may  be  for  the  purpose  of  preventing 
the  possibility  of  this  vapour  from  passing  through  such  bodies,  or  of  running  along  its 
surface,  or  even  of  approaching  very  near  to  it ;  unless  the  action  of  some  other  substance 
adjoined  simultaneously  supervenes  &  assists  it.  Then,  indeed,  when  the  plate  is  thin, 
there  can  be  a  repulsion,  exerted  by  the  particles  of  the  fluid  situated  on  one  of  the  surfaces, 
acting  on  particles  situated  near  the  other  surface.  Still,  it  may  be  that  this  is  not  sufficient 
to  overcome  the  attraction  by  which  the  particles  adhere  to  those  that  are  next  to  them. 
But,  If  this  is  assisted  on  the  one  side  by  the  attraction  of  a  body,  which  is  electric  by 
communication,  moving  towards  it,  &  on  the  other  side  it  is  increased  by  a  fresh  accession 
of  fluid  brought  up  to  the  opposite  surface,  because  this  will  augment  the  repulsive  force 
also  ;  then,  the  repulsive  force  will  overcome  the  attraction.  Now,  when  this  is  the  case, 
part  of  the  fluid  will  flow  off  from  the  further  surface  &  enter  the  new  body  that  has  been 
brought  close  to  It  ;  &  since  part  of  the  repulsive  force  ceases  owing  to  the  removal  of  this 
part  of  the  fluid  (namely,  that  repulsive  force  that  was  exerted  on  the  particles  of  the  nearer 
surface  by  the  part  of  the  fluid  that  flowed  off),  in  consequence,  there  will  adhere  to  the 
nearer  surface  a  greater  amount  of  the  electric  fluid  brought  to  it  by  the  water  or  the  gold  ; 
until,  however,  communication  being  restored  from  without  by  means  of  a  series  of  bodies 
that  are  merely  electric  by  communication,  the  flow  of  the  fluid  from  one  surface  to  the 
other  will  be  unhindered.  Moreover,  this  explanation  is  confirmed  by  the  fact  that,  if 
the  experiment  is  tried  with  a  plate  that  is  too  thick,  it  will  not  succeed.  Further,  the 
fact  that  the  fluid  will  not  pass  through  a  substance  that  is  naturally  electric,  so  that  equality 
is  produced,  can  be  produced  by  the  very  small  distance  over  which  the  huge  attractive 
force  on  the  fluid  substance  extends,  &  the  somewhat  greater  distance  of  its  particles  from 
one  another.  For,  in  this  case,  one  particle  of  the  naturally  electric  substance,  when  it 
has  lost  the  greater  part  of  its  fluid,  will  not  seize  upon  any  great  part  of  the  fluid  surrounding 
another  part,  &  in  close  contact  with  it. 

513.  Whether  these  things  are   indeed  as  stated  cannot  be  determined,  unless  it  can  The  manner  in 
be  shown  at  the  same  time  that  it  is  impossible  for  them  to  be  otherwise.     But  this  fact  which  electric  mat- 

,  ,  m,  i  .       \    .          ,  if-  i          i       ter  seems  to  difier 

is  clear,  that  my  Theory,  always  maintaining  the  same  mode  of  action,  suggests  also  the  from  fire, 
idea  of  these  dispositions  of  matter,  such  as  are  most  of  all  capable  of  explaining  the  difficult 
&  compound  phenomena  of  Nature,  &  the  differences  between  bodies.  I  will  add  but  one 
thing  further  :  since  we  can  detect  a  very  great  analogy  between  the  fiery  substance  & 
the  electric  fluid,  &  also  some  difference,  it  may  possibly  be  that  they  only  differ  from  one 
another  in  the  fact  that  the  one  occurs  in  conjunction  with  actual  fermentation  &  internal 
motion,  due  to  which  it  burns,  heats,  dilates  &  rarefies  substances ;  while  the  other  is  suitable 
to  the  setting  up  of  fermentation,  but  without  that  agitation,  or  at  least  without  an  agitation 
so  great  as  that  produced  by  fermentation  arising  from  a  very  great  mutual  collision,  or 
from  admixture  of  other  substances  that  are  liable  to  fermentation. 

514.  With  regard  to  magnetic  force,  I  will  make  but  the  one  observation,  that  all  Magnetic  force ;  its 
phenomena  with  regard  to  it  reduce  to  a  mere  attraction  of  certain  substances  for  one  tion° defends V^01n 
another.     For  direction,  to  which  both  inclination  &  declination  can  be  reduced,  can  always  the    attraction    & 
be  derived  from  attraction  alone.     We  notice  that  a  magnetic  needle  is  immediately  inclined  fa  ^'"iLsses*  at- 
near  iron  mines ;    &  therefore  within  these  a  magnetic  compass-box  is  of  no  service.     If  tracting. 

there  were  present  at  the  poles,  &  there  only,  immense  masses  of  iron,  every  magnetic  needle 
would  be  directed  towards  those  poles.  But,  since  there  are  iron  mines  in  all  lands,  if 
about  the  poles  there  were  the  same  in  much  greater  abundance  than  in  other  places,  then, 
in  every  case  needles  would  be  directed  towards  the  poles,  but  with  some  deviation  towards 
the  other  masses  scattered  over  the  whole  Earth  ;  this  deviation  could  never  exceed  a 
certain  number  of  degrees,  unless  it  was  taken  too  near  some  one  mine.  Declination  of 


364  PHILOSOPHISE  NATURALIS   THEORIA 

eorum  locorum  positionem  ad  omnes  ejusmodi  massas,  &  vero  etiam  variabitur,  cum  fodinae 
ferri  &  destruantur  in  dies  novae,  &  generentur,  ac  augeantur,  &  minuantur  in  horas. 
Variatio  intra  unum  diem  exigua  erit,  cum  eae  mutationes  in  fodinis  intra  unum  diem 
exiguae  sint  :  procedente  tempore  evadet  major,  eritque  omnino  irregularis  ;  si  mutationes, 
quae  in  fodinis  accidunt,  sint  etiam  ipsae  irregulares. 


Attract  jo  n  e  m,  &  rjr    QUod  autem  ad  attractionem  pertinet  earn  in  particulis  haberi  posse  patet,  & 

polos  cohaerere  cum     ,        J    J  ,   .  ,      f  ... 

hac  Theoria  :  diffi-  an  earum  textu  aebere  penaere  :    plunma  autem  sunt   magnetismi  phaenomena,    quae 
cuitas  de  distantia  ostendant,   mutata   dispositione  particularum  generari  magneticam  vim,   vel  destrui,  & 

ad    quam    visea  ,         ,  .        .         r  ,  .          ,       r   .      .          .  °.  ,        °     . 

extenditur:  conjee-  multo  irequentius  intendi,  vel  remitti,  cujus  rei  exempla  passim  occurrunt  apud  eos,  qm 
tura  de  soiutione  de  magneticis  agunt.  Poli  autem  ex  altera  parte  attractivi,  ex  altera  repulsivi,  qui  haben- 
tur  in  magnetismo  itidem,  cohasrent  cum  Theoria  ;  cum  virium  summa  ex  altera  parte  possit 
esse  major,  quam  ex  altera.  Difficultatem  aliquam  majorem  parit  distantia  ingens,  ad 
quam  ejusmodi  vis  extenditur  :  at  fieri  utique  id  ipsum  potest  per  aliquod  efnuviorum 
intermedium  genus,  quod  tenui-[243]-tate  sua  efTugerit  hue  usque  observantium  oculos, 
&  quod  per  intermedias  vires  suas  connectat  etiam  massas  remotas,  si  forte  ex  sola  diversa 
combinatione  punctorum  habentium  vires  ab  eadem  ilia  mea  curva  expressas  id  etiam 
phenomenon  provenire  non  possit.  Sed  ad  haec  omnia  rite  evolvenda,  &  illustranda  singu- 
lares  tractatus,  &  longae  perquisitiones  requirerentur  ;  hie  mihi  satis  est  indicasse 
ingentem  Theoriae  meae  foecunditatem,  &  usum  in  difficillimis  quibuscunque  Physicas  etiam 
particularis  partibus  pertractandis. 


Superest,  ut  postremo  loco  dicamus  hie  aliquid  de  alterationibus,  &  transforma- 
tria  diversa  prin-  tionibus  corporum.  Pro  materia  mihi  sunt  puncta  indivisibilia,  inextensa,  praedita  vi 
Crove!nire%siu1ntlS  i061^33?  &  viribus  mutuis  expressis  per  simplicem  continuam  curvam  habentem  deter- 
minatas  illas  proprietates,  quas  expressi  a  num.  117,  &  quae  per  aequationem  quoque 
algebraicam  definiri  potest.  An  haec  virium  lex  sit  intrinseca,  &  essentialis  ipsis  indivisi- 
bilibus  punctis  ;  an  sit  quiddam  substantiate,  vel  accidentale  ipsis  superadditum, 
quemadmodum  sunt  Peripateticorum  formae  substantiales,  vel  accidentales  ;  an  sit  libera 
lex  Auctoris  Naturae,  qui  motus  ipsos  secundum  legem  a  se  pro  arbitrio  constitutam  dirigat  : 
illud  non  quaero,  nee  vero  inveniri  potest  per  phaenomena,  quae  eadem  sunt  in  omnibus 
iis  sententiis.  Tertia  est  causarum  occasionalium  ad  gustum  Cartesianorum,  secunda 
Peripateticis  inservire  potest,  qui  in  quovis  puncto  possunt  agnoscere  materiam,  turn  formam 
substantialem  exigentem  accidens,  quod  sit  formalis  lex  virium,  ut  etiam,  si  velint,  destructa 
substantia,  remanere  eadem  accidentia  in  individuo,  possint  conservare  individuum  istud 
accidens,  unde  sensibilitas  remanebit  prorsus  eadem,  &  quas  pro  diversa  combinatione 
ejusmodi  accidentium  pertinentium  ad  diversa  puncta,  erit  diversa.  Prima  sententia 
videtur  esse  plurimorum  e  Recentioribus,  qui  impenetrabilitatem,  &  activas  vires,  quas 
admittunt  Leibnitiani,  &  Newtonian!  passim,  videntur  agnoscere  pro  primariis  materiae 
proprietatibus  in  ipsa  ejus  essentia  sitis.  Potest  utique  hasc  mea  Theoria  adhiberi  in 
omnibus  hisce  philosophandi  generibus,  &  suo  cujusque  peculiari  cogitandi  modo  aptari 
potest. 

Homogeneitas  ele-  517.  Hsec  materia  mihi  est  prorsus  homogenea,  quod  pertinet  ad  legem  virium,  & 

no^admittatuT  argumenta,  quae  habeo  pro  homogeneitate,  exposui  num.  92.  Siqua  occurrent  Naturae 
quanto  piures  com-  phaenomena,  quae  per  unicum  materiae  genus  explicari  non  possint  ;  poterunt  adhiberi 
versas°ieges  virium-  plura  genera  punctorum  cum  pluribus  legibus  inter  se  diversis,  atque  id  ita,  ut  tot  leges 
formam  substantia-  sint,  quot  sunt  binaria  generum,  &  praeterea,  quot  sunt  ipsa  genera,  ut  illarum  singulae 
p^se  Pen^atftiicos!  exprimant  vires  mutuas  inter  puncta  pertinentia  ad  bina  singulorum  binariorum  genera, 
si  yeiint,  agnoscere  &  harum  singulae  vires  mutuas  inter  puncta  pertinentia  ad  idem  genus,  singulae  pro  generibus 
pui  singulis.  Porro  inde  mirum  sane,  quanto  major  [244]  combinationum  numerus  oriretur, 

&  quanto  facilius  explicarentur  omnia  phaenomena.  Possent  autem  illse  leges  exponi 
per  curvas  quasdam,  quarum  aliquae  haberent  aliquid  commune,  ut  asymptoticum  impene- 
trabilitatis  arcum,  &  arcum  gravitatis,  ac  aliae  ab  aliis  possent  distare  magis,  ut  habeantur 
quaedam  genera,  &  quaedam  differentiae,  quae  corporum  elementa  in  certas  classes 
distribuerent  ;  &  hie  Peripateticis,  si  velint,  occasio  daretur  admittendi  materiam  ubique 
homogeneam,  ac  formas  substantiales  diversas,  quae  accidentalem  virium  formam  diversam 
exigant,  &  vero  etiam  piures  accidentales  formas,  quae  diversas  determinent  vires,  ex  quibus 
componatur  vis  totalis  unius  elementi  respectu  sui  similium,  vel  respectu  aliorum. 


A  THEORY  OF  NATURAL  PHILOSOPHY  365 

this  kind  will  be  different  in  different  places,  on  account  of  the  different  situation  of  these 
places  with  respect  to  all  such  masses ;  &  it  will  vary,  since  mines  of  iron  are  destroyed  & 
generated  every  day,  &  are  increased  &  diminished  hourly.  The  variation  within  a  day 
will  be  very  slight,  since  the  daily  change  in  mines  is  very  small ;  as  time  goes  on  it  becomes 
greater,  &  it  will  be  quite  irregular,  if  the  changes  that  take  place  in  mines  are  themselves 
also  irregular. 

515.  With  regard  to  attraction,  it  is  clear  that  this  can  be  had  in  the  particles,  &  that  Attraction,  &  the 
it  must  depend  upon  their  structure.     Moreover,  there  are  very  "many  phenomena  of  terft  ^v  it  h°n^y 
magnetism,  which  will  show  that  magnetic  force  is  generated  by  changing  the  disposition  Theory;    difficulty 
of  the  particles,  or  is  destroyed,  or  more  frequently  is  augmented  or  abated  ;    examples  tcTwhich6  the^force 
of  this  everywhere  come  under  the  observation  of  those  who  study  magnets.     Moreover,  extends  ;     conjee- 
poles  that  are  attractive  on  one  side  &  repulsive  on  the  other,  which  are  also  had  in  magnetism,  tionoTthS  problem! 
agree  with  my  Theory  ;    for,  the  sum  of  the  forces  on  one  side  may  be  greater  than  the 

sum  of  the  forces  on  the  other.  A  somewhat  greater  difficulty  arises  from  the  huge  distance 
to  which  this  kind  of  force  extends.  But  even  this  can  take  place  through  some  intermediate 
kind  of  exhalation,  which  owing  to  its  extreme  tenuity  has  hitherto  escaped  the  notice 
of  observers,  &  such  as  by  means  of  intermediate  forces  of  its  own  connects  also  remote 
masses  ;  if  perchance  this  phenomenon  cannot  be  derived  from  merely  a  different  combination 
of  points  having  forces  represented  by  that  same  curve  of  mine.  But  to  explain  all  these 
things  properly,  &  to  furnish  them  with  illustrations  would  require  separate  treatment 
&  long  investigations.  It  is  enough  for  me  that- 1  have  pointed  out  the  extreme  fertility 
of  my  Theory,  &  its  use  in  any  of  the  most  difficult  &  special  problems  of 
physics. 

516.  It  remains  for   me  here   to  say  a  few  words  finally  about  alterations  &  trans-  The  nature  of  mat- 
formations  of  bodies.     To  me,  matter  is  nothing  but  indivisible  points,  that  are  non-extended,  *®J'  forces  ^"Tkree 
endowed  with  a  force  of  inertia,  &  also  mutual  forces  represented  by  a  simple  continuous  different  principles 
curve  having  those  definite  properties  which  I  stated  in  Art.  117  ;  these  can  also  be  defined  m^a^e1011    they 
by  an  algebraical  equation.     Whether  this  law  of  forces  is  an  intrinsic  property  of  indivisible 

points ;  whether  it  is  something  substantial  or  accidental  superadded  to  them,  like  the 
substantial  or  accidental  shapes  of  the  Peripatetics ;  whether  it  is  an  arbitrary  law  of  the 
Author  of  Nature,  who  directs  those  motions  by  a  law  made  according  to  His  Will ;  this 
I  do  not  seek  to  find,  nor  indeed  can  it  be  found  from  the  phenomena,  which  are  the  same 
in  all  these  theories.  The  third  is  that  of  occasional  causes,  suited  to  the  taste  of  followers 
of  Descartes ;  the  second  will  serve  the  Peripatetics,  who  can  thus  admit  the  existence 
of  matter  at  any  point ;  &  then  a  substantial  form  producing  a  circumstance  (accidens) 
which  becomes  a  formal  law  of  forces ;  so  that,  if  they  wish,  having  destroyed  the  substance, 
that  the  same  circumstances  shall  remain  in  the  individual,  they  can  preserve  that  individual 
circumstance.  Hence  the  sensibility  will  remain  the  same  exactly,  &  such  as  will  be  different 
for  a  different  combination  of  such  circumstances  pertaining  to  different  points.  The 
first  theory  seems  to  be  that  of  most  of  the  modern  philosophers,  who  seem  to  admit 
impenetrability  &  active  forces,  such  as  the  followers  of  Leibniz  &  Newton  all  admit,  as 
the  primary  properties  of  matter  founded  on  its  very  essence.  This  Theory  of  mine  can 
indeed  be  used  in  all  these  kinds  of  philosophising,  &  can  be  adapted  to  the  mode  of  thought 
peculiar  to  any  one  of  them. 

517.  Matter,  in  my  opinion,  is  perfectly  homogeneous ;   what  pertains  to  the  law  of  Homogeneity  of  the 
forces,  &  the  arguments  which  I  have  in  favour  of  homogeneity,  I  have  stated  in  Art.  92.  ^Ie™e0ntts'  admitted! 
If  there  are  any  phenomena  of  Nature,  which  cannot  be  explained  by  a  single  kind  of  matter,  there  will  be  ail  the 
then  we  should  have  to  make  use  of  many  different  kinds  of  points,  with  many  laws  that 

differ  from  one  another  ;  &  this,  too,  in  such  a  manner  that  there  are  as  many  laws  as  there  laws  of  forces ; 

are  pairs  of  kinds  of  points ;  &,  in  addition,  as  many  more  as  there  are  kinds  of  points.     For, 

each  of  the  former  express  the  mutual  forces  between  the  points  belonging  to  two  kinds  substantial  form  & 

of  each  pair,  &  each  of  the  latter  the  mutual  forces  between  points  belonging  to  the  same  theseprintT3  mt° 

kind,  one  for  each  kind.     Further,  from  this  it  is  truly  marvellous  how  much  greater  the 

number  of  combinations  will  become,  &  how  much  more  easily  all  phenomena  can  be 

explained.     Moreover,  the  laws  can  be  expressed  by  curves,  some  of  which  would  have 

something  in  common,  such  as  the  asymptotic  arc  of  impenetrability,  or  the  arc  of  gravitation ; 

while  some  might  be  considerably  different  from  others,  so  that  certain  classes  &' certain 

differences  could  be  obtained,  such  as  would  distribute  the  elements  of  bodies  into  certain 

classes.     This  would  give  the  Peripatetics  an  opportunity,  if  they  so  wished,  of  admitting 

matter  that  was  everywhere  homogeneous,  as  well  as  substantial  forms  of  different  kinds 

such  as  would  necessitate  a  different  accidental  form  of  forces ;   &  also  many  accidental 

forms,  which  determine  different  laws,  from  which  is  compounded  the  total  force  of  one 

element  upon  others  similar  to  it,  or  upon  others  that  are  not. 


366  PHILOSOPHIC  NATURALIS  THEORIA 

Mira  yariatas  con-  518.  Posset  autem  admitti  vis  in  quibusdam  generibus  nulla.  &  tune  substantia  unius 

sectariorum  :  possi-  ••  -i          TI         •  ,.,..,, 

biiitas  quotiibuerit  ex   lls  genenbus  iibernme  permearet  per  substantiam  altenus  sine  ullo  occursu,  qui  m 
Mundorum  in  eo-  numero  finite  punctorum  indivisibilium  nullus  haberetur,  adeoque  transiret  cum  impene- 

dem  spatiocum  ap-          L-T  v      o  •  n-  r 

parent!    compene-  trabilitate  reah,  &  compenetratione  apparente  :    ac  posset  unum  genus   esse  colligatum 
tratione,  sine  uiia  cum  alio  per  kgem  virium,  quam  habeant  cum  tertio,  sine  ulla  lege  virium  mutua  inter 

notitia  unius  cuius-    •  i  j  1111  11  • 

vis  in  aiiis.  JPsa>  vel  possent  ea  duo  genera  nullum  habere  nexum  cum  ullo    tertio  :    atque    in    hoc 

posteriore  casu  haberi  possent  plurimi  Mundi  materiales,  &  sensibiles  in  eodem  spatio  ita 
inter  se  disparati,  ut  nullum  alter  cum  altero  haberet  commercium,  nee  alter  ullam  alterius 
notitiam  posset  unquam  acquirere.  Mirum  sane,  quam  multae  aliae  in  casibus  illius  nexus 
cujuspiam  duorum  generum  cum  tertio  combinationes  haberi  possint  ad  explicanda  Naturae 
phenomena  :  sed  argumenta,  quae  pro  homogeneitate  protuli,  locum  habent  pro  omnibus 
punctis,  cum  quibus  nos  commercium  aliquod  habere  possumus,  pro  quibus  solis  inductio 
locum  habere  potest.  An  autem  sint  alia  punctorum  genera  vel  hie  in  nostro  spatio,  vel 
alibi  in  distantia  quavis,  vel  si  id  ipsum  non  repugnat,  in  aliquo  alio  spatii  genere,  quod 
nullam  habeat  relationem  cum  nostro  spatio,  in  quo  possint  esse  puncta  sine  ulla  relatione 
distantias  a  punctis  in  nostro  existentibus,  nos  prorsus  ignoramus,  nihil  enim  eo  pertinens 
omnino  ex  Naturae  phaenomenis  colligere  possumus,  &  nimis  est  audax,  qui  eorum  omnium, 
quae  condidit  Divinus  Naturae  Fabricator  limitem  ponat  suam  sentiendi,  &  vero  etiam 
cogitandi  vim. 


ffjmfm-  m  homo~  519.  Sed  redeundo    ad    meam   homogeneorum   elementorum  Theoriam,   singulares 

gGIlCllcitlS      SUppOSl~  r  i    •  •  i  11  i  •  • 

tioneessenumerum,  corporum  formae  erunt  combmatio  punctorum  homogeneorum,  quae  habetur  a  distantus 
pun^torumSltl0nua3  ^  positionibus,  ac  praeter  solam  combinationem  velocitas,  &  directio  motus  punctorum 
sunt  radix  'omnium  singulorum  ;    pro  individuis  vero  corporum  massis  accedit  punctorum  numerus.     Dato 
did^ossint^onnae  numero  &  dispositione  punctorum  in  data  massa,  datur  radix  omnium  proprietatum,  quas 
specifics  :    unde  habet  eadem  massa  in  se,  &  omnium  relationum,  [245]  quas  eadem  habere  debet  cum 
transform  t^n S  &  a^*s  massis>   quas  nimirum  determinabunt  numeri,  &  combinationes,   ac  motus  earum, 
&  datur  radix  omnium  mutationum,  quae  ipsi  possunt  accidere.     Quoniam  vero  sunt 
qusedam  combinationes  peculiares,  quae  exhibent  quasdam  peculiares  proprietates  con- 
stantes,  quas  determinavimus,  &  exposuimus,  nimirum  suse  pro  cohaesione,  &  variis  solidi- 
tatum  gradibus,  suse  pro  fluiditate,  suae  pro  elasticitate,  suae  pro  mollitie,  suae  pro  certis 
acquirendis  figuris,  suae  pro  certis  habendis  oscillationibus,  quae  &  per  se,  &  per  vires  sibi 
affixas  diversos  sapores  pariant,  &  diversos  ordores,  &  colorum  diversas  constantes  proprie- 
tates exhibeant,  sunt  autem  aliae  combinationes,  quae  inducunt  motus,  &  mutationes  non 
permanentes,  uti  est  omne  fermentationum  genus ;    possunt  a  primis  illis  constantium 
proprietatum  combinationibus  desumi  specificae  corporum  formas,  &  differentiae,  &  per 
hasce  posteriores  habebuntur  alterationes,  &  transformationes. 

Discrimen      inter  T  -n  •  i«  •  j 

transformation  em,  52°-  inter  illas   autem  proprietates   constantes  possunt   seligi  quaedam,   quae   magis 

&  aiterationem.  constantes  sint,  &  quae  non  pendeant  a  permixtione  aliarum  particularum,  vel  etiam,  quae 
si  amittantur,  facile,  &  prompte  acquirantur,  &  illas  haberi  pro  essentialibus  illi  speciei, 
quibus  constanter  mutatis  habeatur  transformatio,  iisdem  vero  manentibus,  habeatur 
tantummodo  alteratio.  Sic  si  fluidi  particulae  alligentur  per  alias,  ut  motum  circa  se 
invicem  habere  non  possint,  sed  illarum  textus,  &  virium  genus  maneat  idem  ;  conglaciatum 
illud  fluidum  dicetur  tantummodo  alteratum,  non  vero  etiam  mutatum  specifice.  Ita 
alterabitur  etiam,  &  non  specifice  mutabitur  corpus,  aucta  quantitate  materia  igneae,  quam 
in  poris  continet,  vel  aucta  quantitate  materias  igneas,  quam  in  poris  continet,  vel  aucto 
motu  ejusdem,  vel  etiam  aucta  aliqua  suarum  partium  oscillatione,  ac  dicetur  calefactione 
nova  alteratum  tantummodo  :  &  aquae  massa,  quas  post  ebullitionem  redit  ad  priorem 
formam,  erit  per  ipsam  ebullitionem  alterata,  non  transformata  :  figurae  itidem  mutatio, 
ubi  ex  cera,  vel  metallo  diversa  fiunt  opera,  aiterationem  quandam  inducet.  At  ubi  mutatur 
ille  textus,  qui  habebatur  in  particulis,  atque  id  mutatione  constanti,  &  quae  longe  alia 
phaenomena  praebeat ;  turn  vero  dicetur  corrumpi,  &  transformari  corpus.  Sic  ubi  e 
solidis  corporibus  generetur  permanens  aer  elasticus,  &  vapores  elastici  ex  aqua,  ubi  aqua 
in  terram  concrescat,  ubi  commixtis  substantiis  pluribus  arete  inter  se  cohasreant  novo 
nexu  earum  particulae,  &  novum  mixtum  efforment,  ubi  mixti  particulas  separatae  per 
solutionem  nexus  ipsius,  quod  accidit  in  putrefactione,  &  in  fermentationibus  plurimis, 
novam  singulas  constitutionem  acquirant,  habebitur  transformatio. 


A  THEORY  OF  NATURAL  PHILOSOPHY  367 

518.  Also,  in  some  of  these  classes,  the  absence  of  any  force  may  be  admitted ;  &  then  Wonderful  variety 
the  substance  of  one  of  these  classes  will  pas°  perfectly  freely  through  the  substance  of  the  po"sfbiiityCeSof 
another  without  any  collisions  ;  for,  with  a  finite  number  of  indivisible  points,  there  would  anv     number     of 
not  be  any  ;  &  thus  the  substance  would  pass  through  with  real  impenetrability  &  apparent  ing^h^sam^space 
compenetration.     Also  it  would  be  possible  for  one  kind  to  be  bound  up  with  another  by  with  apparent  cpm- 
means  of  a  law  of  forces,  which  they  have  with  a  third,  without  any  mutual  law  of  forces  out^any'^'dication 
between  themselves,  or  these  two  kinds  might  have  no  connection  with  any  third.     In  this  of  the  presence  of 
latter  case  there  might  be  a  large  number  of  material  &  sensible  universes  existing  in  the  same  thJ  others.  * 
space,  separated  one  from  the  other  in  such  a  way  that  one  was  perfectly  independent  of 

the  other,  &  the  one  could  never  acquire  any  indication  of  the  existence  of  the  other.  It 
is  truly  wonderful  how  many  other  combination-  in  cases  of  any  such  connection  of  two 
kinds  with  a  third  could  be  obtained  for  the  purpose  of  explaining  the  phenomena  of  Nature. 
But  the  arguments,  which  I  brought  forward  in  favour  of  homogeneity,  hold  good  for  all 
points,  with  which  we  can  have  any  relation  ;  &  for  these  alone  the  principle  of  induction 
can  hold  good.  Further,  whether  there  may  be  other  kinds  of  points,  either  here  in  the  space 
around  us,  or  somewhere  else  at  a  distance  from  us,  or,  if  the  idea  of  such  a  thing  is  not 
opposed  to  our  reason,  in  some  other  kind  of  space  having  no  relation  with  our  space,  in 
which  there  may  be  points  that  have  no  distance-relation  with  points  existing  in  our  space ; 
of  this  we  can  know  nothing.  For,  nothing  relating  to  it  in  the  slightest  degree  can  be 

fathered  from  the  phenomena  of  Nature  ;  &  it  would  be  great  presumption  for  any  one  to 
x  as  a  limit  his  own  power  of  perception,  or  even  of  imagination,  of  all  the  things  that 
the  Divine  Author  of  Nature  has  founded. 

519.  But,  to  return  to  my  Theory  of  homogeneous  elements,  the  several  forms  of  Form  in  the  hyp°- 
bodies  will  consist  of  a  combination  of  homogeneous  points,  which  comes  from  their  distances  ity  uf  the  number  & 
&  positions,  &,  in  addition  to  combination  alone,  the  velocity  &  direction  of  the  motion  disposition  of   the 
of  each  of  the  points ;  also  for  individual  masses  of  bodies  there  is  to  be  added  the  number  Ftitute'the  basis  of 
of  points  that  form  them.     Given  the  number  &  disposition  of  the  points  in  a  given  mass,  ail  properties ;  what 

i_      v      •       r     11   •  •  i_-i_  •    i_  •        i  ••  OI-L  r     11    may  be  sald  about 

the  basis  or  all  its  properties,  which  are  inherent  in  the  mass,  is  given  ;   &  also  that  ot  all  specific  form ;  hence, 

the  relations  that  the  same  mass  must  have  with  other  masses ;  that  is  to  say,  those  determined  alterations  &  trans- 

by  their  numbers,  combinations  &  motions  ;    moreover,  the  basis  of  all  changes  that  can 

happen  to  it  is  also  given.     Now,  since  there  are  certain  special  combinations,  representing 

certain  special  constant  properties,  which  we  have  determined  &  explained,  namely,  those 

corresponding  to  cohesion,  &  various  degrees  of  solidity,  those  for  fluidity,  for  elasticity, 

for  softness,  for  the  acquisition  of  certain  shapes,  for  the  existence  of  certain  oscillations,  which 

combinations,  both  of  themselves  &  through  forces  connected  with  them,  produce  different 

tastes  &  different  smells,  &  exhibit  the  different  constant  properties  of  colours ;  &  also  there 

are  other  combinations  which  induce  motions  &  changes  that  are  not  permanent,  like  all 

sorts  of  fermentations  ;    there  can  be  derived  from  the  primary  combinations  of  constant 

properties  the  specific  forms  of  bodies  &  their  differences,  &  from  the  latter  also  can  be 

obtained  alterations  &  transformations  in  these  forms. 

520.  Now,  amongst  these  constant  properties  there  may  be  chosen,  some  that  are  more  Distinction  between 
constant  than  others  ;   such  as  do  not  depend  upon  admixture  with  other  particles,  &.  also  ^e^°ion?'tl0n     & 
such  as,  if  they  should  be  lost,  would  be  easily  &  quickly  acquired.     These  propertiep  could 

be  considered  to  be  essential  to  the  specie'? ;  &  if  such  properties  suffered  a  permanent  change, 
we  should  have  a  transformation  ;  whereas,  if  they  persisted,  there  would  only  be  an  alteration. 
Thus,  if  the  particles  of  a  fluid  were  bound  together  by  other  particles,  so  that  they  could 
have  no  motion  about  one  another,  but  their  structure  &  the  kind  of  forces  corresponding 
to  them  remained  the  same,  the  fluid  thus  congealed  would  be  said  to  have  been  merely 
altered,  &  not  to  have  been  specifically  changed  as  well.  Thus  also,  a  body  would  be  said 
to  be  altered,  but  not  specifically  changed,  if  the  quantity  of  fiery  matter  which  it  contains 
in  its  pores  is  increased ;  or  if  there  is  an  increase  in  its  motion,  or  even  in  some  oscillation 
of  its  parts ;  similarly,  it  would  be  said  to  be  merely  altered  by  a  fresh  accession  of  heat. 
A  mass  of  water,  which  after  ebullition  returns  to  its  original  form,  will  be  altered  by  that 
ebullition,  but  not  transformed ;  &  a  change  of  shape,  as  when  different  things  are  made 
from  wax  &  metal,  gives  some  sort  of  alteration.  But  when  the  structure  in  the  particles 
is  changed,  &  the  change  is  such  as  will  give  far  different  phenomena,  then  the  body  would 
be  said  to  have  been  broken  down  &  transformed.  Thus,  when  from  solid  bodies  there  is 
generated  a  permanent  elastic  gas,  &  elastic  vapour  from  water,  when  water  is  congealed 
into  earth,  when  several  substances  are  intimately  mixed  with  one  another  &  in  consequence 
adhere  with  some  fresh  connection  between  their  particles,  &  form  a  new  mixture,  when 
the  mixed  particles,  separated  by  the  breaking  of  this  connection,  as  happens  in  the  case 
of  putrefaction  &  in  most  fermentations,  severally  acquire  fresh  constitutions ;  then  a 
transformation  takes  place. 


368  PHILOSOPHIC  NATURALIS  THEORIA 

Quid  requireretur  521.  Si    possemus   inspicere   intimam   particularum    constitutionem,    &   textum,    ac 

formamnSPint!mam)  distinguere  a  se  invicem  particulas  ordinum  gradatim  altiorum  a  punctis  elementaribus 

unde  liceret  a  priori  ad  haec  nostra  corpora  ;    fortasse  inveniremus  aliqua  particularum  genera  [246]  ita  suae 

genera"5 ^specLsl  f°rmse  tenacia,  ut  in  omnibus  permutationibus  ea  nunquam  corrumpantur,  sed  mutentur 

quid    praestandum,  quorundam   altiorum    ordinum   particulse   per   solam    mutationem    compositionis,    quam 

ia  '      habent  a  diversa  dispositione  particularum  constantium  ordinis  inferioris ;    liceret  multo 

certius  dividere  corpora  in  suas     species,  &  distinguere  elementa  quaedam,  quse   haberi 

possent    pro    simplicibus,   &  inalterabilibus   vi   Naturae,    turn    compositiones    mixtorum 

specifkas,    &   essentiales    ab    accidentalibus    proprietatibus    discernere.     Sed   quoniam   in 

intimum  ejusmodi  textum  penetrare  nondum  licet ;    eas  proprietates  debemus  diligenter 

notare,  quae  ab  illo  intimo  textu  proveniunt,  &  nostris  sensibus  sunt  perviae,  quae  quidem 

omnes  consistunt  in  viribus,  motu,  &  mutatione  dispositionis  massularum  grandiuscularum, 

quae  sensibus  se  nostris    objiciunt,  &  constanter  habitas,  vel    facile,  &  brevi    recuperatas 

distinguere  a  transitoriis,  vel  facile,  &  constanter  amissas,  &  ex  illarum  aggregate  distinguere 

species,  hasce  vero  habere  pro  accidentalibus. 

Videri,    nos    nun-  ^22.  Verum  quod  ad  omne  hoc  areumentum  pertinet,  non  erit  abs  re,  si  postremo 

quam  posse  devenire   ,  J.  ,     ^  ,,  °.  r,  .  .      .  r .       .. 

a  d    cognoscendam  loco  hue  transferam  ex  Stayana  Recentiore  Philosophia,  ac  meis  in  earn  adnotatiombus, 
intimam    substan-  iiluj  quod  habeo  ad  versum  547  libri  i  :  "  Quamvis  intrinsecam  corporum  naturam  intueri 

tiam,  &   essentiam,  '  T.  ....        ,*T»  J*  .  r 

acdiscrimina  speci-  non  liceat,  non  esse  adjiciendum,  amrmat,  .Naturae  investigandae  studium  :  posse  ex  exterms 
fica-  illis  proprietatibus  plures  detegi  in  dies  :  ad  ipsum  summae  laudi  esse  :  ideam  sane,  quam 

habemus  confusam  substantiae  eas  habentis  proprietates,  proprietatibus  ipsis  auctis  exten- 
dimus.  Rem  illustrat  aptissimo  exemplo  ejus  substantiae,  quam  aurum  appellamus,  ac 
seriem  proprietatum  eo  ordine  proponit,  quo  ipsas  detectas  esse  verosimiliter  arbitratur  ; 
colorem  fulvum,  pondus  gravissimum,  ductilitatem,  fusilitatem,  quod  in  fusione  nihil 
amittat,  quod  rubiginem  non  contrahat.  Diu  his  tantummodo  proprietatibus  auri  sub- 
stantiam  contineri  est  creditum,  sero  additum,  solvi  per  illam,  quam  dicunt  aquam  regiam, 
&  praecipitari  immisso  sale.  Porro  &  aliae  supererunt  plurimae  ejusmodi  proprietates  olim 
fortasse  detegendae  :  quo  plures  detegimus  eo  plus  ad  confusam  illam  naturae  auri 
cognitionem  accedimus  :  a  clara,  atque  intima  ipsius  naturae  contemplatione  adhuc  absumus. 
Idem,  quod  in  hoc  vidimus  peculiar!  corpore,  de  corporis  in  genere  natura  affirmat. 
Investigandas  proprietates,  quibus  detectis  ilium  intimum  proprietatum  fontem  attingi 
nunquam  posse  :  nil  nisi  inania  proferri  vocabula,  ubi  intimae  proprietates  investigantur." 


Quid  tamen  praes-  523.  Haec  ego  quidem  ex  illo  :    turn  meam  hanc  ipsam  Theoriam  respiciens,  quam 

generaies)SSpropri^  &  ipse  Hbro  io  exposuit  nondum  edito,  sic  persequor  :  "  Quid  autem,  si  partim  observatione 

tates,  &  generaiia  partim  ratiocinatione  adhibita,  constaret  demum,  materiam  homogeneam  esse,  ac  omne 

hic^pKestitum.  e  &  discrimen  inter  corpora  prove-[247]-nire  a  forma,  nexu,  viribus,  &  motibus  particularum, 

quae  sint  intima  origo  sensibilium  omnium  proprietatum.     Ea  nostros  sensus  non  alia 

effugiunt  ratione,  nisi  ob  nimis  exiguam  particularum  molem  :    nee  nostrae  mentis  vim, 

nisi  ob  ingentem  ipsarum  multitudinem,  &  sublimissimam,  utut  communem,  virium  legem, 

quibus  fit,  ut  ad  intimam  singularum  specierum  compositionem  cognoscendam  aspirare 

non  possimus.     At  generalium  corporis  proprietatum,  &  generalium  discriminum  explica- 

tionem  libro   io  ex  intimis  iis  principiis  petitam,  exhibebimus  fortasse  non  infeliciter  : 

peculiarium  corporum  textum  olim  cognosci,  difficillimum  quidem  esse,  arbitror,  prorsus 

impossibile,  affirmare  non  ausim." 

QUO  pacto  interea  524.  Demum  ibidem  illud  addo,  quod  pertinet  ad  genera,  &  species  :  "  Interea  specificas 

species  tmgua-  naturas  sestimamus,  &  distinguimus  a  collectione  ilia  externarum  proprietatum,  in  quo 
plurimum  confert  ordo,  quo  deteguntur.  Si  quaedam  collectio,  quae  sola  innotuerat, 
inveniatur  simul  cum  nova  quadam  proprietate  conjuncta,  in  aliis  fere  aequali  numero 
cum  alia  diversa ;  earn,  quam  pro  specie  infima  habebamus,  pro  genere  quodam  habemus 
continente  sub  se  illas  species,  &  nomen,  quod  prius  habuerant,  pro  utraque  retinemus. 
Si  diu  invenimus  cunjunctam  ubique  cum  aliqua  nova,  deinde  vero  alicubi  multo  posterius 
inveniatur  sine  ilia  nova  :  turn,  nova  ilia  jam  in  naturae  ideam  admissa,  hanc  substantiam 
ea  carentem  ab  ejusmodi  natura  arcemus,  nee  ipsi  id  nomen  tribuimus.  Si  nunc  inveniretur 
massa,  quae  ceteras  omnes  enumeratas  auri  proprietates  haberet,  sed  aqua  regia  non  solveretur, 


A  THEORY  OF  NATURAL  PHILOSOPHY  369 

521.  If  we  could  inspect  the  innermost  constitution  of  particles  &  their  structure,  What  is   required 
&  distinguish  particles  from  one  another  &  separate  them  into  classes,  step  by  step  of  hieher  *°  enaj)le  ?s  to  look 

,  j?  *-  .  11-  i  i  *•      \        i\   r~     i    °  m">  the  innermost 

orders,  from  elementary  points  up  to  our  own  bodies  ;  then,  perhaps,  we  should  find  some  constitution,™ 
classes  of  particles  to  be  so  tenacious  of  their  form  that  in  all  changes  they  would  never  order  .tl£lt  j^0"^,1* 
be  broken  down  ;    but  the  particles  of  higher  orders  would  be  changed  by  mere  change  fronTfirst  principles 
of  the  composition  that  they  have  owing  to  a  different  disposition  of  the  particles  of  a  lower  *?  reduc«  matter  to 
order  from  which  they  are  formed.     It  would  then  be  possible  to  divide  with  far  greater  what  is  to  be  done! 
certainty  bodies  into  their  species,  &  to  distinguish  certain  elements  which  could  be  taken  since  such  a  thing  is 
as    the    simple    elements,  unalterable  by  any  force  in  Nature ;    &  then  to  distinguish 
the  specific  &  essential  compositions  of  mixtures  from   accidental  properties.     But,  since 
we  cannot  as  yet  penetrate  into  the  innermost  structure  of  this  sort,  we  must  carefully 
observe  those  properties,  that  arise  from  this  innermost  structure,  &  are  accessible  to  our 
senses ;    these  indeed  all  consist  of  the  forces,  motion  &  change  of  disposition  of  those 
comparatively  large,  though  really  small,  masses  that  meet  our  senses ;  &  we  must  distinguish 
between  those  properties  that  are  constantly  possessed,  or  easily  &  quickly  recovered,  & 
those  that  are  transitory,  or  easily  lost  &  lost  for  good  ;  &  from  the  aggregate  of  the  former 
to  distinguish  the  species,  while  considering  the  latter  as  accidental  properties. 

522.  But,  with  respect  to  all  this  argument,  it  will  not  be  out  of  place  if,  in  the  last  it  is  thus  to  be  seen 
place,  I  here  quote  from  Stay's  Recentior  Pbilosopbia,  &  my  notes  thereon,  that  which  I  tha:t  we  J*11  ne7U 

f  -r,      t    -T       tc   AII          i  •  .  arrive    at    a     full 

have  written  on  verse  547  of  Book  1.        Although  we  cannot  peer  into  the  intrinsic  nature  knowledge  of    the 
of  bodies,  the  endeavour  to  investigate  Nature,  he  states,  must  not  be  abandoned.     Many  "™ermost  &  essen- 

,  .  i    i   -i      r  i  mi  .     .       •       ,  '    tial    substance,    or 

things  can  be  detected  daily  from  those  external  properties.     This  is  worthy  of  all  praise ;  the  distinction  be- 

we  truly  extend  the  idea,  which  we  have  in  a  confused  form  of  a  substance  possessing  these  tween  sPecies- 

properties,  if  the  properties  are  increased.     He  illustrates  the  matter  with  a  very  fitting 

example  of  the  substance,  which  we  call  gold,  &  enunciates  the  series  of  properties  in  the 

order  in  which  he  considers  that  in  all  probability  they  were  detected  : — yellow  colour, 

very  heavy  weight,  ductility,  fusibility,  that  nothing  is  lost  in  fusion,  that  it  does  not  rust. 

For  a  long  time  it  was  believed  that  the  substance  of  gold  was  comprised  in  these  properties 

only  ;  later,  there  was  added,  that  it  was  dissolved  by  what  Is  called  aqua  regia,  &  precipitated 

from  the  solution  by  salt.     Moreover,  there  will  be  in  addition  very  many  other  properties 

of  this  kind,  perhaps  to  be  detected  in  the  future  ;   &  the  more  of  these  we  find  out,  the 

nearer  we  shall  approach  to  that  hazy  knowledge  of  the  nature  of  gold ;    but  we  are  still 

far  from  obtaining  a  clear  &  intimate  view  of  this  nature.     He  asserts  the  same  thing  about 

the  nature  of  a  body  In  general,  as  we  have  seen  in  the  case  of  this  particular  body.     He* 

states  that  the  properties  should  be  investigated,  although  from  their  detection  the  inmost 

source  of  the  properties  can  never  be  reached ;  that  nothing  except  empty  words  can  be 

produced,  when  fundamental  properties  are  investigated." 

523.  These  were  my  words  in  that  book ;    then  considering  my  own  Theory,  which  What  may,   how- 
he  also  explained  in  Book  10,  not  yet  published,  I  went  on  thus  : — "  But  what  if,  partly  euShed^thareC'ard 
by  observation  &  partly  by  using  deduction,  it  should  finally  be  established  that  matter  to  general  properties 
is  homogeneous,  &  that  all  distinction  between  bodies  comes  from  form,  connection,  forces,  *  iefen<has     been 
&  motions  of  the  particles,  such  as  may  be  the  fundamental  origin  of  all  sensible  properties  ?  done  'in  this  work. 
These  escape  our  senses  for  no  other  reason  than  the  exceedingly  small  volume  of  the 

particles ;  nor  are  they  beyond  the  powers  of  our  intelligence,  except  on  account  of 
their  huge  number,  &  the  very  complicated,  though  general,  law  of  forces.  Owing 
to  these,  we  cannot  hope  to  obtain  an  intimate  knowledge  of  the  composition  of  each 
species.  But  we  will  present,  perhaps  not  unsuccessfully,  in  book  10,  an  explanation 
of  the  general  properties  of  a  body  &  the  general  distinctions  between  them,  derived  from 
such  fundamental  principles.  I  consider  that  the  attainment  of  a  knowledge  of  the  structure 
of  particular  bodies  in  the  future  will  be  very  difficult ;  that  it  will  be  altogether  impossible, 
I  will  not  dare  to  assert." 

1524.  Lastly,  I  add  this  in  the  same  connection,  relating  to  classes  &  species  : — "  Amongst  T  n  e      manner, 

11-  '         .  •/-  „     v     '         -i      i  e  i  11        •          t  3,    amongst      other 

other  things,  we  estimate  specific  natures,  &  distinguish  them  from  the  collection  ot  external  things,  in  which  we 
properties ;  &  In  this  the  order  in  which  they  are  detected  is  of  special  assistance.  If  any  shall  distinguish 
collection,  which  had  alone  been  observed,  should  be  discovered  conjoined  with  some  fresh 
property,  &  in  others  of  nearly  equal  number  conjoined  with  something  different ;  then 
that,  which  we  had  considered  as  a  fundamental  species,  we  should  now  consider  as  a  class 
containing  within  it  both  these  species  ;  &  the  name  that  they  had  originally,  we  should 
retain  for  both  species.  If  for  some  time  we  found  it  conjoined  with  some  fresh  property, 
&  then  at  another  time  much  later  it  is  found  without  that  fresh  property ;  then,  this 
fresh  property  being  admitted  into  the  idea  of  nature,  we  should  exclude  the  substance 
lacking  this  property  from  a  nature  of  this  kind,  &  should  not  give  it  that  name.  If  now 
a  mass  should  be  found,  which  had  all  the  other  enumerated  properties  of  gold,  but  was 
not  dissolved  by  aqua  rcgia,  we  should  say  that  it  was  not  gold.  If  at  the  beginning  it  was 

P  P 


370  PHILOSOPHIC  NATURALIS  THEORIA 

earn  non  esse  aurum  diceremus.  Si  initio  compertum  esset,  alias  ejusmodi  massas  solvi, 
alias  non  solvi  per  aquam  regiam,  sed  per  alium  liquorem,  &  utrumque  in  sequali  fere  earum 
massarum  numero  notatum  esset,  putatum  fuisset,  binas  esse  auri  species,  quarum  altera 
alterius  liquoris  ope  solveretur." 

Haec  ego  ibi ;    unde  adhuc  magis  patet,  quid  specifics  formae  sint,  &  inde,  quid  sit 
transformatio.     Sed  de  his  omnibus  jam  satis. 


A  THEORY  OF  NATURAL  PHILOSOPHY  371 

discovered  that  certain  masses  of  the  same  sort  were  dissolved  by  aqua  regia,  but  that  others 
were  not,  but  were  dissolved  by  another  liquid ;  &  each  of  the  two  phenomena  was 
observed  in  an  approximately  equal  number  of  masses ;  then,  it  would  be  considered  that 
there  were  two  sorts  of  gold,  &  that  one  sort  was  dissolved  by  one  liquid,  &  the  other  by 
the  other." 

Those  are  my  words ;  &  from  them  it  can  be  easily  seen  what  specific  forms  are ; 
&  from  that,  what  transformation  is.     But  I  have  now  said  sufficient  on  the  point. 


[248]  APPENDIX 

AD  META  PHYSIC  AM  PERTINENS 

DE  ANIMA  ET  DEO 

525'  Quae  Pertment  a&  discrimen  animae  a  materia,  &  ad  modum,  quo  anima  in  corput 
sit  addita.'  3git,  rejecta  Leibnitianorum  harmonia  praestabilita,  persecutus  jam  sum  in  parte  prima 

a  num.  153.  Hie  primum  &  id  ipsum  discrimen  evolvam  magis,  &  addam  de  ipsius  animae, 
&  ejus  actuum  vi,  ac  natura,  nonnulla,  quae  cum  eodem  operis  argumento  arctissime  con- 
nectuntur  :  turn  ad  eum  colligendum,  qui  semper  maximus  esse  debet  omnium  philosophic- 
arum  meditationum  fructus,  nimirum  ad  ipsum  potentissimum,  ac  sapientissium  Auctorem 
Naturae  conscendam. 

Discrimen  inter  am-  r26.  Imprimis  hie  iterum  patet,  quantum  discrimen  sit  inter  corpus.  &  animam,  ac 

mam  &  corpus  :in.-'  .     *     ..     .   A  .  ....-?  .  ,  ' 

hoc  omnia  peragi  inter  ea,  quae  corporeae  matenae  tnbuimus,  &  quae  in  nostra  spintuali  substantia  expenmur. 
per  distantias  lo-  j^j  omnia  perfecimus  tantummodo  per  distantias  locales,  &  motus,  ac  per  vires,  quae  nihil 

cales,      motus,     ac  r  .  .    ,  .  i  i        i          •  i  t 

vires  inducentes  ahud  sunt,  nisi  determmationes  ad  motus  locales,  sive  ad  mutandas,  vel  conservandas  locales 
motum  locaiem.  distantias  certa  lege  necessaria,  &  a  nulla  materias  ipsius  libera  determinatione  pendentes. 
Nee  vero  ullas  ego  repraesentativas  vires  in  ipsa  materia  agnosco,  quarum  nomine  haud 
scio,  an  ii  ipsi,  qui  utuntur,  satis  norint,  quid  intelligant,  nee  ullum  aliud  genus  virium, 
aut  actionum  ipsi  tribuo,  praster  illud  unum,  quod  respicit  locaiem  motum,  &  accessus 
mutuos,  ac  recessus. 

in      anima      nos          527.  At  in  ea  nostra  substantia,  qua  vivimus,  nos  quidem  intimo  sensu,  &  reflexione, 

expend  sensationes,     i      i  t»    j  •  •  o  •  i  j«   • 

&  cogitationes,  ac  duplex  aliud  operationum  genus  expenmur,  &  agnoscimus,  quarum  alterum  dicimus 
voiitiones:  vim  sensationem,  alterum  coeitationem,  &  volitionem.  Profecto  idea,  quam  de  illis  habemus 

esse  in  nobis  inna-    ..  -  •  i  t  T  i     •  i  11  11* 

tam,  qua  videamus  mtimam,  &  prorsus  cxperimcntalem,  est  longe  diversa  ab  idea,  quam  habemus,  locahs 
nanim  discrimina,  &  distantiae,  &  motus.  Et  quidem  illud  mihi,  ut  in  prima  parte  innui,  omnino  persuasum 

relationem        quam  .  .  .  j  •  •  j  o     -11  11 

habent  ad  sutetan-  est,  inesse  animis  nostns  vim  quandam,  qua  ipsas  nostras  ideas,  &  illos,  non  locales,  sed 
tias,  a  quibus  animasticos  motus,  quos  in  nobis  ipsis  inspicimus,  intime  cognoscamus,  &  non  solum  similes 

procedunt  essential-          •,•     •      *vi  •  j«  •          r  ..... 

fter  diversas.  •  a  dissimihbus  possimus  discerncrc,  quod  omnino  facimus,  cum  post  equi  visi  ideam,  se 
nobis  idea  piscis  objicit,  &  hunc  dicimus  non  esse  equum  ;  vel  cum  in  [249]  primis  prin- 
cipiis  ideas  conformes  affirmando  conjungimus,  difformes  vero  separamus  negando  ;  verum 
etiam  ipsorum  non  localium  motuum,  &  idearum  naturam  immediate  videamus,  atque 
originem  ;  ut  idcirco  nobis  evidenter  constet  per  sese,  alias  oriri^in  nobis  a  substantia  aliqua 
externa  ipsi  animo,  &  admodum  discrepante  ab  ipso,  utut  etiam  ipsi  conjuncta,  quam 
corpus  dicimus,  alias  earum  occasione  in  ipso  animo  exurgere,  atque  enasci  per  longe  aliam 
vim  :  ac  primi  generis  esse  sensationes  ipsas,  &  directas  ideas,  posterioris  autem  omne 
reflexionum  genus,  judicia,  discursus,  ac  voluntatis  actus  tam  varies  :  qua  interna  evidentia, 
&  conscientia  sua  illi  etiam,  qui  de  corporum,  de  aliorum  extra  se  objectorum  existentia 
dubitare  vellent,  ac  idealismum,  &  egoismum  affectant,  coguntur  vel  inviti  internum 
ejusmodi  ineptissimis  dubitationibus  assensum  negare,  &  quotiescunque  directe,  &  vero 
etiam  reflexe,  ac  serio  cogitant,  &  loquuntur,  aut  agunt,  ita  agere,  loqui,  cogitare,  ut  alia 
etiam  extra  se  posita  sibi  similia,  &  spiritualia,  &  materialia  entia  agnoscant  :  neque  enim 
libros  conscriberent,  &  ederent,  &  suam  rationibus  confirmare  sententiam  niterentur ; 
nisi  illis  omnino  persuasum  esset,  existere  extra  ipsos,  qui,  quae  scripserint,  &  typis 
vulgaverint,  perlegant,  qui  eorum  rationes  voce  expressas  aure  excipiant,  &  victi  demum 
se  dedant. 

Duo  genera  actuum 

no^^pWs^icfmus!  52^'  ^  vero  ex  motibus  quibusdam  localibus  in  nostro  corpore  factis  per  impulsum 
sensationes,  &  cogi-  ab  externis  corporibus,  vel  per  se  etiam  eo  modo,  quo  ab  externis  fierent,  ac  delatis  ad 
c^^as^ss^mus  cerebrum  (in  eo  enim  alicubi  videtur  debere  esse  saltern  praecipua  sedes  animae,  ad  quam 
etiam  sine  corpore  nimirum  tot  nervorum  fibrae  pertingunt  idcirco,  ut  impulsiones  propagatae,  vel  per  succurn 

37? 


APPEND  I  X 

RELATING    TO    METAPHYSICS 
THE  MIND  AND  GOD 

525.  What  relates  to  the  distinction  between  the  mind  &  matter,  &  the  manner  in  The  theme  of  this 
which  the  mind  acts  on  the  body,  I  have  already  investigated  in  the  First  part,  from  Art.  r^o'rf'for  adding 
153  on,  after  rejecting  the  pre-established  harmony  of  the  followers  of  Leibniz.     Here  I  will  it. 

first  of  all  consider  more  fully  this  distinction  ;  &  I  will  add  something  with  regard  to  the 
mind  itself,  the  force  of  its  actions,  &  its  nature ;  these  are  closely  connected  with  the  very 
theme  of  this  work.  After  that,  I  will  proceed  to  consider  that  which  always  ought  to  be 
the  most  profitable  of  all  philosophical  meditations,  namely,  the  power  &  wisdom  of  the 
Author  of  Nature. 

526.  Here,  in  the  first  place,  it  is  clear  how  great  a  distinction  there  is  between  the  Distinction      be- 
body  &  the  mind,  &  between  those  things  that  we  term  corporeal  matter  &  those  which  ^f^ty*.  lathis 
we  feel  in  our  spiritual  substance.     In  Art.  153,  we  did  everything  by  the  sole  means  of  everything    is   ac- 
local  distances  &  motions,  &  by  forces  that  are  nothing  else  but  propensities  to  local  motions,  ^°™a3n1cse^ed  b&  'mo- 
or propensities  to  change,  or  preserve,  local  distances  in  accordance  with  a  certain  necessary  tions,  &  forces  mdu- 
law  ;  &  these  do  not  depend  on  any  free  determination  of  the  matter  jtself.     But  I  do  not  cins local  motlons- 
recognize  any  representative  forces  in  matter  itself — I  do  not  know  whether  those,  who 

use  the  term,  are  really  sure  of  what  they  mean  by  it — nor  do  I  attribute  to  it  any  other 
type  of  forces  or  actions  besides  that  one  which  has  to  do  with  local  motions  &  mutual 
approach  &  recession. 

527.  But  in  this  substance  of  ours,  by  which  we  live,  we  feel  &  recognize,  by  an  inner  in    the    mind  we 
sense  &  thought,  another  twofold  class  of  operations  ;    one  of  which  we  call  sensation,  &  *£ei  ^sensations, 
the  other  thought  or  will.     Without  any  doubt,  the  idea  which  we  have  within  us,  which  pose ;  force  is  in- 
is  altogether  the  result  of  experience,  of  the  former,  is  far  different  to  that  which  we  have  ^which'we'see  the 
of  local  distance  &  motion.     Indeed  I  am  quite  of  the  opinion,  as  I  remarked  in  the  First  differences  between 
Part,  that  there  is  in  our  minds  a  certain  force,  by  means  of  which  we  obtain  full  cognition  ^^o^tha't^tey 
of  our  very  ideas  &  those  non-local,  but  mental,  motions  that  we  observe  in  our  own  selves ;  bear  to  essentially 
&  we  can  distinguish  between  like  &  unlike,  as  we  assuredly  do,  when  after  the   idea  gta^ge^m 

of  a  horse  that  has  been  seen  there  presents  itself  the  idea  of  a  fish,  &  we  say  that  this  is  they  proceed. 

not  a  horse ;    or  when,  in  elementary  principles,  we  join  together  affirmatively  like  ideas, 

&  separate  unlike  ideas  with  a  negation.     Indeed,  we  also  see  immediately  the  nature  & 

origin  of  these  non-local  motions  &  ideas.     Hence,  it  is  self-evident  to  us  that  some  of  them 

arise  through  a  substance  external  to  the  mind,  &  altogether  different  from  it,  but  yet  in 

connection  with  it,  which  we  call  the  body ;   &  that  others  take  rise  from  direct  encounter 

with  the  mind  itself,  &  spring  from  a  far  different  force.     We  see  that  to  the  first  class 

belong  sensations  &  direct  ideas,  &  to  the  second  all  kinds  of  reflections,  decisions,  trains 

of  reasoning,  &  the  numerous  different  acts  of  the  will.     By  this  internal  evidence,  &  their  own 

consciousness,  even  those,  who  would  like  to  doubt  the  existence  of  bodies,  &  other  objects 

external  to  themselves,  &  affect  idealism  &  egoism,  are  forced  to  refuse,  though  unwillingly, 

their  inward  assent  to  such  very  absurd  doubts.     As  often  as  directly,  or  even  reflectively 

&  seriously,  they  think,  speak,  or  act,  they  are  forced  so  to  act,  speak,  or  think,  that  they 

recognize  other  entities  situated  external  to  themselves,  which  are  like  to  themselves,  both 

spiritual  &  material.     For,  they  would  not  write  &  publish  books,  or  try  to  corroborate  their 

theory  with  arguments ;    unless  they  were  fully  persuaded  that,  external  to  themselves, 

there  exist  those  who  will  read  what  they  have  written  &  published  in  printed  form,  & 

those  who  will  hear  the  reasons  they  have  spoken,  &  at  length  acknowledge  themselves 

convinced. 

528.  Now,  certain  local  motions  in  our  body  are  engendered  by  impulse  from  external  J^^"^  ^e  ~..T. 
bodies,  or  even  self-produced  by  the  manner  in  which  they  come  from  without,  &  these  ceive  in  ourselves, 
are  carried  to  the  brain.     For  in  the  brain,  somewhere,  it  seems  that  the  seat  of  the  mind  ^fought*  10orS  'wilt 
must  be  situated ;   &  that  is  why  so  many  nerve-fibres  extend  to  it,  so  that  the  impulses  which  we  can  exer- 
can  be  carried  to  it,  propagated  either  by  a  volatile  juice  or  by  rigid  fibres  in  all  directions,  ?^e  even   wlthou 

373 


374  PHILOSOPHIC  NATURALIS  THEORIA 

volatilem,  vel  per  rigidas  fibras  quaquaversus  deferri  possint,  &  inde  imperium  in  universum 
exerceri  corpus)  exurgunt  motus  quidam  non  locales  in  animo,  nee  vero  liberi,  &  ideae 
coloris,  saporis,  odoris,  soni,  &  vero  etiam  doloris,  qui  oriuntur  quidem  ex  motibus  illis 
localibus  ;  sed  intima  conscientia  teste,  qua  ipsorum  naturam,  &  originem  intuemur,  longe 
aliud  sunt,  quam  motus  ipsi  locales  :  sunt  nimirum  vitales  actus,  utut  non  liberi.  Praeter 
hos  autem  in  nobis  ipsis  illud  aliud  etiam  operationum  genus  perspicimus  cogitandi,  ac 
volendi,  quod  alii  &  brutis  itidem  attribuunt,  cum  quibus  illud  primum  operationum 
genus  commune  nobis  esse  censent  jam  omnes,  praeter  Cartesianos  paucos,  Philosophi  : 
nam  &  Leibnitiani  brutis  ipsis  animam  tribuunt,  quanquam  non  immediate  agentem  in 
corpus  :  sed  ex  iis,  qui  ipsam  cogitandi,  &  volendi  vim  brutis  attribuunt,  in  iis  agnoscunt 
passim  omnes,  qui  sapiunt,  nostra  inferiorem  longe,  &  ita  a  materia  pendentem,  ut  sine 
ilia  nee  vivere  possint,  nee  agere  ;  dum  nostras  animas  etiam  a  corpore  separatas  credimus 
posse  eosdem  seque  cogitationis,  &  volitionis  actus  exercere. 

Si   ea  brutis  con-  [250]  529.  Porro    ex    his,     qui     cogitationem,    &    voluntatem     brutis     attribuunt,    alii 

hnj^rfectiora1tnltiis  utli°Lue  gcneri  applicant  nomen  spiritus,  sed  distinguunt  diversa  spirituum  genera,  alii 

essedebeant,  &  quid  vocem  spiritualis  substantise  tribuunt  illis  solis,  quae  cogitare,  &  velle  possint  etiam  sine 

"e  s^rttus        ullo  nexu  cum  corpore  &  sine  ulla  materiae  organica  dispositione,  &  motu,  qui  necessarius 

est  brutis,  ut  vivant.    Atque  id  quidem  admodum  facile  revocari  potest  ad  litem  de  nomine, 

&  ad  ideam,  quae  affigatur  huic  voci  spiritus,  vel  spiritualis,  cujus  vocis  latina  vis  originaria 

non  nisi  tenuem  flatum   significat  :    nee    magna    erit   in  vocum  usurpatione  difficultas  ; 

dummodo  bene  distinguantur  a  se  invicem  materia  expers  omni  &  sentiendi,  &  cogitandi, 

ac  volendi  vi,  a  viventibus  sensu  praeditis  ;   &  in  viventibus  ipsis  anima  immortalis,  ac  per 

se  ipsam  etiam  extra  omne  organicum  corpus  capax  cogitationis,  &  voluntatis,  a  brutis 

longe    imperfectioribus,    vel    quia    solum    sentiendi   vim    habeant    omnis    cogitationis,  & 

voluntatis  expertia,  vel  quia,  si  cogitent,  &  velint,  longe  imperfectiores  habeant  ejusmodi 

operationes,    ac   dis^oluto    per   organici  corporis   corruptionem   nexu  cum  ipso   corpore, 

prorsus  dispereant. 


inter  c-0>  Ceterum  longe  aliud  profecto  est  &  tenuitas  lamellae,   quae  determinat  hunc 

motus,      a      quibus  .  JJ  ...  P  r,.  ,         n  '.  *  . 

idea  excitatur,  &  potms,  quam  ilium  coloratum  radium  ad  renexionem,  ut  ad  oculos  nostros  devemat,  in 
ideam     ipsam:  quo  sensu  adhibet  coloris  nomen  vulgus,  &  opifices  ;  &  dispositio  punctorum  componentium 

quatuor  acceptiones    ^       .      .  .    .  .°  .  '          ff..  '  .£..       .     r  r    . 

vocis  color.  particulam  lumims,  quae  certum  ipsi  conciliat  reirangibilitatis  gradum,  certum  in  certis 

circumstantiis  intervallum  vicium  facilioris  reflexionis,  &  facilioris  transmissus,  unde  fit, 
ut  certam  in  oculi  fibris  impressionem  faciat,  in  quo  sensu  nomen  coloris  adhibent  Optici  ; 
&  impressio  ipsa  facta  in  oculo,  &  propagata  ad  cerebrum,  in  quo  sensu  coloris  nomen 
Anatomici  usurpare  possunt  ;  &  longe  aliud  quid,  &  diversum  ab  iis  omnibus,  ac  ne  analogum 
quidem  illis,  saltern  satis  arcto  analogiae,  &  omnimodae  similitudinis  genere,  est  idea  ilia, 
quae  nobis  excitatur  in  animo,  &  quam  demum  a  prioribus  illis  localibus  motibus  determi- 
natam  intuemur  in  nobis  ipsis,  ac  intima  nostra  conscientia,  &  animi  vis,  de  cujus  vera  in 
nobis  ipsis  existentia  dubitare  omnino  non  possumus,  evidentissima  voce  admonent  ea 
de  re,  &  certos  nos  reddunt. 


Commercium    ani-  r^  porro  commercium  illud  inter  animam,  &  corpus,  quod  unionem  appellamus, 

mae     cum     corpore       .      fj.         .  ,.  ,  .  ,.  ,     r 

continere  triaiegum  tria  habet  inter  se  diversa  legum  genera,  quarum  bina  sunt  prorsus  diversa  ab  ea  etiam, 
genera:  quae   sint  quae  habetur  inter  materiae  puncta,  tertium  in  aliquo  genere  convenit  cum  ipsa,  sed  ita 

pnora  duo.  J1  ........  ,.r  .   v  ?      •  •  •+.       T>  • 

longe  in  alns  plunmis  ab  ea  distat,  ut  a  material!  mechamsmo  pemtus  remotum  sit.  rnores 
sunt  in  ordine  ad  motus  locales  organici  nostri  corporis,  vel  potius  ejus  partis,  sive  ea  sit 
fluidum  quoddam  tenuissimum,  sive  sint  solidae  fibrae  ;  &  ad  motus  non  locales,  sed 
animasticos  nostri  a-[25i]-nimi,  nimirum  ad  excitationem  idearum,  &  ad  voluntatis  actus. 
Utroque  legum  genere  ad  quosdam  motus  corporis  excitantur  quidam  animi  actus,  & 
vice  versa,  &  utrumque  requirit  inter  cetera  positionem  certam  in  partibus  corporis  ad 
se  invicem,  &  certam  animae  positionem  ad  ipsas  :  ubi  enim  laesione  quadam  satis  magna 
organici  corporis  ea  mutua  positio  partium  turbatur,  ejusmodi  legum  observantia  cessat  : 
nee  vero  ea  locum  habere  potest,  si  anima  procul  distet  a  corpore  extra  ipsum  sita. 

in  aitero  ex  iis  532.  Sunt  autem  ejusmodi  legum  duo  genera  :    alterum  genus  est  illud,  cujus  nexus 

A^co^uTnecSsS-  est  necessarius,  alterum,  cujus    nexus  est  liber  :    habemus  enim  &  liberos,  &  necessaries 
ius,  in  aitero  liber  :  motus,  &  saepe  fit,  ut  aliquis  apoplexia  ictus  amittat  omnem,  saltern  respectu  aliquorum 
exponuntur  ambo.     membrorum,  facultatem  liberi  motus  ;   at  necessarios,  non  eos  tantum,  qui  ad  nutritipnem 
pertinent,  &  a  sola  machina  pendent,  sed  &  eos,  quibus  excitantur  sensationes,  retineat. 


APPENDIX  375 

&  from  it  control  can  be  exercised  over  the  whole  body.  From  these  local  motions  there 
arise  certain  non-local  motions  in  the  mind,  that  are  not  indeed  free  motions,  such  as  the 
ideas  of  colour,  taste,  smell,  sound,  &  even  grief,  all  of  which  indeed  arise  from  such  local 
motions.  But,  on  the  evidence  of  our  inner  consciousness,  by  means  of  which  we  observe 
their  nature  &  origin,  they  are  something  far  different  to  these  local  motions ;  that  is  to 
say,  they  are  vital  actions,  although  not  voluntary.  Besides  these  we  also  perceive  in  our 
own  selves  that  other  kind  of  operations,  those  of  thinking  &  willing.  This  kind  some 
people  also  attribute  to  brutes  as  well ;  &  all  philosophers,  except  a  few  of  the  Cartesians, 
already  believe  that  the  first  kind  of  operations  is  common  to  the  brutes  &  ourselves.  The 
followers  of  Leibniz  attribute  a  mind  even  to  the  brutes,  although  one  that  does  not  act 
directly  on  the  body.  But  of  those  who  attribute  to  the  brutes  the  power  of  thinking  & 
willing,  all  those  that  have  any  understanding  admit  that  in  the  brutes  it  is  far  inferior  to 
our  own  ;  &  so  dependent  on  matter,  that  without  it  they  cannot  live  or  act ;  while  they 
believe  that  our  minds,  even  if  separated  from  the  body,  are  capable  of  exercising  the  same 
acts  of  thought  &  will  just  as  well. 

529.  Again,  of  those  who  attribute  to  brutes  the  power  of  thought  &  will,  some  apply  if  these  powers  are 
to  either  class  the  term  "  spirit,"  but  distinguish  between  two  different  kinds  of  spirits ;  ^e  ^rStls^they 
others  attribute  the  name  of  spiritual  substance  to  those  only  that  can  think  &  will  without  must  be  much  more 
any  connection  with  the  body,  &  without  any  organic  disposition  of  matter,  &  the  motion 

that  is  necessary  to  the  brutes  in  order  that  they  may  live.  This  may  quite  easily  be  reduced 
to  a  quarrel  over  a  mere  term,  &  the  idea  that  is  assigned  to  the  word  spirit,  or  spiritual, 
of  which  the  original  Latin  signification  is  merely  "  a  tenuous  breath."  There  will  not  be 
any  great  difficulty  over  the  use  of  the  terms,  so  long  as  matter  (which  is  devoid  of  all  power 
of  feeling,  thinking  &  willing)  &  living  things  possessed  of  feeling  are  carefully  distinguished 
from  one  another  ;  &  also  amongst  living  things,  the  immortal  mind,  &,  on  account  of  it, 
in  addition  also  every  organic  body  capable  of  thinking  &  willing,  from  the  far  more  imperfect 
brutes ;  either,  because  they  have  the  power  of  feeling  only,  &  are  unable  to  think  or  will ; 
or  because,  if  they  do  think  &  will,  they  have  these  powers  far  more  imperfectly,  &,  if  the 
connection  with  the  body  is  destroyed  by  some  corruption  of  the  organic  body,  they  perish 
altogether. 

530.  Besides,  there  is  certainly  a  very  great  difference  between  thinness  of  the  plate,  Distinction      be- 
which  determines  one  coloured  ray  of  light  rather  than  another  to  be  reflected,  so  that  it  *ween,  .the  m°tjon 

,.,',.  °  ,  ,  „     by  which    an  idea 

comes  to  the  eyes,  in  which  sense  ordinary  people  &  craftsmen  use  the  term  colour;    &  is  excited  &  the  idea 
the  disposition  of  the  points  forming  a  particle  of  light,  to  which  corresponds  a  definite  itself;  four  accepca- 

"  ,        ..  .....     r    ..  .  °     .  r  °  ,   '-    .  .   .      -r  i       r          r  tions   of  the   term 

degree  of  refrangibility,  &  in  certain  circumstances  a  definite  interval  between  the  fits  of  colour. 
easier  reflection  &  easier  transmission,  whence  there  arises  the  fact  that  it  makes  a  definite 
impression  upon  the  nerves  of  the  eyes,  in  which  sense  the  term  colour  is  used  by  investigators 
in  Optics ;  &  the  impression  itself  that  is  made  upon  the  eyes,  &  propagated  to  the  brain, 
in  which  sense  anatomists  may  employ  the  term  ;  &  something  far  different,  &  of  a  diverse 
nature  to  all  the  foregoing,  being  not  even  analogous  to  them,  or  only  with  a  kind  of  analogy, 
&  total  similitude  that  is  sufficiently  close,  is  the  idea  itself,  which  is  excited  in  our  minds, 
&  which,  determined  at  length  by  the  former  local  motions,  we  perceive  within  ourselves ; 
&  our  inner  consciousness,  &  the  force  of  the  mind,  concerning  the  existence  of  which  within 
us  there  cannot  be  the  slightest  doubt,  warn  us  with  no  uncertain  voice  about  the  matter, 
&  make  us  acquainted  with  it. 

;-?i.  Now,  the  intercourse  between  the  mind  &  the  body,  which  we  term  union,  has  The  intercourse  of 

i  !•     i        c  i  i-n-  r  i  „       f     i  i  •         V/T  the  mind  with  the 

three  kinds  of  laws  different  from  one  another  ;    &  of  these,  two  are  also  quite  different  body    contains 
also  from  that  which  obtains  between  points  of  matter  ;  while  the  third  in  some  sort  agrees  three  kinds  of  laws ; 

.....  -,.._  -  .    \  ,  ....  ,  the  nature  of   the 

with  it,  but  is  so  far  different  from  it  m  very  many  other  ways  that  it  is  altogether  remote  first  two. 
from  any  material  mechanism.  The  two  former  are  especially  applicable  to  local  motions, 
of  our  organic  bodies,  or  rather  of  part  of  them,  whether  that  part  consists  of  a  very  tenuous 
fluid,  or  of  solid  fibres ;  &  to  motions  that  are  not  local  motions,  but  to  mental  motions 
of  our  minds,  such  as  the  excitation  of  ideas,  &  acts  of  the  will.  According  to  each  of  these 
laws,  certain  acts  of  the  mind  are  transmitted  to  certain  motions  of  the  body,  &  vice  versa  ; 
&  each  kind  demands,  amongst  other  things,  a  certain  relative  situation  of  parts  of  the  body, 
&  a  certain  situation  of  the  mind  with  regard  to  these  parts.  For,  when  this  mutual 
situation  between  the  parts  is  sufficiently  disturbed  by  a  sufficiently  great  lesion  of  the 
organic  body,  observance  of  these  laws  ceases ;  nor  indeed  does  it  hold,  if  the  mind  is  far 
away  from  the  body  situated  outside  it.  in  one  of  these,  the 

532.  Moreover,  of  such  laws  there  are  two  kinds ;  the  one  kind  is  that  in  which  the  connection  between 
connection  is  necessary,  while  in  the  other  the  connection  is  free.  For,  we  have  both  j^dy  is  Of  a 
necessary  &  free  motions ;  &  it  often  happens  that  one  who  is  stricken  with  apoplexy  loses  necessary  nature, 
all  power  of  free  motions,  at  least  with  respect  to  some  of  his  limbs ;  while  he  retains  the  £ee  ;°  explanation 
necessary  motions,  not  only  those  which  relate  to  nutrition,  &  depend  solely  upon  a  mechanism,  °f  each  of  them. 


3/6  PHILOSOPHIC  NATURALIS  THEORIA 

Unde  apparet  &  illud,  diversa  esse  instrumenta,  quibus  ad  ea  duo  diversa  motuum  genera 
utimur.  Quanquam  &  in  hoc  secundo  legum  genere  fieri  posset,  ut  nexus  ibi  quidem 
aliquis  necessarius  habeatur,  sed  non  mutuus.  Ut  nimirum  tota  libertas  nostra  consistat 
in  excitandis  actibus  voluntatis,  &  eorum  ope  etiam  ideis  mentis,  quibus  semel  libero 
animastico  motu  intrinseco  excitatis,  per  legem  hujus  secundi  generis  debeant  illico  certi 
locales  motus  exoriri  in  ea  corporis  nostri  parte,  quae  est  primum  instrumentum  liberorum 
motuum,  nulli  autem  sint  motus  locales  partis  ullius  nostri  corporis,  nullae  ideae  nostrae 
mentis,  qux  animum  certa  lege  determinent  ad  hunc  potius,  quam  ilium  voluntatis  liberum 
actum  ;  licet  fieri  possit,  ut  certa  lege  ad  id  inclinent,  &  actus  alios  aliis  faciliores  reddant, 
manente  tamen  semper  in  animo,  in  ipsa  ilia  ejus  facultate,  quam  dicimus  voluntatem, 
potestate  liberrima  eligendi  illud  etiam,  contra  quod  inclinatur,  &  efficiendi,  ut  ex  mera 
sua  determinatione  praeponderet  etiam  illud,  quod  independenter  ab  ea  minorem  habet 
vim.  In  eodem  autem  genere  nexus  quidam  necessarii  erunt  itidem  inter  motus  locales 
corporis,  ac  ideas  mentis,  cum  quibusdam  indeliberatis  animi  affectionibus,  quae  leges, 
quam  multas  sint,  quam  variae,  &  an  singula  genera  ad  unicam  aliquam  satis  generalem 
reduci  possint,  id  vero  nobis  quidem  saltern  hue  usque  est  penitus  inaccessum. 


533-  Tertium  legum  genus  convenit  cum  lege  mutua  punctorum  in  hoc  genere,  quod 
nexu  mutuo  inter  ad  motum  localem  pertinet  animae  ipsius,  ac  certam  ejus  positionem  ad  corpus,  &  ad  certam 
mUIquo  a^ecT^i'ui^  organorum  dispositionerri.  Durante  nimirum  dispositione,  a  qua  pendet  vita,  anima 
mum  difierat.  necessario  debet  mutare  locum,  dum  locum  mutat  corpus,  atque  id  ipsum  quodam  necessario 

nexu,  non  libero  :  si  enim  praeceps  gravitate  sua  corpus  ruit,  si  ab  alio  repente  impellitur, 
si  vehitur  navi,  si  ex  ipsius  ani-[252]-mae  voluntate  progreditur,  moveri  utique  cum  ipso 
debet  necessario  &  anima,  ac  illam  eandem  respectivam  sedem  tenere,  &  corpus  comitari 
ubique.  Dissolute  autem  eo  nexu  organicorum  instrumentorum,  abit  illico,  &  a  corpore, 
jam  suis  inepto  usibus,  discedit.  At  in  eo  haec  virium  lex  localem  motum  animae  respiciens 
plurimum  differt  a  viribus  materiae,  quod  nee  in  infinitum  protenditur,  sed  ad  certam 
quandam  satis  exiguam  distantiam,  nee  illam  habet  tantam  reciprocationem  determinationis 
ad  accessum,  &  recessum  cum  tot  illis  limitibus,  vel  saltern  nullum  earum  rerum  habemus 
indicium.  Fortasse  nee  in  minimis  distantiis  a  quovis  materiae  puncto  determinationem 
ullam  habet  ad  recessum,  cum  potius  ipsa  compenetrari  cum  materia  posse  videatur  :  nam 
ex  phaenomenis  nee  illud  certo  colligi  posse  arbitror,  an  cum  ullo  materiae  puncto  com- 
penetretur.  Deinde  nee  hujusmodi  vires  habet  perennes,  &  immutabiles,  pereunt  enim 
destructa  organizatione  corporis,  nee  eas  habet,  cum  suis  similibus,  nimirum  cum  aliis 
animabus,  cum  quibus  idcirco  nee  impenetrabilitatem  habet,  nee  illos  nexus  cohaesionum, 
ex  quibus  materiae  sensibilitas  oritur.  Atque  ex  iis  tarn  multis  discriminibus,  &  tarn 
insignibus,  satis  luculenter  patet,  quam  longe  haec  etiam  lex  pertinens  ad  unionem  animae 
cum  corpore  a  materiali  mechanismo  distet,  &  penitus  remota  sit. 


Ubisit  sedesanimsE,  53^.  Ut>i  sit  animae  sedes,  ex  puris  pbcenomenis  certo  nosse  omnino  non  possumus  :   an 

"  nimirum  ea  sit  praesens  certo  cuidam  punctorum  numero,  &  toti  spatio  intermedio  habens 
virtualem  illam  extensionem,  quam  num.  84  in  primis  materiae  elementis  rejecimus,  an 
compenetretur  cum  uno  aliquo  puncto  materiae,  cui  unita  secum  ferat  &  necessaries  illos, 
&  liberos  nexus,  ut  vel  illud  punctum  cum  aliis  etiam  legibus  agat  in  alia  puncta  quaedam, 
vel  ut,  enatis  certis  quibusdam  in  eo  motibus,  caetera  fiant  per  virium  legem  toti  materiae 
communem  ;  an  ipsa  existat  in  unico  puncto  spatii,  quod  a  nullo  materiae  puncto  occupetur, 
&  inde  nexum  habeat  cum  certis  punctis,  respectu  quorum  habeat  omnes  illas  motuum 
localium,  &  animasticorum  leges,  quas  diximus  ;  id  sane  ex  puris  Nature  phanomenis,  & 
vero  etiam,  ut  arbitror,  ex  reflexione,  &  meditatione  quavis,  quae  fiat  circa  ipsa  phenomena, 
nunquam  nobis  innotescet. 

Demonstratur  id 
ipsum  producendo, 
quid  oporteret  nosse  .  .  .  .  ,  . 

ad  resoivendam  535.  Nam  ad  id  determmandum  ex  phaenomenis  utcunque  consideratis,  oporteret 

-  nosse,  an  ea  phaenomena  possint  haberi  eadem  quovis  ex  iis  modis,  an  potius  requiratur 
aliquis  ex  iis  determinatus  ut  conjunctio,  localis  etiam,  animae  cum  magna  corporis  parte, 


APPENDIX  377 

but  also  those  by  which  sensations  are  excited.  From  which  it  is  also  clear  that  the 
instruments  which  we  employ  to  produce  the  two  different  kinds  of  motions  must  be 
different.  Also,  although  in  the  second  kind  of  these  laws  it  may  happen  that  there  is, 
even  in  it,  some  sort  of  necessary  connection,  yet  it  is  not  a  mutual  connection.  Thus, 
the  whole  of  our  power  of  free  action  consists  of  the  excitation  of  acts  of  the  will,  &  by 
means  of  these  of  ideas  of  the  mind  also  ;  once  these  have  been  excited  by  a  free  &  intrinsic 
motion  of  the  mind,  owing  to  a  law  of  this  second  kind  there  must  immediately  arise  certain 
local  motions  in  that  part  of  the  body  which  is  the  prime  instrument  of  free  motions ; 
but  there  may  be  no  motions  of  any  part  of  the  body,  no  motions  of  the  mind,  which 
determine  the  mind  to  this  rather  than  to  that  free  act  of  the  will.  It  may  happen,  possibly, 
that  by  a  certain  law  there  is  an  inclination  to  one  thing  &  that  the  motions  produce  some 
acts  more  easily  than  others ;  &  yet,  because  there  always  remains  in  the  mind  &  that 
faculty  of  it  which  we  call  the  will  a  perfectly  free  power  of  choosing  even  that  thing  against 
which  it  is  naturally  inclined,  there  will  even  be  a  power  of  bringing  it  about  that,  due 
merely  to  its  own  determination,  the  thing,  which  independently  of  this  determination 
would  have  the  less  force,  will  preponderate.  However,  in  this  same  kind  of  law,  there 
will  be  also  certain  connections  of  the  necessary  type  between  the  local  motions  of  the  body 
&  the  ideas  of  the  mind,  together  with  some  involuntary  affections  of  the  mind  ;  &  how 
many  of  these  laws  there  may  be,  &  how  different  they  may  be,  &  whether  all  the  several 
kinds  can  be  reduced  to  a  single  law  of  fair  generality,  is  indeed,  at  least  up  till  now,  quite 
impossible  to  determine. 

533.  The  third  kind  of  law  agrees  with  the  mutual  law  of  points  in  the  fact  that  it  The     points      in 
pertains  to  local  motion  of  the  mind  itself,  to  a  definite  position  which  it  has  with  regard  %££hot  ^  a^£ 
to  the  body,  &  to  the  definite  arrangement  of  the  organs.     Thus,  while  the  arrangement  with   the    mutual 
persists,  upon  which  life  depends,  the  mind  must  of  necessity  change  its  position,  as  the  tweenCtl<points  b  of 
body  changes  its  position,  &  that  on  account  of  some  connection  of  the  necessary  type,  matter:  and  those 
&  not  a  free  connection.     For,  if  the  body  rushes  headlong  through  its  own  gravity,  or  UJ^I*  £0£  most 
is  vigorously  impelled  by  another,  or  if  it  is  borne  on  a  ship,  or  if  it  progresses  through  the 

will  of  the  mind  itself,  in  every  case  the  mind  also  must  necessarily  move  along  with  the 
body,  &  keep  to  its  seat  with  respect  to  the  body,  &  accompany  the  body  everywhere.  But 
if  this  connection  of  the  organic  instruments  is  dissolved,  straightway  it  goes  off  &  leaves 
the  body  which  is  now  useless  for  its  purposes.  But  this  law  of  forces  governing  the 
local  motion  of  the  mind  differs  greatly  from  the  law  of  forces  between  points  of  matter 
in  this,  that  it  does  not  extend  to  infinity,  but  only  to  a  fairly  small  distance,  &  that  it  does 
not  contain  that  great  alternation  of  propensity  for  approach  &  recession,  going  with 
as  many  limit-points ;  or  at  least  we  have  no  indication  of  these  things.  Perhaps  too,  even 
at  very  small  distances  from  any  point  of  matter,  it  has  no  propensity  for  recession,  since 
it  seems  rather  to  have  a  power  of  compenetration  with  matter.  For,  I  do  not  think  that 
it  can  with  certainty  be  decided  from  phenomena,  whether  there  is  compenetration  with 
any  point  of  matter  or  not.  Secondly,  it  has  no  lasting  &  unvarying  forces  of  this  kind ; 
for  they  are  destroyed  as  soon  as  the  organization  of  the  body  is  destroyed ;  nor  are  there 
forces  with  things  like  itself,  that  is  to  say  other  minds,  &  so  there  can  be  no  impenetrability 
existing  between  them  ;  nor  can  there  be  those  connections  of  cohesion  from  which  the 
sensibility  of  matter  arises.  From  the  number  of  these  differences  &  special  characteristics, 
it  is  fairly  evident  how  far  even  this  law  pertaining  to  the  union  of  the  mind  with  the  body 
differs  from  a  material  mechanism,  &  that  it  is  something  of  quite  a  different  nature. 

534.  We  are  quite  unable  to   ascertain  with  any  certainty  from  phenomena  alone  the  it  is  not  possible 
position  of  the  seat  of  the  mind.     That  is  to  say,  we  cannot  ascertain  whether  it  is  present  f*™e  toPdetemSn8 
in  any  definite  number  of  points,  &  has  such  a  virtual  extension  through  the  whole  of  the  the  position  of  the 
intermediate  space,   as,  in  Art.  84,  we  rejected  in  the  case  of  the  primary  elements  of  sea  ° 

matter.     It  cannot  be  ascertained  whether  it  has  compenetration  with  some  one  point  of 

matter,  &,  united  with  this,  bears  along  with  itself  those  necessary  &  free  motions,  so  that 

either  this  point  acts  on  certain  other  points  with  even  other  laws,  or  so  that,  certain  definite 

motions  being  produced  in  this  point,  others  take  place  on  account  of  the  law  of  forces 

that  is  common  to  the  whole  of  matter.     It  cannot  be  ascertained  whether  it  exists  in  a 

single  point  of  space,  which  is  unoccupied  by  any  point  of  matter,  &  on  that  account  has 

a  connection  with  certain  definite  points,  with  respect  to  which  it  has  all  those  laws  of 

local  &  mental  motions,  of  which  we  have  spoken.     We  can  never    become    acquainted 

with  any  of  these  points  from  the  phenomena  of  Nature  alone  certainly,  &  indeed,  as  I  think, 

neither  can  we  by  reflection  or  any  consideration  whatever,  that  may  be  made  with  regard  xhis  is  proved  by 

to  these  phenomena.  setting  forth  what 

535.  For,  in  order  to  determine  it  from  any  consideration  of  phenomena  in  any  way.  Wnown  mV0rder  to 
it  would  be  necessary  to  know  whether  these  phenomena  could  happen  in  any  of  these  obtain  a  solution  of 
ways,  or  rather  some  particular  one  of  them  is  required,  determined  as  a  conjunction,  also  phenomena!" 


378  PHILOSOPHIC  NATURALIS   THEORIA 

vel  etiam  cum  toto  corpora.  Ad  id  autem  cognoscendum  oporteret  distinctam  habere 
notitiam  earum  legum,  quas  secum  trahit  conjunctio  animae  cum  corpora,  &  totius 
dispositionis  punctorum  omnium,  quae  corpus  constituunt,  ac  legis  virium  mutuarum 
inter  materias  puncta,  turn  etiam  ha-[253]-bere  tantam  Geometriae  vim,  quanta  opus  est 
ad  determinancies  omnes  motus,  qui  ex  sola  mechanica  distributione  eorundem  punctorum 
oriri  possint.  lis  omnibus  opus  esset  ad  videndum,  an  ex  motibus,  quos  anima  imperio 
suae  voluntatis,  vel  necessitate  suae  naturae  induceret  in  unicum  punctum,  vel  in  aliqua 
determinata  puncta,  consequi  deinde  possent  per  solam  legem  virium  communem  punctis 
materiae  omnes  reliqui  spirituum,  &  nervorum  motus,  qui  habentur  in  motibus  nostris 
spontaneis,  &  omnes  motus  tot  particularum  corporis,  ex  quibus  pendent  secretiones, 
nutritio,  respiratio,  ac  alii  nostri  motus  non  liberi.  At  ilk  omnia  nobis  incognita  sunt, 
nee  ad  illud  adeo  sublime  Geometrise  genus  adspirare  nobis  licet,  qui  nondum  penitus 
determinare  potuimus  motus  omnes  trium  etiam  massularum,  quae  certis  viribus  in  se 
invicem  agant. 

• 

Faisitas  plurium  536.  Fuerunt,  qui  animam  concluserint  intra  certam  aliquam  exiguam  corporis  nostri 

opmionum  de  ejus  particulam,  ut  Cartesius  intra  glandulam  pinealem  :    at  deinde  compertum  est,  ea  parte 

sede :    non  proban,    ±  '..  °  f  .    .  .       .  .        ^ ......  . 

earn  non  extendi  sola  non  contmeri  :  nam  ea  parte  dempta,  vita  superfuit  :  sic  sine  pineali  glandula  aliquando 
per  totum  corpus.  vitam  perdurasse,  compertum  jam  est,  ut  animalia  aliqua  etiam  sine  cerebro  vitam  produx- 
erunt.  Alii  diffusionem  animse  per  totum  corpus  impugnant  ex  eo,  quod  aliquando 
homines,  rescissa  etiam  manu,  dixerint,  se  digitorum  dolorem  sentire,  tanquam  si  adhuc 
haberent  digitos ;  qui  dolor  cum  sentiatur  absque  eo  quod  anima  ibi  digitis  sit  praesens  : 
inde  inferri  posse  arbitrantur,  quotiescumque  digitorum  sentimus  dolorem,  illam  sentiri 
sine  praesentia  animas  in  digitis.  At  ea  ratio  nihil  evincit  :  fieri  enim  posset,  ut  ad  habendum 
prima  vice  sensum,  quern  in  digitorum  dolore  experimur,  requireretur  praesentia  animae 
in  ipsis  digitis,  sine  qua  ejus  doloris  idea  primo  excitari  non  possit,  possit  autem  efformata 
semel  per  ejusmodi  praesentiam  excitari  iterum  sine  ipsa  per  eos  motus  nervorum,  qui 
cum  motu  fibrarum  digiti  in  primo  illo  sensu  conjunct!  fuerant  :  praeterquam  quod  adhuc 
remanet  definiendum  illud,  an  ad  nutritionem  requiratur  praesentis  animae  impulsus  aliquis, 
an  ea  per  solum  mechanismum  obtineri  possit  tota  sine  ulla  animae  operatione. 


Conclusio  pro  ignor-  537.  Haec  omnia  abunde  ostendunt,  phaenomenis  rite  consultis  nihil  satis  certo  definiri 

rn^topcwJit  Sse.U°"  posse  circa  animae  sedem,  nee  ejus  diffusionem  per  magnam  aliquam  corporis  partem,  vel 
etiam  per  totum  corpus  excludi.  Quod  si  vel  per  ingentem  partem,  vel  etiam  per  totum 
corpus  protendatur,  id  ipsum  etiam  cum  mea  theoria  optime  conciliabitur.  Poterit  enim 
anima  per  illam  virtualem  extensionem,  de  qua  egimus  a  num.  83,  existere  in  toto  spatio, 
quo  continentur  omnia  puncta  constituentia  illam  partem,  vel  totum  corpus  :  atque  eo 
pacto  adhuc  magis  in  mea  theoria  differet  anima  a  materia  ;  cum  simplicia  materiae  elementa 
non  nisi  in  singulis  spatii  punctis  existant  singula  singulis  momentis  temporis,  anima  autem 
licet  itidem  sim-[2S4]-plex,  adhuc  tamen  simul  existet  in  punctis  spatii  infinitis^conjungens 
cum  unico  momento  temporis  seriem  continuam  punctorum  spatii,  cui  toti  simul  erit 
praesens  per  illam  extensionem  virtualem,  ut  &  Deus  per  infinitam  Immensitatem  suam 
praesens  est  punctis  infinitis  spatii  (&  ille  quidem  omnibus  omnino),  sive  in  iis  materia 
sit,  sive  sint  vacua. 

Nunquamproduciab  538.  Et  haec  quidem  de  sede  animae  :  illud  autem  postremo  loco  addendum  hie  censeo 

z™SerT°ir?mpartes  &  legibus  omnibus  constituentibus  ejus  conjunctionem  cum  corpore,  quod  est  observa- 

oppositas:  quid  inde  tionibus  conforme,  quod  diximus  num.  74,  &  387,  nunquam  ab  anima  produci  motum  in 

consequatur.  unQ  j^gj.^  puncto,  quin  in  alio  aliquo  sequalis  motus  in  partem  contrariam  producatur, 

unde  fit,  ut  nee  liberi,  nee  necessarii  materiae  motus  ab  animabus  nostris  orti  perturbent 

actionis,    &   reactionis    aequalitatem,    conservationem    ejusdem    status    centri    communis 

gravitatis,  &  conservationem  ejusdem  quantitatis   motus  in  Mundo  in  eandem  plagam 

computari. 


Transitus  ad  Auc-  53^  Haec  quidem  de  anima  :  jam  quod  pertinet  ad  ipsum  Divinum  Naturae  Opificem, 

cujusp^rfectionesTn  in  hac  Theoria  elucet  maxime  &  necessitas  ipsum  omnino  admittendi,  &  summa  ipsius, 
hac  Theoria  elucent  atque  infinita  Potentia,  Sapientia,  Providentia,  quae  venerationem  a  nobis  demississimam, 

m^v-imp  *  *  * 


maxime. 


APPENDIX  379 

local,  of  the  mind  with  a  great  part  of  the  body,  or  even  with  the  whole  of  the  body.  But 
to  know  this,  it  would  be  necessary  to  have  a  clear  knowledge  of  their  laws,  which  conjunction 
of  the  mind  with  the  body  necessitates ,  &  also  a  knowledge  of  the  entire  disposition  of 
all  the  points  constituting  the  body,  &  the  laws  for  the  mutual  forces  between  points  of 
matter.  In  addition,  there  would  be  the  necessity  for  as  great  geometrical  powers,  as 
would  be  enough  to  determine  all  the  motions,  which  might  be  produced  merely  on  account 
of  the  mechanical  distribution  of  these  points.  All  of  these  would  be  needed  for 
perceiving  whether,  from  the  motions,  which  the  mind  could  induce,  by  the  power  of  its 
own  will  or  the  necessity  of  its  nature,  on  a  single  point,  or  on  certain  given  points,  by 
means  of  the  single  law  of  forces  common  to  points  of  matter,  there  could  follow  all  the  other 
motions  of  the  spirits  &  nerves,  such  as  take  place  in  our  voluntary  motions ;  as  well  as 
all  those  different  motions  of  particles  of  the  body  upon  which  depend  secretions,  nutrition, 
respiration,  &  other  motions  of  ours  that  are  not  voluntary.  But  all  these  are  unknown 
to  us ;  nor  may  we  aspire  to  such  a  sublime  kind  of  geometry,  for  as  yet  we  cannot  altogether 
determine  all  the  motions  of  even  three  little  masses,  which  act  upon  one  another  with 
forces  that  are  known. 

536.  There  have  been  some  who  would  confine  the  mind  to  some  very  small  portion  of  Fa-}s&y  °f  several 
the  body  ;  for  instance,  Descartes  suggested  the  pineal  gland.     But,  later,  it  was  discovered  seat'of^the  mind": 
that  it  could  not  be  contained  in  that  part  alone  ;   for,  if  that  part  were  removed,  life  still  it  cannot  be  proved 
went  on.      It  has  been  already  discovered  that  life  endured  for  some  time  without  the  extend  throughout 
pineal  gland,  just  as  some  animals  produced  life  even  without  a  brain.     Others  argued  *h^  whole  °f  the 
against  the  diffusion  of  the  mind  throughout  the  whole  of  the  body,  from  the  fact  that 

sometimes  men,  after  the  hand  had  been  cut  off,  said  that  they  could  still  feel  the  pain  in 
the  fingers,  as  if  they  still  had  fingers ;  &  smce  this  pain  is  felt,  although  in  this  case  there  is 
not  the  fact  that  the  mind  is  present  in  the  fingers,  they  thought  that  it  could  be  inferred 
that,  as  often  as  we  feel  a  pain  in  the  fingers,  we  feel  it  without  the  presence  of  the 
mind  in  the  fingers.  But  such  argument  proves  nothing  at  all ;  for  it  might  happen  that, 
in  order  that  there  should  be  in  the  first  place  that  feeling,  which  we  experience  of  pain 
in  the  fingers,  there  were  required  the  presence  of  the  mind  in  the  fingers,  without  which 
it  would  be  impossible  that  an  idea  of  the  pain  could  be  excited  in  the  first  place ;  but, 
once  this  idea  had  been  formed,  it  might  be  possible  that  it  could  once  more  be  excited, 
without  the  presence  of  the  mind  in  the  fingers,  by  the  motions  of  the  nerves,  which  had 
been  conjoined  with  a  motion  of  the  fibres  of  the  finger  when  the  pain  was  first  felt.  Be-  . 
sides,  it  still  remains  to  be  decided  whether  any  impulse  of  a  present  mind  is  required 
for  nutrition,  or  whether  this  can  be  obtained  wholly  without  any  operation  of  the  mind, 
by  means  of  a  mere  mechanism  alone. 

537.  All  these  things  show  fully  that  nothing  certain  can  be  stated  with  regard  to  the  Conclusion  that  the 
seat  of  the  mind  from  a  due  consideration  of  phenomena  ;  nor  that  its  diffusion  through-  Unknown  •*  "he r'e 
out  any  great  part  of  the  body,  or  even  throughout  the  whole  body,  is  excluded.     But  if  &  in  what  manner 
it  should  extend  throughout  a  great  part,  or  even  the  whole,  of  the  body,  that  also  would 

fit  in  excellently  with  my  Theory.  For,  by  means  of  such  virtual  extension  as  we  discussed 
in  Art.  83,  the  mind  might  exist  in  the  whole  of  the  space  containing  all  the  points  which 
form  that  part  of  the  body,  or  that  form  the  whole  body.  With  this  idea,  in  my  Theory, 
the  mind  will  differ  still  more  from  matter  ;  for  the  simple  elements  of  matter  cannot  exist 
except  in  single  points  of  space  at  single  instants  of  time,  each  to  each,  while  the  mind  can 
also  be  one-fold,  &  yet  exist  at  one  &  the  same  time  in  an  infinite  number  of  points  of  space, 
conjoining  with  a  single  instant  of  time  a  continuous  series  of  points  of  space  ;  &  to  the 
whole  of  this  series  it  will  at  one  &  the  same  time  be  present  owing  to  the  virtual  extension 
it  possesses ;  just  as  God  also,  by  means  of  His  own  infinite  Immensity,  is  present  in  an 
infinite  number  of  points  of  space  (&  He  indeed  in  His  entirety  in  every  single  one), 
whether  they  are  occupied  by  matter,  or  whether  they  are  empty. 

538.  These  things  indeed  relate  to  the  seat  of  the  mind;    but  I  think  there  should  Motion  can  never 
be  added  here  in  the  last  place,  concerning  all  the  laws  governing  its  conjunction  with  the  nTinPdr,°tnkssbyitthis 
body,  that  which  is  in  conformity  with  the  observations  that  I  made  in  Art.  74  &  Art.  equal  in  opposite 
387  ;  namely,  that  motion  can  never  be  produced  by  the  mind  in  a  point  of  matter,  without 

producing  an  equal  motion  in  some  other  point  in  the  opposite  direction.  Whence  it  comes 
about  that  neither  the  necessary  nor  the  free  motions  of  matter  produced  by  our  minds  can 
disturb  the  equality  of  action  &  reaction,  the  conservation  of  the  same  state  of  the  centre 
of  gravity,  &  the  conservation  of  the  same  quantity  of  motion  in  the  Universe,  reckoned 
in  the  same  direction. 

539.  So  much  for  the  mind  ;   now,  as  regards  the  Divine  Founder  of  Nature  Himself,  m 
there  shines  forth  very  clearly  in  my  Theory,  not  only  the  necessity  of  admitting  His  existence  the  perfections  of 
in  every  way,  but  also  His  excellent  &  infinite  Power,  Wisdom,  &  Foresight ;    which  demand  ^hoic1le 

from  us  the  most  humble  veneration,  along  with  a  grateful  heart,  &  loving  affection.     The  Theory. 


380 


PHILOSOPHISE  NATURALIS  THEORIA 


vanam  sine  re. 


&  simul  gratum  animum,  atque  amorem  exposcant  :  ac  vanissima  illorum  somnia  corruunt 
penitus,  qui  Mundum  vel  casu  quodam  fortuito  putant,  vel  fatal!  quadam  necessitate 
potuisse  condi,  vel  per  se  ipsum  existere  ab  aeterno  suis  necessariis  legibus  consistentem. 
Error   tnbuentium  r  ^O-  j?t  primo  quidem  quod  ad  casum  pertinet,  sic  ratiocinantur  :   finiti  terminorum 

Munch    onginem  JT  . -T        .      ^  n       ~    .  i  •         •  •    r-    • 

casui   fortuito:  numeri  combmationes  numero  nmtas  habent,  combmationes  autem  per  totam  innmtam 
casum   esse  vocem  aeternitatem  debent  extitisse  numero  infinitae  :   etiamsi  nomine  combinationum  assumamus 

vanam  cin*»  m  * 

totam  seriem  pertinentem  ad  quotcunque  millenos  annos.  Quamobrem  in  fortuita 
atomorum  agitatione,  si  omnia  se  aequaliter  habuerint,  ut  in  longa  fortuitorum  serie  semper 
accidit,  debuit  quaevis  ex  ipsis  redire  infinitis  vicibus,  adeoque  infinities  major  est  prob- 
abilitas  pro  reditu  hujus  individuae  combinationis,  quam  habemus,  quocunque  finite 
numero  vicium  redeuntis  mero  casu,  quam  pro  non  reditu.  Hi  quidem  inprimis  in  eo 
errant,  quod  putent  esse  aliquid,  quod  in  se  ipso  revera  fortuitum  sit ;  cum  omnia  deter- 
minatas  habeant  in  Natura  causas,  ex  quibus  profluunt,  &  idcirco  a  nobis  fortuita  dicantur 
quaedam,  quia  causas,  a  quibus  eorum  existentia  determinatur,  ignoramus. 


Numerurn  cpmbina-  r  jj    ge(j  eo  omisso,  falsissimum  est,  numerum  combinationum  esse  finitum  in  terminis 

tionum  in  terminis  JT     ,...  .  .  i    n/r        i-  •• 

etiam  numero  finitis  numero  nnitis  :    si  omnia,  quae  ad  Mundi  constitutionem  necessana  sunt,  perpendantur. 
esse  infinitum  :   si  Est  nuidem  finitus  numerus  combinationum,  si  nomine  combinationis  assumatur  tantum- 

nte     omnia   expen-  ,  ^        ,  .  ,  ...  .    .  ..    '  .  .  .  .  ...     ,         . 

dantur.  modo  ordo  quidam,  quo  alii  termini  post  alios  jacent  :  nine  ultro  agnosco  mud  :  si  omnes 

litterae,  quae  [255]  Virgilii  poema  componunt,  versentur  temere  in  sacco  aliquo,  turn 
extrahantur,  &ordinentur  omnes  litterae,  aliae  post  alias,  atque  ejusmodi  operatic  continuetur 
in  infinitum,  redituram  &  ipsam  combinationem  Virgilianam  numero  vicium  quenvis 
determinatum  numerum  superante.  At  ad  Mundi  constitutionem  habetur  inprimis 
dispositio  punctorum  materiae  in  spatio  patente  in  longum,  latum,  &  profundum  :  porro 
rectas  in  uno  piano  sunt  infinitae,  plana  in  spatio  sunt  infinita,  &  pro  quavis  recta  in  quovis 
piano  infinita  sunt  curvarum  genera,  quae  cum  eadem  ex  dato  puncto  directione  oriantur, 
in  quarum  singularum  classibus  infinities  plures  sunt,  quae  per  datum  punctorum  numerum 
non  transeant.  Quare  ubi  seligenda  sit  curva,  quae  transeat  per  omnia  materiae  puncta, 
jam  habemus  infinitum  saltern  ordinis  tertii.  Praeterea,  determinata  ejusmodi  curva, 
potest  variari  in  infinitum  distantia  puncti  cujusvis  a  sibi  proximo  :  quamobrem  numerus 
dispositionum  possibilium  pro  quovis  puncto  materiae  adhuc  ceteris  manentibus  est  infinitus, 
adeoque  is  numerus  ex  omnium  mutationibus  possibilibus  est  infinitus  ordinis  expositi  a 
numero  punctorum  aucto  saltern  ternario.  Iterum  velocitas,  quam  habet  dato  tempore 
punctum  quodvis,  potest  variari  in  infinitum,  &  directio  motus  potest  variari  in  infinitum 
ordinis  secundi  ob  directiones  infinitas  in  eodem  piano,  &  plana  infinita  in  spatio. 
Quare  cum  constitutio  Mundi,  &  sequentium  phaenomenorum  series  pendeat  ab  ipsa  velo- 
citate,  &  directione  motus ;  numerus,  qui  exprimit  gradum  infiniti,  ad  quern  assurgit 
numerus  casuum  diversorum,  debet  multiplicari  ter  per  numerum  punctorum  materise. 


aeternitate. 


Cujus  ordinis  infini-  54.2.  Est  igitur  numerus  casuum  diversorum  non  finitus,  sed  infinitus  ordinis  expositi 

aWsshni,  ^irTinv  a  quarta  potentia  numeri  punctorum  aucta  saltern  ternario,  atque  id  etiam  determinata 
mensum  aitioris  curva  virium,  quae  potest  itidem  infinitis  modis  variari.  Quamobrem  numerus  combina- 
tionum  relativarum  ad  Mundi  constitutionem  non  est  finitus  pro  dato  quovis  momento 
temporis,  sed  infinitus  ordinis  altissimi,  respectu  infiniti  ejus  generis,  cujus  generis  est 
infinitum  numeri  punctorum  spatii  in  recta  quapiam,  quae  concipiatur  utrinque  in  infinitum 
producta.  At  huic  infinite  est  analogum  infinitum  momentorum  temporis  in  tota  utraque 
aetemitate,  cum  unicam  dimensionem  habeat  tempus.  Igitur  numerus  combinationum 
est  infinitus  ordinis  in  immensum  aitioris  ordine  infiniti  momentorum  temporis,  adeoque 
non  solum  non  omnes  combinationes  non  debent  redire  infinities  :  sed  ratio  numeri  earum, 
quae  non  redeunt,  est  infinita  ordinis  altissimi,  quam  nimirum  exponit  quarta  potentia 
numeri  punctorum  aucta  saltern  binario,  vel,  si  libeat  variare  virium  leges,  saltern  ternario. 
Quamobrem  ruit  futile  ejusmodi,  atque  inane  argumentum. 


in    ipso    immense  54.3.  Sed  hide  etiam  illud  eruitur,  in  immenso  isto  com-[2S6]-binationum  numero 

mero'S  immensuni  infinities  esse  plures  pro  quovis  genere  combinationes  inordinatas,  quae  exhibeant  incertum 
plures  esse  combina-  chaos,  &  massam  temere  volitantium  punctorum,  quam  quae  exhibeant  Mundum  ordinatum, 
quam  orduTata?*^'  &  certis  constantem  perpetuis  legibus.     Sic  ex.  gr.  ad  efformandas  particulas,  quae  constanter 
suam  formam  retineant,  requiritur  collocatio  in  punctis  illis,  in  quibus  sunt  limites,  & 


APPENDIX  381 

truly  groundless  dreams  of  those,  who  think  that  the  Universe  could  have  been  founded 
either  by  some  fortuitous  chance  or  some  necessity  of  fate,  or  that  it  existed  of  itself  from 
all  eternity  dependent  on  necessary  laws  of  its  own,  all  these  must  altogether  come  to  nothing. 

540.  Now  first  of  all,  the  argument  that  it  is  due  to  chance  is  as  follows.     The  The  error  made  by 
combinations  of  a  finite  number  of  terms  are  finite  in  number;    but  the  combinations  Jha^ the° Universe 
throughout  the  whole  of  infinite  eternity  must  have  been  infinite  in  number,  even  if  we  was   produced   by 
assume  that  what  is  understood  by  the  name  of  combinations  is  the  whole  series  pertaining  f^^nce  -^u^an 
to  so  many  thousands  of  years.     Hence,  in  a  fortuitous  agitation  of  the  atoms,  if  all  cases  empty  phrase  with- 
happen  equally,  as  is  always  the  ca^e  in  a  long  series  of  fortuitous  things,  one  of  them  is  out  a  ^iP8^*0  cor" 
bound  to  recur  an  infinite  number  of  times  in  turn.     Thus,  the  probability  of  the  recurrence 

of  this  individual  combination,  which  we  have,  is  infinitely  more  probable,  in  any  finite 
number  of  succeeding  returns  by  mere  chance,  than  of  its  non-recurrence.  Here,  first  of 
all,  they  err  in  the  fact  that  they  consider  that  there  is  anything  that  is  in  itself  truly 
fortuitous ;  for,  all  things  have  definite  causes  in  Nature,  from  which  they  arise  ;  &  there- 
fore some  things  are  called  by  us  fortuitous,  simply  because  we  are  ignorant  of  the  causes  by 
which  their  existence  is  determined. 

541.  But,  leaving  that  out  of    account,  it  is  quite  false  to  say  that  the  number  of  The     number    of 
combinations  from  a  finite  number  of  terms   is   finite,  if  all   things  that  are  necessary  to  amongst 'terms' that 
the  constitution  of  the  Universe  are  considered.     The  number  of  combinations  is  indeed  are  even  finite  in 
finite,  if  by  the  term   combination    there  is   implied   merely  a  certain  order,  in  which  jf'th^a 

some  of  the  terms  follow  the  others.     I  readily  acknowledge   this   much ;     that,   if   all  ly  considered. 

the  letters  that  go    to  form   a  poem  of    Virgil  are  shaken  haphazard  in   a  bag,  &  then 

taken  out  of  It,  &  all  the  letters  are  set  in  order,  one  after  the  other,  &  this  operation  is 

carried  on   indefinitely,  that  combination  which  formed  the    poem  of  Virgil  will  return 

after  a  number  of  times,  if  this  number  is  greater  than  some  definite  number.     But,  for 

the  constitution  of  the  Universe,  we  have  first  of  all  the  arrangement  of  the  points  of  matter, 

in  a  space  that  extends  in  length,  breadth  &  depth  ;   further,  there  are  an  infinite  number 

of  straight  lines  m  any  one  plane,  an  infinite  number  of  planes  in  space,  &  for  any  straight 

line  in  any  plane  there  are  an  infinite  number  of  classes  of  curves,  which  will  start  from 

a  given  point  in  the  same  direction  as  the  straight  line  ;    &  in  every  one  of  these  classes 

there  are  infinitely  more  which  do  not  pass  through  a  given  number  of  points.     Hence, 

when  a  curve  has  to  be  selected  which  shall  pass  through  all  points  of  matter,  we  now  have 

an  infinity  of  at  least  the  third  order.     Besides,  after  any  curve  has  been  chosen,  the  distance 

of  each  point  from  the  one  next  to  it  can  be  varied  indefinitely ;    hence  the  number  of 

possible  arrangements  for  any  one  point  of  matter,  while  the  rest  remain  fixed,  is  infinite. 

Therefore  it  follows  that  the  number  derived  from  the  possible  changes  m  all  of  these  things 

is  infinite,  of  the  order  determined  by  the  number  of  points  increased  at  least  three  times. 

Again,  the  velocity  which  any  point  has  at  a  given  time  can  be  varied  indefinitely ;   &  the 

direction  of  motion  can  be  varied  to  an  infinity  of  the  second  order,  on  account  of  the 

infinity  of  directions  in  the  same  plane  &  the  infinity  of  planes  in  space.     Hence,  since  the 

constitution  of  the  Universe,  &  the  series  of  consequent  phenomena,  depend  on  the  velocity 

&  the  direction  of  motion ;  the  number,  which  expresses  the  degree  of  infinity  to  which 

the  number  of  different  cases  mounts  up,  must  be  multiplied  three  times  by  the  number 

of  points  of  matter. 

542.  Therefore  the  number  of  cases  is  not  finite,  but  infinite  of  the  order  expressed  The  order  of  the 
by  the  fourth  power  of  the  number  of  points  increased  threefold  at  least  ;  &  that  is  so,  even  ceedfng'iy*  "igh" 
if  there  is  a  definite  curve  of  forces  which  also  can  be  varied  in  an  infinity  of  ways.     Hence  immensely    higher 
the  number  of  relative  combinations  necessary  to  the  formation  of   the  Universe  is  not  instants*  of"  timer  in 
finite  for  any  given  instant  of  time  ;    but  it  is  infinite,  of  an  exceedingly  high  order  with  the  whole  of  eter- 
respect  to  an  infinity  of  the  kind  to  which  belongs  the  infinity  of  the  number  of  points  mty' 

of  space  in  any  straight  line,  which  is  conceived  to  be  produced  to  infinity  in  both  directions. 
To  this  infinity  the  infinity  of  the  instants  in  the  whole  of  eternity  past  &  present  is  analogous ; 
for  time  has  but  one  dimension.  Hence,  the  number  of  combinations  is  infinite  of  an  order 
that  is  immensely  higher  than  the  order  of  the  Infinity  of  instants  of  time ;  &  thus,  not 
only  does  it  follow  that  not  all  the  combinations  are  not  bound  to  return  an  infinite  number 
of  times,  but  the  ratio  even  of  those  that  do  not  return  is  infinite,  of  a  very  high  order, 
namely  that  which  is  expressed  by  the  fourth  power  of  the  number  of  points  increased 
twofold  at  least,  or  threefold  at  least  if  we  choose  to  vary  the  laws  of  forces.  Hence,  the 
arguments  of  this  sort  that  are  brought  forward  are  futile  &  worthless. 

543.  Moreover  from  this  it  also  follows  that,  in  this  immense  number  of  combinations,  in    this    immense 
there  will  be,  for  any  kind,  infinitely  more  irregular  combinations,  such  as  represent  indefinite  atio^even^here 
chaos  &  a  mass  of  points  flying  about  haphazard,  than  there  are  of  those  that  exhibit  the  are      immensely 
regular  combinations  of  the  Universe,  which  follow  definite  &  everlasting  laws.     For  instance,  """j,^ 

in  order  to  form  particles  which  continually  maintain  their  form,  there  is  required  their  there  are  regular. 


382  PHILOSOPHIC  NATURALIS  THEORIA 

quorum  numerus  debet  esse  infinities  minor,  quam  numerus  punctorum  sitorum  extra 
ipsos  :  nam  intersectiones  curvae  cum  axe  debent  fieri  in  certis  punctis,  &  inter  ipsa  debent 
intercedere  segmenta  axis  continua,  habentia  puncta  spatii  infinita.  Quamobrem  nisi  sit 
aliquis,  qui  ex  omnibus  seque  per  se  possibilibus  seligat  unam  ex  ordinatis  ;  infinities 
probabilius  est,  infinitate  ordinis  admodum  elevati,  obventuram  inordinatam  combinationum 
seriem,  &  chaos,  non  ordinatam,  &  Mundum,  quern  cernimus,  &  admiramur.  Atque  ad 
vincendam  determinate  earn  infinitam  improbabilitatem,  requiritur  infinita  vis  Conditoris 
Supremi  seligentis  unam  ex  iis  infinitis. 

Non  determinari  ab  r A*    Nec  vero  illud  obiici  potest,  etiam  hominem,   qui  statuam  aliquam  effingat, 

homine  individuum:    .,    .    -/'T   ..  .,,         •    j-    -j  t  -IT    i         •  •    c    •  11- 

sed  eo  determinante  finita  vi  eligere  illam  individuam  formam,  quam  illi  dat,  inter  mfimtas,  quas  naben  possunt. 
intraiimites,adquos  Nam  imprimis  ille  earn  individuam  non  eligit,  sed  determinat  modo  admodum  confuso 
reifquamTn defer-  figuram  quandam,  &  individua  ilia  oritur  ex  Naturae  legibus,  &  Mundi  constitutione  ilia 
minationem  yinci  individua,  quam  naturae  Opifex  Infinitus  infinitam  indeterminationem  superans  deter- 

ab  Ente  in  infimtum        .  ,       .        *    ,  .  .,,.  .  , F        ,  ..       .       . 

Ubero.  mmavit,  per  quam  ab  ejus  voluntatis  actu  onuntur  mi  certi  motus  in  ejus  brachns,  &  ab 

hisce  motus  instrumentorum.  Quin  etiam  in  genere  idcirco  tarn  multi  Philosophi 
determinationem  ad  individuum,  &  determinationem  ad  omnes  illos  gradus,  ad  quos 
cognitio  creati  determinantis  non  pertingit,  rejecerunt  in  Deum  infinita  cognoscendi,  & 
discernendi  vi  praeditum,  necessaria  ad  determinandum  unum  individuum  casum  ex 
infinitis  ad  idem  genus  pertinentibus ;  cum  creatae  mentis  cognitio  ad  finitum  tantummodo 
graduum  diversorum  numerum  distincte  percipiendum  extendi  possit  :  sine  ullo  autem 
determinante  ex  casibus  infinitis,  &  quidem  tanto  infinitatis  gradu,  individuus  unus  prae 
aliis  per  se,  aut  per  fortuitam  eventualitatem  prodire  omnino  non  potest. 

Hunc  ordinem  non  CAC,  Sed  nee  dici  potest,  hunc  ipsum  ordinem  necessarium  esse,  &  aeternum  ac   per 

posse    dici    per    se  ,    .  .  j  .  •  ,  ,  .  .r 

necessarium:  prima  se  subsistere,  casu  quovis  sequentc  determinate  a  proxime  praecedente,  &  a  lege  vinum 
impugnatio  a  nuUo  intrinseca,  &  necessaria  iis  individuis  punctis,  &  non  aliis.  Nam  contra  hoc  ipsum  miserum 

nexu,     qui     videtur  a      •  i  •  T         •      •        j         j  j-fc    -i 

haberi  inter  distan-  sane  ettugmm  quamplunma  sunt,  quae  opponi  possunt.  Inprimis  admodum  dirncile  est, 
tiam,  &  vim,  quae  ut  homo  sibi  serio  persuadeat,  hanc  unam  virium  legem,  quam  habet  hoc  individuum 

idcirco  liberum  de-  ,      •         •    j-    -j    •  r    •  'Li  o  •  •     • 

terminantem  requi-  punctum  respectu  hujus  individui  puncti,  fuisse  possibiJem,  &  necessanam,  ut  nimirum 
"jnt-  in  hac  individua  [257]  distantia  se  potius  attrahant,  quam  repellant,  &  se  attrahant  tanta 

potius  attractione,  quam  alia.  Nulla  appaiet  sane  connexio  inter  distantiam  tantam, 
&  tantam  talis  speciei  vim,  ut  ibi  non  potuerit  esse  alia  quaevis,  &  ut  hanc  potius,  quam 
aliam  pro  hisce  punctis  non  selegerit  arbitrium  entis  habentis  infinitam  determinativam 
potentiam,  vel  pro  hisce  punctis  id,  si  libeat,  ex  natura  sua  petentibus,  non  posuerit  alia 
puncta  illam  aliam  petentia  ex  sua  itidem  natura. 

Secunda  a  numero  54.6.  Praeterea  cum  &  infinitum,  &  infinite  parvum  in  se  determinatum,  &  in  se  tale, 

qu^determinantem  m  creatis  sit  impossibile  (quod  de  infinite  in  extensione  demonstravi  (')  pluribus  in  locis, 

voluntatem  poscit.     nee  una  tantum  demonstratione,  ut  in  dissertatione  De  Natura,   &  usu  infinitorum,   & 

infinite  -parvorum,  ac  in  dissertatione  adjecta  meis  Sectionum  Conicarum  Elementis,  Element. 

torn.  3) ;   finitus  est  numerus  punctorum  materiae,  vel  saltern  in  communi  etiam  sententia 

finita  est  materiae  existentis  massa,  quae  finitum  spatium  occupare  debet,  &  non  in  infinitum 

(t)  En  unam  ex  ejusmodi  demonstrationibus.  Sit  in  fig.  71  spatium 
a  C  versus  AE  infinitum,  &  in  eo  angulus  rectilineus  ACE  bifariam 
sectus  per  rectum  CD.  Sit  autem  GH  parallela  CA,  ques  occurrat  CD 
in  H,  ac  producatur  ita,  ut  HF  fiat  dupla  GH,  ducaturque  CF,  y  omnes 
CA,  CB,  CD,  CE  in  infinitum  producantur.  Inprimis  totum  spatium 
infinitum  ECD  debet  esse  eequale  infinite  ACD  :  nam  ob  angulum  ACE 
bifariam  sectum  sibi  invicem  congruerent.  Deinde  triangulum  HCF  est 
duplum  HCG,  ob  FH  duplum  HG.  Eodem  pacto  ductis  aliis  ghf  ipsi 
parallelis,  hCf  erit  duplam  hCg,  adeoque  &  area  FHhf  dupla  HGgh. 
Quare  W  summa  omnium  FHhf  dupla  summte  omnium  HGgh,  nimirum 
tota  area  infinita  BCD  dupla  infinite  DCE,  adeoque  dupla  ACD,  nimi- 
rum pars  dupla  totius,  quod  est  absurdum.  Porro  absurdum  oritur  ab 
ipsa  infinitate,  si  enim  sint  arcus  circulares  GMI,  gmi  centra  C  ;  sector 
ICM  erit  aqualis  GCM,  y  triangulum  FCH  duplum  GCH.  Donee 
sumus  in  quantitatibus  finitis,  res  bene  procedit,  qui  a  FCH  non  est  pars 
ICM,  sicut  BCD  est  pars  ACD,  nee  MCG,  tf  HCG  sunt  unum,  W 
idem,  ut  DCE  est  unicum  infinitum  absolutum  contentum  cruribus  CD, 
CE.  Absurdum  oritur  tantummodo,  ubi  sublatis  prorsus  limitibus,  a 
quibus  oriuntur  discrimina  spatiorum  inclusorum  iisdem  angulis  ad  C,  sit 
suppositio  infiniti  absoluti,  qute  contradictionem  involvit. 


APPENDIX  383 

grouping  together  in  those  points  in  which  there  are  limit-points ;  &  of  these  the  number 
must  be  infinitely  less  than  the  number  of  points  situated  without  them.  For  the 
intersections  of  the  curve  with  the  axis  must  take  place  in  certain  points ;  &  between  these 
points  there  must  lie  continuous  segments  of  the  axis,  having  on  them  an  infinite  number 
of  points  of  space.  Hence,  unless  there  were  One  to  select,  from  among  all  the  combinations 
that  are  equally  possible  in  themselves,  one  of  the  regular  combinations,  it  would  be  infinitely 
more  probable,  the  infinity  being  of  a  very  high  order,  that  there  would  happen  an  irregular 
series  of  combinations  &  chaos,  rather  than  one  that  was  regular,  &  such  an  Universe  as  we 
see  &  wonder  at.  Then,  to  overcome  definitely  this  infinite  improbability,  there  would 
be  required  the  infinite  power  of  a  Supreme  Founder  selecting  one  from  among  those 
infinite  combinations. 

544.  Nor  can  the  argument  be  raised  that  even  man,  when  he  fashions  a  statue,  with  The    individual  is 
but  a  finite  force  selects  that  individual  form  which  he  gives  to  it,  from  among  an  infinite  m°antefri™whenbit 
number  which  are  possible.     For,  first  of  all,  the  man  does  not  select  that  individual  form  ;  has  been  determined 
he  determines  in  a  very  confused  way  a  certain  shape,  &  that  individual  form  arises  from  which1  maV-s'know0 
the    laws  of  Nature,  &  from   that   individual   constitution   of   the  Universe  which   the  ledge   attains,  the 
Infinite  Founder  of  Nature,  overcoming  the  infinite  lack  of  determination,  has  determined  ;  termined*  S  "over- 
through  which,  by  an  act  of  his  will,  arise  those  definite  motions  in  the  arms  of  the  man,  come  by  a  Being 
&  from  these  the  motions  of  his  tools.     Moreover,  in  general,  on  this  account,  so  many  £e°         infinitely 
philosophers  have  thrown  back  individual  determination,  &  a  determination  for  all  those 

stages  to  which  the  knowledge  of  a  determining  created  thing  cannot  attain,  upon  a  God 
endowed  with  an  infinite  power  of  knowing  &  distinguishing,  such  as  is  necessary  for  the 
task  of  determining  one  individual  case  from  among  an  infinite  number  pertaining  to  the 
same  class.  For  the  knowledge  of  a  created  mind  can  only  be  extended  to  perceiving 
distinctly  a  finite  number  of  different  stages.  But,  unless  there  is  someone  to  determine 
it,  one  individual  cannot  of  itself,  or  through  fortuitous  happening,  possibly  come  forth  in 
preference  to  others,  from  among  an  infinite  number  of  cases,  let  alone  from  an  infinity  of 
such  a  high  degree. 

545.  No  more  can  it  be  said  that  this  very  regularity  is  necessary,  everlasting,  &  self-  This    regularity 
sustained,  any  one  case  following  the  one  next  before  it  &  determined  by  it,  &  by  a  law  of  b^^ecessa^  in 
forces  that  is  intrinsic  &  necessary  to  those  individual  points  &  to  no  others.     For  against  this  itself  ;    first,  be- 
really  worthless  subterfuge  there  are  very  many  arguments  that  can  be  brought  forward,  parent  °fabsenceaof 
First  of  all,  it  is  very  difficult  to  see  how  a  man  can  seriously  persuade  himself  that  one  any     connection 
particular  law  of  forces,  which  one  particular  point  has  with  regard  to  another  particular  ^force"  th^atter 
point,  should  be  possible  &  necessary,  so  that,  for  instance,  at  one  particular  distance  the  therefore    requires 
points  should  attract  one  another  rather  than  repel  one  another,  &  attract  one  another  lio^ee  determma" 
with  an  attraction  that  is  so  much  greater  than  that  with  which  they  attract  others.     In 

truth,  there  is  apparently  no  connection  between  so  great  a  distance  &  so  great  a  force  of 
such  a  sort,  that  there  could  not  be  any  other  in  the  circumstances ;  &  that  the  will  of  a 
Being  having  infinite  determinative  power  should  not  select  one  in  particular  rather  than 
another  for  these  points ;  or  should  not  substitute,  for  these  points  that  from  their  very 
nature,  if  you  like  to  say  so,  require  the  first,  other  points  that  also  from  their  nature  require 
that  other  connection. 

546.  Besides,  the  infinite  &  the  infinitely  small,  self-determined  &  such  of  themselves,  Second    argument 
is  impossible   in   created  things ;    as  I  proved  concerning  the  infinite  in  extension  (*)  in  fofce^number  *of 
several  places,  &  with  more  than  one  proof,  for  instance,  in  the  dissertation  De  Natura,  points,    which  re- 
y  usu  infinitorum,  £f?  infinite  parvorum,  &  in  a  dissertation  added  to  my  Sectionum  Conicarum  j^g"^/1  determln~ 
Elementa,  Elem.  Vol.  3.      It  therefore  follows  that  the  number  of  points  of  matter  is  finite ; 

or  at  least,  even  in  the  commonly  accepted  opinion,  the  mass  of  existing  matter  is  finite ; 

(t)  Here  is  one  of  these  -proofs.  In  Fig.  71,  let  the  space  from  C  in  the  direction  of  A,  E  be  infinite  ;  y  in 
this  space,  let  the  rectilineal  angle  ACE  be  bisected  by  the  straight  line  CD.  Also  let  GH  be  parallel  to  CA,  meeting 
CD  in  H  ;  y  let  it  be  produced  so  that  HF  is  double  GH  ;  join  CF,  fcf  let  all  the  straight  lines  CA,  CB,  CD, 
CE  be  produced  to  infinity.  Now,  first  of  all,  the  whole  of  the  infinite  space  ECD  must  be  equal  to  the  infinite  space 
ACD ,-  for,  on  account  of  the  bisection  of  the  angle  ACE,  they  will  be  congruent  with  one  another.  Secondly,  the 
triangle  HCF  is  double  the  triangle  HCG,  sinee  FH  is  double  HG.  In  the  same  way,  if  other  parallels  like  ghf  are 
drawn,  hCf  will  be  double  hCg  ,-  y  thus  the  area  FHhf  will  be  double  HGgh.  Hence,  the  sum  of  all  such  areas  as  FHhf 
will  be  double  the  sum  of  all  such  as  HGgh  ;  that  is  to  say,  the  whole  of  the  infinite  area  BCD  will  be  double  the 
infinite  area  DCE,  y  there f 01  e  double  ACD  ;  the  part  double  the  whole,  which  is  impossible.  Further,  the  impossibility 
springs  from  the  supposition  of  infinity  ;  for,  if  GMI,  gmi  are  circular  arcs  whose  centre  is  C,  the  sector  ICM  will 
be  equal  to  GCM,  y  the  triangle  FCH  will  be  double  GCH.  So  long  as  we  are  dealing  with  finite  quantities,  the 
matter  goes  on  quite  correctly,  because  FCH  is  not  a  part  of  ICM,  as  BCD  is  a  part  of  ACD,  nor  are  MCG  y 
HCG  one  y  the  same,  as  DCE  is  the  unique  infinite  absolute  content  of  the  arms  CD,  DE.  The  impossibility  only 
arises  when,  all  limits  being  taken  away,  from  which  arise  the  differences  between  the  spaces  included  by  the  same 
angles  at  C,  the  supposition  is  made  of  absolute  infinity,  which  involves  the  contradiction. 


PHILOSOPHIC  NATURALIS   THEORIA 


protendi.  Porro  cur  hie  sit  potius  numerus  punctorum,  haec  potius  massse  quantitas  in 
Natura,  quam  alia  ;  nulla  sane  ratio  esse  potest,  nisi  arbitrium  entis  infinita  determinativa 
potentia  praediti,  &  nemo  sanus  sibi  facile  serio  persuadebit,  in  quodam  determinate  numero 
punctorum  haberi  necessitatem  existentiae  potius,  quam  in  alio  quovis. 

Tertiaabaetemitate,  547.  Accedit  illud,  quod  si  Mundus  cum  hisce  legibus  fuisset  ab  aeterno  ;    extitissent 

motusT^um^iinea  ']am  motus  seterni,  &  lineae  a  singulis  punctis  descriptae  debuissent  fuisse  jam  in  infinitum 
necessario  infinita  :  productae :  nam  in  se  ipsas  non  redeunt  sine  arbitrio  entis  infinitam  improbabilitatem  vincentis, 
cum  demonstraverim  supra  pluribus  in  locis  infinities  improbabilius  esse,  [258]  aliquod 
punctum  redire  aliquando  ad  locum,  quem  alio  temporis  momento  occupaverit,  quam 
nullum  redire  unquam.  Porro  infinitum  in  extensione  impossibile  prorsus  esse,  ego  quidem 
demonstravi,  uti  monui,  &  ilia  impossibilitas  pertinere  debet  ad  omne  genus  linearum, 
quae  in  infinitum  productae  sint.  Potest  utique  motus  continuari  in  infinitum  per 
aeternitatem  futuram,  quia  si  aliquando  coepit,  nunquam  habebitur  momentum  temporis, 
in  quo  jam  fuerit  existentia  infinitae  lineae  :  secus  vero,  si  per  aeternitatem  praecedentem 
jam  extiterit  :  nee  in  eo  futuram  aeternitatem  cum  prseterita  prorsus  analogam  esse 
censeo,  ut  illud  indefinitum  futurae  non  sit  verum  quoddam  infinitum  praeteritae.  Quod 
si  linea  infinita  non  fuerit,  &  quies  est  infinities  adhuc  improbabilior,  quam  regressus  pro 
unico  temporis  momento  ad  idem  spatii  punctum,  ac  multo  magis  aeterna  quies  :  utique 
nee  motum  habuit  aeternum  materia,  nee  existere  potuit  ab  seterno,  cum  sine  &  quiete,  & 
motu  existere  non  potuerit,  adeoque  creatione  omnino,  &  Creatore  fuit  opus,  qui  idcirco 
infinitam  haberet  effectivam  potentiam,  ut  omnem  creare  posset  materiam,  ac  infinitam 
determinativam  vim,  ut  libero  arbitrio  suo  utens  ex  omnibus  infinitis  possibilibus  momentis 
totius  aeternitatis  in  utramque  partem  indefinitae  illud  posset  seligere  individuum  momentum, 
in  quo  materiam  crearet,  ac  ex  omnibus  infinitis  illis  possibilibus  statibus,  &  quidem  tarn 
sublimi  infinitatis  gradu,  seligere  ilium  individuum  statum,  complectentem  unam  ex  illis 
curvis  per  omnia  puncta  dato  ordine  accepta  transeuntibus,  ac  in  ea  determinatas  illas 
distantias,  ac  determinatas  motuum  velocitates,  &  directiones. 


Validissima  ab  im- 
possibilitate  seriei 
infinitae  terminorum, 
in  qua  alii  ab  aliis 
determinentur  ad 
existendum  sine  ex- 
trinseco  determin- 
ants ;  ea  hie  demon- 
stratur. 


548.  Verum  hisce  omnibus  etiam  omissis,  est  illud  a  determinatione  itidem  necessaria 
repetitum,  in  qua  vis  Theoria  validissimum,  sed  adhuc  magis  in  mea,  in  qua  omnia  phaenomena 
pendent  a  curva  virium,  &  inertiae  vi.  Nimirum  materia  licet  ponatur  ejusmodi,  ut  habeat 
necessariam,  &  sibi  essentialem  vim  inertiae,  &  virium  activarum  legem  ;  adhuc  ut  quovis 
dato  tempore  posteriore  habeat  determinatum  statum,  quem  habet,  debet  determinari 
ad  ipsum  a  statu  praecedenti,  qui  si  fuisset  diversus,  diversus  esset  &  subsequens ;  neque 
enim  lapis,  qui  sequent!  tempore  est  in  Tellure,  ibi  esset ;  si  immediate  antecedent!  fuisset 
in  Luna.  Quare  status  ille,  qui  habetur  tempore  sequent!,  nee  a  se  ipso,  nee  a  materia, 
nee  ab  ullo  ente  materiali  turn  existente,  habet  determinationem  ad  existendum,  &  pro- 
prietates,  quas  habet  materia  perennes,  indifferentiam  per  se  continent,  nee  ullam 
determinationem  inducunt.  Determinationem  igitur,  quam  habet  ille  status  ad  existendum, 
accipit  a  statu  praecedenti.  Porro  status  praecedens  non  potest  determinare  sequentem, 
nisi  quatenus  ipse  determinate  existit.  Ipse  autem  nullam  itidem  in  se  habet  determin- 
ationem ad  existendum,  sed  illam  accipit  a  praecedente.  [259]  Ergo  nihil  habemus  adhuc 
in  ipso  secundum  se  considerato  determinationis  ad  existendum  pro  postremo  illo  statu. 
Quod  de  secundo  diximus,  dicendum  de  tertio  praecedente,  qui  determinationem  debet 
accipere  a  quarto,  adeoque  in  se  nullam  habet  determinationem  pro  existentia  sui,  nee 
idcirco  ullam  pro  existentia  postremi.  Verum  eodem  pacto  progrediendo  in  infinitum, 
habemus  infinitam  seriem  statuum,  in  quorum  singulis  habemus  merum  nihil  in  ordine 
ad  determinatam  existentiam  postremi  status.  Summa  autem  omnium  nihilorum  utcunque 
numero  infinitorum  est  nihil ;  jam  diu  enim  constitit,  ilium  Guidonis  Grandi,  utut  summi 
Geometrae,  paralogismum  fuisse,  quo  ex  expressione  seriei  parallelae  ortae  per  divisionem 

intulit  summam  infinitorum  zero  esse  revera  aequalem  dimidio.     Non  potest  igitur 

ilia  series  per  se  determinare  existentiam  cujuscunque  certi  sui  termini,  adeoque  nee  tota 
ipsa  potest  determinate  existere,  nisi  ab  ente  extra  ipsam  posito  determinetur. 


In   quo    hoc   argu- 

communi  adhibente  549-  Hoc  quidem  argumento  jam  ab  annis  multis  uti  soleo,  quod  &  cum  aliis  pluribus 

impossibiiitatem  se-  communicavi,  neque  id  ab  usitato  argumento,  quo  reiicitur  series  contingentium  infinita 

riei      contingentium      .  .       *         ,  •  •  ••••vj-rr  j 

sine  eote  necessario.  sine  ente  extrinseco  dante  existentiam  seriei  toti,  in   alio  dittert,  quam   in   eo,   quod   a 


APPENDIX  385 

&  this  must  occupy  finite  space  &  cannot  extend  indefinitely.  Now,  there  is  truly  no 
possible  reason  why  there  should  be  this  finite  number  of  points,  or  this  quantity  of  matter 
in  Nature,  rather  than  that  ;  except  the  will  of  a  Being  possessed  of  infinite  determinative 
power.  No  one  in  his  right  senses  will  easily  persuade  himself  seriously  that  there  is  any 
necessity  for  existence  in  any  one  number  of  points,  rather  than  m  any  other. 

547.  In  addition,  if  the  Universe  had  gone  on  with  these  laws  from  eternity,  then  A  third  argument 
already  there  would  have  been  eternal  motions,  &  straight  lines  described  by  the  several  n^d^n^wMch 
points  would  already  have  been  produced  to  infinity.  For  they  do  not  re-enter  themselves,  motions  have  last- 


except  by  the  will   of   a  Being  who  overcomes  the  infinite  improbability  ;    since  I  have  necesSu36  infinite13 

proved  above  in  several  places  that  it  is  infinitely  more  improbable  that  any  point  should  the  impossibility  of 

return  at  some  time  to  the  same  place  as  it  had  occupied  at  some  other  instant,  than  that  thls- 

no  point  should  ever  return.     Moreover,  I  have  proved  that  infinity  in  extension  is  quite 

impossible,  as  I  have  already  observed  ;    &  this  impossibility  must  pertain  to  every  kind 

of  lines  that  have  been  produced  indefinitely.     Anyhow,  the  motion  can  be  continued 

indefinitely  throughout  future  eternity  ;  for,  if  it  commenced  at  any  one  instant  there  never 

would  be  an  instant  of  time,  in  which  there  has  already  been  the  existence  of  an  infinite 

line  ;   but  otherwise,  if  it  has  already  existed  throughout  past  eternity.     However,  in  this 

connection,  I  do  not  think  that  future  eternity  is  quite  analogous  with  past  eternity  ;    so 

that  this  indefinite  of  the  future  is  not  really  the  same  thing  as  an  infinite  of  the  past. 

But  if  there  has  not  been  an  infinite  line  (&  absolute  rest  is  still  more  infinitely  improbable 

than  a  return  for  a  single  instant  to  the  same  point  of  space,  &  eternal  rest  is  even  more 

improbable  still),  then  it  certainly  follows  that  matter  cannot  have  had  eternal  motion,  nor 

can  it  have  existed  from  eternity.     For,  it  could  not  have  existed  without  both  rest  & 

motion  ;    &  thus,  there  was  altogether  a  need  for  creation,  &  a  Creator,  &  therefore  He 

would  have  an  infinite  effective  power,  so  that  He  could  create  all  matter,  &  an  infinite 

determinative  force  ;  so  that,  out  of  all  the  possible  instants,  infinite  in  number,  in  the 

whole  of  eternity  indefinitely  prolonged  in  either  direction,  He  could  choose  of  His  Own 

untrammelled  will  that   particular  instant  in  which  to  create   matter  ;    &  out  of  all  the 

infinite  number  of  possible  states,  &  this  to  such  a  high  degree  of  infinity,  He  could  select 

that  one  particular  state,  which  involves  one  of  those  curves  passing  through  all  the  points 

taken  in  a  certain  order  ;  &  in  it  could  choose  those  determinate  distances,  &  the  determinate 

velocities  &  directions  of  the  motions. 

548.  But,  leaving  all  these  things  out  of  the  question,  there  is  a  very  strong  argument  A    very  strong 
in  any  Theory,  derived  also  from  a  necessity  for  determination  ;    but  especially  strong  in  "o^the  impoS 
my  Theory,  where  all  phenomena  depend  on  a  curve  of  forces,  &  the  force  of  inertia.     Thus,  biiity  of  an  infinite 
although  matter  may  be  assumed  to  be  of  such  a  nature  as  to  have  a  necessary  &  essential  *£^  °theerideter° 
force  of  inertia  &  a  law  of  active  forces  ;  yet,  in  order  that  at  any  subsequent  time  it  may  minatkm  to  exist 
have  the  determinate  state,  which  it  actually  has,  it  must  be  determined  to  that  state,  from  that^o^ano^her 
the  state  just  preceding  ;  &  if  this  preceding  state  had  been  different,  the  subsequent  state  without  something 
would  also  have  been  different.     For  a  stone,  which  at  a  subsequent  instant  is  on  the  Earth,  enc^from'witho'ut1" 
would  not  have  been  there  at  the  instant,  if  at  the  instant  immediately  preceding  it  had  been  proof  of  the  impos- 
on  the  Moon.     Hence  the  state  which  occur?  at  the  subsequent  instant,  neither  of  itself,  slbUltv  here  &lven- 
nor  from  matter,  nor  from  any  material  entity  then  existing,  has  any  determination  to  exist  ; 

&  the  properties,  which  matter  has  unvarying,  contain  of  themselves  indifference  nor  do 
they  lead  to  any  determination.  The  determination,  then,  which  that  state  has  to  exist,  is 
derived  from  the  state  preceding  it.  Further,  a  preceding  state  cannot  determine  the  one 
which  follows  it,  except  in  so  far  as  it  itself  has  existed  determinately.  Moreover,  this 
preceding  state  also  has  no  determination  in  itself  to  exist,  but  derives  it  from  one  that 
precedes  it.  Consequently,  we  have  as  yet  nothing  in  this,  considered  by  itself,  yielding 
determination  to  exist  for  that  last  state.  What  has  been  said  with  regard  to  this  second  state, 
is  to  be  said  also  about  the  third  preceding  state  ;  this  must  receive  its  determination  from  a 
fourth,  &  so  in  itself  has  no  determination  for  its  own  existence,  nor  on  that  account  has  it 
any  for  the  existence  of  the  last  state.  Now,  going  on  indefinitely  in  the  same  manner,  we 
have  an  infinite  series  of  states,  in  each  of  which  we  have  absolutely  nothing  for  the  purpose 
of  determining  the  existence  of  the  last  state.  Moreover,  the  sum  of  all  these  nothings,  no 
matter  how  infinite  the  number  of  them,  is  nothing  also.  For,  it  has  been  long  ago  made 
clear  that  Guide  Grandi,  although  a  very  eminent  geometer,  enunciated  a  fallacy  when,  from 
an  expression  of  a  parallel  series  derived  by  division  of  I  by  (i  +  i),  he  deduced  that  the  sum 
of  an  infinite  number  of  zeros  was  really  equal  to  -J-.  Therefore,  that  series  of  states  T 

,  •!  f  •  •  i  .     .In     what     this 

cannot  determine  the  existence  of  any  particular  one  of  its  terms,  &  so  neither  can  the  whole  argument      differs 
of  it  exist  determinately,  unless  it  be  determined  by  a  Being  situated  without  itself.  *rom  tj1.6  usual  °°e' 

T  ,  iii-  /•  ,    T  i  •  i    •      depending  upon  the 

549.  1  have  employed  this  argument  for  many  years  past,  &  I  have  communicated  it  impossibility  of  a 
to  several  others  ;    it  does  not    differ  from  the  usual  argument  employed,  which  denies  se.ri^s  of  events 

..  ...  .          .     ~    .  .          -  .  .  .  .  ,     _r  .  -  '        .    .  .  without  a  necessary 

the  possibility  of  an  innnite  series  of  contingents  without  an  outside  Being  giving  existence  being. 

c  c 


386  PHILOSOPHIC  NATURALIS  THEORIA 

contingentia  res  ad  determinationem  est  translata,  &  a  defectu  determinationis  pro  sua 
cujusque  existentia  res  est  translata  ad  defectum  determinationis  pro  existentia  unius 
determinati  status  assumpti  pro  postremo  ;  id  autem  praestiti,  ne  eludatur  argumentum 
dicendo,  in  tota  serie  haberi  determinationem  ad  ipsam  totam,  cum  pro  quovis  termino 
habeatur  determinatio  intra  eandem  seriem,  nimirum  in  termino  praecedente.  Ilia 
reductione  ad  vim  determinativam  existentiae  postremi  quaesitam  per  omnem  seriem, 
devenitur  ad  seriem  nihilorum  respectu  ipsius  quorum  summa  adhuc  est  nihilum. 


m  "habere  55°'  Jam  vero  ^oc  ens  extrinsecum  seriei  ipsi,  quod  hanc  seriem  elegit  prae  seriebus 

debet.  aliis  infinitis  ejusdem  generis,  infinitam  habere  debet  determinativam,  &  electivam  vim, 

ut  unam  illam  ex  infinitis  seligat.  Idem  autem  &  cognitionem  habere,  debuit,  & 
sapientiam,  ut  hanc  seriem  ordinatam  inter  inordinatas  selegerit  :  si  enim  sine  cogni- 
tione,  &  electione  egisset,  infinities  probabilius  fuisset,  ab  illo  determinatum  iri  aliquam 
ex  inordinatis,  quam  unam  ex  ordinatis,  ut  hanc  ;  cum  nimirum  ratio  inordinatarum  ad 
ordinatas  sit  infinita,  &  quidem  ordinis  altissimi,  adeoque  &  excessus  probabilitatis  pro 
cognitione,  &  sapientia,  ac  libera  electione  supra  probabilitatem  pro  caeco  agendi  modo, 
fatalismo,  &  necessitate,  sit  infinitus,  qui  idcirco  certitudinem  inducit. 


brutas    "Ser°hic  55  J<  Atque  n^c  notandum  &  illud,  pro  quovis  indivi-[26o]-duo  statu  respondente 

occurrit,'a  quo  unp  cuivis  momento  temporis,  &  multo  magis  pro  quavis  individua  serie  respondente  cuivis 

rum'  !L°soio'  infinite  contmuo   tempori,    improbabilitas    determinatae    ipsius    existentias    est    infinita,    &  nos 

Ubero.  deberemus  esse  certi  de  ejus  non  existentia,  nisi  determinaretur  ab  infinite  determinantes 

&  nisi  ejus  determinationis  notitiam  nos  haberemus.     Sic  si  in  urna  sint  nomina  centum, 

&  unum,  &  agatur  de  uno  determinate,  an  extractum  inde  prodierit,  centuplo  major  est 

improbabilitas  ipsi   contraria  :    si   mille,  &  unum,   millecupla  :   si  numerus   sit   infinitus  ; 

improbabilitas   erit   infinita,    quae    in   certitudinem   transit  :    sed   si   quis   viderit   extrac- 

tionem,  &  nobis   nunciet  ;    tota  improbabilitas  ilia  repente    corruit.      Verum  &  in  hoc 

exemplo  individua  ilia  determinatio  a  create  agente  non  habebitur  inter  infinitas  possi- 

biles,  nisi  ex  legibus  ab  infinite  determinante  jam  determinatis  in  Natura,  &  ab  ejusdem 

determinatione  ad  individuum,  uti  paullo  ante  dicebamus  de  individuae  figurae  electione 

pro  statua. 

Quanta  sapientia  ^52.  Porro   qui  aliquanto  diligentius   perpenderit  vel  ilia  pauca,   quae  adnotavimus 

opus    fuerit    ad  -'-'..        ,.   ~  .,       .   »  ,    *  ~  r  IT  r     •      i     -1 

seiigendum   numer-  necessaria  in  distnbutione  punctoru-m  ad  efformanda  diversa  particularum  genera,  quae 
um    &   ordinem  exhibeant  diversa  corpora,  videbit  sane  quanta  sapientia,  &  potentia  sit  opus  ad  ea  omnia 

punctorum,  &  legem  ..,..,  r  ,->    •  i  S  «  i   •     •  T»II 

virium.  perspicienda,  eligenda,  praestanda.     Quid  vero,  ubi  cogitet,  quanta  altissimorum  rroble- 

matum  indeterminatio  occurrat  in  infinite  illo  combinationum  possibilium  numero,  & 
quanta  cognitione  opus  fuerit  ad  eligendas  illas  potissimum,  quae  necessariae  erant  ad  hanc 
usque  adeo  inter  se  connexorum  phaenomenorum  seriem  exhibendam  ?  Cogitet,  quid 
una  lux  praestare  debeat,  ut  se  propaget  sine  occursu,  ut  diversam  pro  diversis  coloribus 
refrangibilitatem  habeat,  &  diversa  vicium  intervalla,  ut  calorem  &  igneas  fermentationes 
excitet  :  interea  vero  aptandus  fuit  corporum  textus,  &  laminarum  crassitudo  ad  ea 
potissimum  remittenda  radiorum  genera,  quae  illos  determinates  colores  exhiberent  sine 
ceterarum  &  alterationum,  &  transformationum  jactura,  disponendse  oculorum  partes, 
ut  imago  pingeretur  in  fundo,  &  propagaretur  ad  cerebrum,  ac  simul  nutritioni  daretur 
locus,  &  alia  ejusmodi  praestanda  sexcenta.  Quid  unus  aer,  qui  simul  pro  sono,  pro 
respiratione,  &  vero  etiam  nutritione  animalium,  pro  diurni  caloris  conservatione  per 
noctem,  pro  ventis  ad  navigationem,  pro  vaporibus  continendis  ad  pluvias,  pro  innumeris 
aliis  usibus  est  conditus  ?  Quid  gravitas,  qua  perennes  fiunt  planetarum  motus,  &  comet- 
arum,  qua  omnia  compacta,  &  coadunata  in  ipsorum  globis,  qua  una  suis  maria  continentur 
littoribus,  &  currunt  fluvii,  imber  in  terram  decidit,  &  earn  irrigat,  ac  foecundat,  sua  mole 
aedificia  consistunt,  temporis  mensuram  exhibent  pendulorum  oscillationes  ?  [261]  si  ea 
repente  deficeret  ;  quo  noster  incessus,  quo  situs  viscerum,  quo  aer  ipse  sua  elasticitate 
dissiliens  ?  Homo  hominem  arreptum  a  Tellure,  &  utcunque  exigua  impulsum  vi,  vel 
uno  etiam  oris  flatu  impetitum,  ab  hominum  omnium  commercio  in  infinitum  expelleret, 
nunquam  per  totam  aeternitatem  rediturum. 

Congeries    e  o  r  u  m, 
quae  evincunt  in  eli- 

fan^tiamentrovi-  553*  ^ed  <lu^  eg°  nsec  singularia  persequor  ?   quanta  Geometria  opus  fuit  ad  eas  com- 

dentiam,  immensas, 


APPENDIX  387 

to  the  whole  series,  except  in  the  detail  that  the  matter  is  altered  from  a  contingence  to 
a  determination,  £  from  a  lack  of  determination  of  the  existence  of  any  thing  in  itself  the 
question  is  transferred  to  a  lack  of  determination  for  the  existence  of  one  determined  state 
assumed  as  the  last  of  the  series.  But  my  argument  is  superior  to  the  usual  one,  in  that  it 
cannot  be  evaded  by  saying  that  there  is  in  the  whole  series  a  determination  to  the  series 
as  a  whole  ;  since  for  any  term  there  is  a  determination  within  the  same  series,  namely 
one  derivable  from  the  preceding  term.  By  my  reduction  to  a  force  determining  existence 
of  the  last  term  throughout  the  whole  series,  the  result  is  a  series  of  zeros  with  regard  to 
this  last  term,  &  the  sum  of  these  is  still  zero. 

550.  Now,  the  Being  external  to  the  series,  which  chooses  this  series  in  preference  The  necessary 
to  all  others  of  the  infinite  number  in  the  same  class,  must  have  infinite  determinative  &  attributes   of    the 
elective  force,  in  order  that  He  may  select  this  one  out  of  an  infinite  number.     Also  He 

must  have  knowledge  &  wisdom,  in  order  to  select  this  regular  series  from  among  the 
irregular  series ;  for,  if  He  had  acted  without  knowledge  &  selection,  it  would  have  been 
infinitely  more  probable  that  there  would  have  been  a  determination  by  Him  of  one  of 
the  irregular  series,  than  of  one  of  the  regular  series,  such  as  the  one  in  question.  For 
the  ratio  of  the  number  of  irregular  series  to  the  number  of  regular  serie?  is  infinite,  &  that 
too  of  a  very  high  order  ;  &  thus,  the  excess  of  the  probability  in  favour  of  knowledge, 
wisdom,  &  arbitrary  selection  is  infinitely  greater  than  the  probability  in  favour  of  blind 
choice,  fatalism.  &  necessity ;  &  this  therefore  leads  to  a  certainty. 

551.  Here  also  it  is  to  be  observed  that  for  any  individual  state  corresponding:  to  any  The  sort  of  Being 

J  .  t     •  o  -L  f  •      i  f  •  who     could     over- 

given  instant  01  time,  &  much  more  tor  any  particular  series  corresponding  to  a  given  come  the    infinite 

continuous  time,  the  improbability  of  a  self-determined  existence  is  infinite  ;   &  we  ought  improbability 

i  .,.  A.  '     ,  ,  -11  •    «    • .        i  •  .     which  here  occurs : 

to  be  certain  of  its  non-existence,  unless  it  were  determined  by  an  infinite  determmator,  &  it  could  be  accom- 

we  had  evidence  of  the  determination.     Thus,  if  in  an  urn  there  are  a  hundred  &  one  names,  P1131"^  OI^y  .  b/ 

&  it  is  a  question  with  regard  to  one  determined  name,  whether  it  has  been  drawn  from  nitely  free. 

the  urn,  the  improbability  is  a  hundredfold  to  the  contrary ;   &  if  there  were  a  thousand 

&  one  names,  a  thousandfold  ;    if   the   number  of  names   is    infinite,   the   improbability 

will  be  infinite  ;  &  this  passes  into  a  certainty.     But  if  anyone  should  have  seen  the  drawing 

&  give  us  information,  then  the  whole  of  the  improbability  would  immediately  be  destroyed. 

Again,  in  this  example,  the  particular  determination  by  a  created  agent  will  not  be  from 

among  an  infinite  number  of  possibles,except  on  account  of  laws  already  determined  in  Nature 

by  an  infinite  Determinator  and  from  the  determination  to  the  individual  by  the  same  power ; 

as  I  said,  a  little  earlier,  when  speaking  of  the  selection  of  a  particular  form  for  a  statue. 

552.  Now,  if  anyone  will  consider  a  little  more  carefully  even  the  few  things  I  have  H.°w  great  the 

J.J         ,  '       .          ,  ,      i  .  ,         ',         ,  ...     °,.ff  Wisdom      would 

mentioned  as  necessary  in  the  arrangement  of  the  points  for  the  formation  of  the  different  have  to  be  to  select 
kinds  of  particles,  which  different  bodies  exhibit,  he  must  perceive  how  great  the  wisdom  &  the  nuniber  & 
power  must  needs  be,  to  comprehend,  select  &  establish  all  these  things.  What  then,  pomts.'Tthe  °aw  of 
when  he  considers  how  great  an  indeterminateness  in  problems  of  very  high  degree  occurs  forces- 
through  the  infinite  number  of  possible  combinations ;  &  how  great  the  knowledge  would 
have  to  be  to  select  those  of  them  especially,  which  were  necessary  to  yield  this  series  of 
phenomena  so  far  connected  with  one  another  ?  Let  him  consider  what  properties  the 
single  substance  called  light  must  exhibit,  such  that  it  is  propagated  without  collision,  that 
it  has  different  refrangibilities  for  different  colours,  &  different  intervals  between  its  fits,  that  it 
should  excite  heat,  &  fiery  fermentations.  At  the  same  time  the  texture  of  bodies  &  the 
thickness  of  plates  had  to  be  made  suitable  for  the  giving  forth  of  those  kinds  of  rays  especially, 
which  were  to  exhibit  determinate  colours,  without  sacrificing  other  alterations  and  trans- 
formations ;  the  arrangement  of  parts  of  the  eyes,  so  that  an  image  is  depicted  at  the  back 
&  propagated  to  the  brain  ;  &  at  the  same  time  place  should  be  given  to  nutrition,  &  thou- 
sands of  other  things  of  the  same  sort  to  be  settled.  What  the  properties  of  the  single 
substance  called. air,  which  at  one  &  the  same  time  is  suitable  for  sound,  for  breathing,  even 
for  the  nutrition  of  animals,  for  the  preservation  during  the  night  of  the  heat  received 
during  the  day,  for  holding  rain-clouds,  &  innumerable  other  uses.  What  those  of  gravity, 
through  which  the  motions  of  the  planets  &  comets  go  on  unchanged,  through  which  all 
things  became  compacted  &  united  together  within  their  spheres,  through  which  each  sea 
is  contained  within  its  own  bounds,  &  rivers  flow,  the  rain  falls  upon  the  earth  &  irrigates 
it,  &  fertilizes  it,  houses  stand  up  owing  to  their  own  mass,  &  the  oscillations  of  pendu- 
lums yield  the  measure  of  time.  Consider,  if  gravity  were  taken  away  suddenly,  what  would 
become  of  our  walking,  of  the  arrangement  of  our  viscera,  of  the  air  itself,  which  would  fly 
off  in  all  directions  through  its  own  elasticity.  A  man  could  pick  up  another  from  the  Groups  of  points. 
Earth,  &  impel  him  with  ever  so  slight  a  force,  or  even  but  blow  upon  him  with  his  breath,  which  prove 

o     j    •         i  •        e  •  -1111  •  •/-•  conclusively  the  im- 

&  drive  him  from  intercourse  with  all  humanity  away  to  infinity,  nevermore  to  return  measurabiiityofthe 
throughout  all  eternity.  power  wisdom  and 

553.  But  why  do  I  enumerate  these  separate  things  ?     Consider  how  much  geometry     rmp 


388  PHILSOPHI^  NATURALIS  THEORIA 

binationes  inveniendas,  quae  tot  organica  nobis  corpora  exhiberent,  tot  arbores,  &  flores 
educerent,  tot  brutis  animantibus,  &  hominibus  tarn  multa  vitae  instrumenta  submini- 
starent  ?  Pro  fronde  unica  efformanda  quanta  cognitione  opus  fuit,  &  providentia,  ut 
motus  omnes  per  tot  saecula  perdurantes,  &  cum  omnibus  aliis  motibus  tarn  arete  connexi 
illas  individuas  materiae  particulas  eo  adducerent,  ut  illam  demum,  illo  determinato  tempore 
frondem  illius  determinatae  curvaturae  producerent  ?  quid  autem  hoc  ipsum  respectu 
eorum,  ad  quae  nulli  nostri  sensus  pervadunt,  quae  longissime  supra  telescopiorum,  &  infra 
microscopiorum  potestatem  latent  ?  Quid  respectu  eorum,  quae  nulla  possumus 
contemplatione  assequi,  quorum  nobis  nullam  omnino  licet,  ne  levissimam  quidem  con- 
jecturam  adipisci,  de  quibus  idcirco,  ut  phrasi  utar,  quam  alibi  ad  aliquid  ejusdem  generis 
exprimendum  adhibui,  de  quibus  inquam,  hoc  ipsum,  ignorari  ea  a  nobis,  ignoramus  ? 
Ille  profecto  unus  immensam  Divini  Creatoris  potentiam,  sapientiam,  providentiam  humanae 
mentis  captum  omnem  longissime  superantes,  ignorare  potest,  qui  penitus  mente  cascutit, 
vel  sibi  ipsi  oculos  eruit,  &  omnem  mentis  obtundit  vim,  qui  Natura  altissimis  undique 
inclamante  vocibus  aures  occludit  sibi,  ne  quid  audiat,  vel  potius  (nam  occludere  non  est 
satis)  &  cochleam,  &  tympanum,  &  quidquid  ad  auditum  utcunque  confert, '  proscindit, 
dilacerat,  eruit,  ac  a  se  longissime  projectum  amovet. 

Quid  prospiciendum  CCA.  Sed  in  hac  tanta  eligentis,  ac  omnia  providentis  Supremi  Conditoris  sapientia, 

fuent   pro  nostra  JJT  •  .  j      •        •     j    i  o  •      -11     f      ji 

existentia,  &  nostris  atque  exsequentis  potentia,  quam  admirari  debemus  perpetuo,  &  veneran,  mud  adhuc 
commodis: quantum  magis  cogitandum  est  nobis,  quantum  inde  in  nostros  etiam  usus  promanarit,  quos  utique 

ipsi  inde  simus  ob-  •     -11  -j  •       o    /*  -i  •  •  •  •      o 

stricti.  respexit  me,  qurvidet  omnia,  &  fines  sibi  istos  omnes  constituit,  qui  per  ea  omnia  &  nostrae 

ipsi  existentiae  viam  stravit,  ac  nos  prae  infinitis  aliis  hominibus,  qui  existere  utique  poterant, 
elegit  ab  ipso  Mundi  exordio,  motus  omnes,  ad  horum,  quibus  utimur,  organorum  forma- 
tionem  disposuit,  praeter  ea  tarn  multa  quae  ad  tuendam,  &  conservandam  hanc  vitam, 
ad  tot  commoda,  &  vero  etiam  voluptates  conducerent.  Nam  illud  omnino  credendum 
firmissime,  non  solum  ea  omnia  vidisse  unico  intuitu  Auctorem  Naturae,  sed  omnes  eos 
animo  sibi  constitutes  habuisse  fines,  ad  quos  conducunt  media,  quae  videmus  adhibita. 

Mundum  non  esse  err.  Haud  ego  quidem  Leibnitianis,  &  aliis  quibuscunque  [262]  Optimismi  defen- 

possibilium     optim-  •,  •  •**»       j  i  •  •    •  o  • 

um,  cum  in  possibi-  sonbus  assentior,  qui  Mundum  nunc,  in  quo  vivimus,  &  cujus  pars  sumus,  omnium 
litms  nuiius  terminus  perfectissimum  esse  arbitrantur,  ac  Deum  faciunt  natura  sua  determinatum  ad  id  creandum 
officere  sapiential  qu°d  perfectissimum  sit,  ac  eo  ordine,  qui  perfectissimus  sit.  Id  sane  nee  fieri  posse  arbitror  : 
ac  bonitati  infinite,  cum  nimirum  in  quovis  possibilium  genere  seriem  agnoscam  finitorum  tantummodo, 

quod  non  fecerit,  nee  .      •     ~    .  r  r,  ..  ^  •       j*  ^       ._•'      j 

potentia,  quod  non  quanquam  in  mfimtum  productam,  ut  num.  90  exposui,  in  qua,  ut  in  distantns  duorum 
potuent  id  facere.  punctorum  nulla  est  minima,  nulla  maxima  ;  ita  ibidem  nulla  sit  perfectionis  maximse, 
nulla  minimae,  sed  quavis  finita  perfectione  utcunque  magna,  vel  parva,  sit  alia  perfectio 
major,  vel  minor  :  unde  fit,  ut  quancunque  seligat  Naturae  Auctor,  necessario  debeat 
alias  majores  omittere  :  nee  vero  ejus  potentiae  illud  ofHcit,  quod  creare  nonpossit  optimum, 
aut  maximum,  ut  nee  officit,  quod  non  possit  simul  creare  totum,  quodcunque  creare 
potest  :  nam  id  eo  evadit,  ut  non  possit  se  in  eum  statum  redigere,  in  quo  nihil  melius, 
aut  majus,  vel  absolute  nihil  aliud  creare  possit  :  nee  officit  aut  sapientiae,  aut  bonitati 
infinitae,  quod  optimum  non  seligat,  ubi  optimum  est  nullum. 


Quam  multa  pessima  cc6.  £x  ajia   parte  determinatio  ilia  ad  optimum,  &  libertatem   Divinam  tollit,  & 

consectaria  secum..  •  •  •       r 

trahat  sententia  contmgentiam  rerum  omnium,  cum,  quae  existunt,  necessana  riant,  quae  non  existunt, 
Mundi  perfectissimi.  evadant  impossibilia  ;  ac  praeterea  nobis  quodammodo  in  ilia  hypothesi  debemus,  quod 
existimus,  non  illi.  Qui  enim  potuit  non  existere  id,  quod  habuit  pro  sua  existentia 
rationem  praevalentem,  quam  Naturae  Auctor  cum  viderit,  non  potuerit  non  sequi,  nee 
vero  potuerit  non  videre  ?  Qui  existere  potuit  id,  quod  eandem  habuit  non  existendi 
necessitatem  ?  Quid  vero  illi  pro  nostra  existentia  debeamus,  qui  nos  condidit  idcirco, 
quia  in  nobis  invenit  meritum  majus,  quam  in  iis,  quos  omisit,  &  a  sua  ipsius  natura  necessario 
determinatus  fuit,  &  adactus  ad  obsequendum  ipsi  huic  nostro  intrinseco,  &  essential! 
merito  praevalenti  ?  Distinguendum  est  inter  hsec  duo  :  unum  esse  alio  melius,  &  esse 
melius  creare  potius  unum,  quam  aliud.  Illud  primum  habetur  ubique,  hoc  secundum 
nusquam,  sed  aeque  bonum  est  creare,  vel  non  creare  quodcunque,  quod  physicam  bonitatem 
quancunque  habeat,  utcunque  majorem,  vel  minorem  alio  quovis  omisso  :  solum  enim 


APPENDIX  389 

was  needed  to  discover  those  combinations  which  were  to  display  to  us  so  many 
organic  bodies,  produce  so  many  trees  &  flowers,  &  supply  so  many  instruments 
of  life  to  living  brutes  &  men.  For  the  formation  of  a  single  leaf,  how  great 
was  the  need  for  knowledge  &  foresight,  in  order  that  all  those  motions,  lasting 
for  so  many  ages,  &  so  closely  connected  with  all  other  motions,  should  so  bring 
together  those  particular  particles  of  matter,  that  at  length,  at  a  certain  deter- 
minate time,  they  should  produce  that  leaf  with  that  determinate  curvature.  What 
is  this  in  comparison  with  those  things  to  which  none  of  our  senses  can  penetrate,  things 
that  lie  hidden  far  &  away  beyond  the  power  of  telescopes,  &  too  small  lor  the 
microscope  ?  What  of  those  which  we  can  never  understand  no  matter  how  hard  we  think 
about  them,  of  which  we  can  never  attain  not  even  the  slightest  idea  ;  concerning  which 
therefore,  to  use  a  phrase  I  have  elsewhere  employed  to  express  something  of  the  same  sort, 
of  which  I  say  this : — "  We  do  not  know  the  very  fact  of  our  ignorance.  "  Undoubtedly 
he  alone  can  be  ignorant  of  the  immeasurable  power,  wisdom  &  foresight  of  the  Divine 
Creator,  far  surpassing  all  comprehension  of  the  human  intellect,  whose  mind  is 
altogether  blind,  or  who  tears  out  his  eyes,  &  dulls  every  mental  power,  who  shuts 
his  ears  to  Nature,  so  that  he  shall  not  hear  her  as  she  proclaims  in  accents  loud 
on  every  side,  or  rather  (for  to  shut  them  is  not  enough)  cuts  away,  tears  up  &  destroys, 
&  hurls  far  from  him  the  cochlea  &  the  tympanum  &  anything  else  that  helps  him  to  hear. 

554.  But,  in  this  great  wisdom  of  selection  &  universal  foresight  on  the  part  of  the  HOW  our  existence 
Supreme  Founder,  &  the  power  of  carrying  it  out,  there  i«  still  another  thing  for  us  to  and   our  c°nveni- 

r  .  ,  ,       ,  ,     /  111-1  i        ences    would    have 

consider  ;   namely,  how  much  proceeds  from  it  to  meet  the  needs  of  us,  who  are  all  under  to    be    provided 
the  care  of  Him  Who  sees  all  things,  &  has  imposed  on  Himself  the  accomplishment  of  all  for;  what  a  debt 

\Tn       i  ill  i        r  »  -11  11     n     r  i        we     are     therefore 

those  purposes ;  Who  has  smoothed  the  path  of  our  existence  with  them  all,  &  from  the  under  to  Him. 
commencement  of  the  Universe  has  chosen  us  in  preference  to  an  infinite  number  of  other 
human  beings  that  might  have  existed  ;  Who  has  planned  all  the  motions  nece>sary  for  the 
formation  of  the  organs  we  employ,  besides  all  the  many  things  that  should  conduce  towards 
the  protection  &  preservation  of  this  life,  to  its  many  conveniences,  nay,  even  to  its  pleasures. 
For,  it  cannot  be  but  a  matter  of  the  firmeat  belief,  not  only  that  the  Author  of  Nature  saw 
all  these  things  with  a  single  intuition,  but  also  that  He  had  settled  in  his  mind  all 
those  purposes,  to  which  the  means  that  we  see  employed  conduce. 

555.  I  do  not  indeed  agree  with  the  followers  of  Leibniz,  or  with  any  of  the  upholders  The  Universe  is  not 
of  Optimism,  who  consider  that  this  Universe,  in  which  we  live  &  of  which  we  are  part,  'the  . best  toi  a11 
is  the  most  perfect  of  all ;    &  who  thus  make  God  determined  by  His  own  nature  for  the  amongst  e  possibles 
creation  of  that  which  is  the  most  perfect,  &  in  that  order  which  is  the  most  perfect.      In  there    is    no . last 
truth,  I  think  that  such  a  thing  would  be  impossible ;    for,  I  recognize,  in  any  kind  of  argument*  Eigauist 
possibles,  a  series  of  finites  only,  although  prolonged  to  infinity,  as  I  explained  in  Art.  90  ;  infinite  wisdom  & 
&  in  this  series,  just  as  in  the  case  of  the  distances  between  two  points,  there  is  no  greatest  He'dkTn'ot  mTue^t 
or  least,  here  also  there  is  no  case  of  greatest  or  of  least  perfection  ;    but,  for  any  finite  so :  nor  against  His 
perfection,  however  great  or  small,  there  is  another  perfection  that  is  greater  or  smaller.  HeWwas  unaWe^to 
Hence  it  comes  about  that,  whatever  the  Author  of  Nature  should  select,  He  would  have  make  i<:  so- 

to  omit  some  that  were  of  greater  perfection.  But,  neither  is  it  an  argument  against  His 
power,  that  He  cannot  create  the  best  or  the  greatest  ;  nor  similarly  is  it  an  argument 
against  His  power  that,  whatever  He  could  create,  He  could  not  create  it  as  a  whole  at  one  & 
the  same  time.  For,  it  would  come  to  this,  that  He  would  put  Himself  in  the  position 
where  He  could  create  nothing  better,  nothing  greater,  or  absolutely  nothing  else.  Similarly, 
it  is  no  argument  against  His  infinite  wisdom  &  goodness,  that  He  did  not  select  the 
best,  when  there  is  no  best. 

556.  On  the  other  hand,  that  determination  for  the  best  takes  away  altogether  the  The     number    of 
freedom  of  God,  &  the  contingency  of  all  things ;  for,  those  things  which  exist  become  Fosi?  imperfections 

0      i  ,          -,  i  ..  ,          ¥»     « i  i         ,    °      i      .  involved  in  the  idea 

necessary,  &  those  that  do  not  are  impossible.  Besides,  on  that  hypothesis,  we  should  be  Of  a  most  perfect 
under  some  sort  of  obligation  to  ourselves,  &  not  to  Him,  for  the  fact  that  we  exist.  For  Universe- 
how  was  it  possible  that  a  thing  should  not  exist,  which  had  a  powerful  reason  for  its  exist- 
ence ;  for,  when  the  Author  of  Nature  saw  this  reason,  He  could  not  fail  to  follow  it, 
nor  indeed  could  He  fail  to  see  it  ?  How  could  a  thing  exist  which  had  a  like  need  for  non- 
existence  ?  For  what  should  we  have  to  thank  Him,  if  He  had  created  us  for  the  simple 
reason  that  in  us  He  found  a  greater  merit  than  in  those  whom  He  omitted,  if  He  was 
necessarily  determined  by  His  own  nature,  &  driven  by  it  to  submit  to  our  mere 
intrinsic  &  essential  overpowering  merit  ?  We  must  mark  the  distinction  between  the 
two  dictums  : — (i)  this  thing  is  better  than  that,  (2)  it  would  be  better  to  create  this 
thing  than  to  create  that.  There  is  a  possibility  of  the  first  in  all  cases,  but  never  any  of 
the  second.  It  is  an  equally  good  thing  to  create  or  not  to  create  anything  whatever, 
which  has  any  physical  goodness,  however  much  greater  or  less  than  anything  else  which 
has  been  omitted.  The  exercise  of  Divine  freedom  alone  is  infinitely  more  perfect  than 


390  PHILOSOPHIC  NATURALIS   THEORIA 

Divinse  libertatis  exercitium  infinities  perfectius  est  quavis  perfectione  creata,  quae  idcirco 
nullum  potest  offerre  Divinae  libertati  meritum  determinativum  ad  se  creandum. 

Media  tamen  idonea  557.  Cum  ea  infinita  libertate  Divina  componitur  tamen  illud,  quod  ad  sapientiam 

-  pertinet,  ut  ad  eos  fines,  quos  sibi  pro  liberrimo  suo  arbitrio  praefixit  Deus,  media  semper 
ad  fines  sibi  apta  debeat  seligerc,  quae  finem  propositum  frustrari  non  sinant.     Porro  haec  media  etiam 
1"  m  nostrum  bonum  selegit  plurima,  dum  totam  Naturam  conderet,  quod  quern  a  nobis 
exigat  beneficiorum  memorem,  &  gratum  animum,  quern  etiam  tan-[263]-tae  beneficentiae 
respondentem  amorem  cum  ingenti  ilia  admiratione,  &  veneratione  conjunctum,  nemo 
non  videt. 
Deduci  nos  inde  ad  rrg>  Superest  &  illud  innuendum.  neminem  sanae  mentis  hominem  dubitare  posse, 

revelationem,      quae          .     JJ      .        r  .  ,.  .  ,         .  ,.  .      r     .  . 

tamen  hue  non  quin,  qui  tantam  in  ordmanda  JNatura  providentiam  ostendit,  tantam  erga  nos  in  nobis 
pertineat,  ad  opus  seligendis,  in  consulendo  nostris  &  indigentiis.  &  commodis  beneficentiam,  illud  etiam 

mmirum  pure  philo-  ,         .  ......     °.  ,   . 

sophicum.  prasstare  voluerit,  ut  cum  adeo  imbecilla  sit,  &  nebes  mens  nostra,  &  ad  ipsius  cogmtionem 

per  sese  vix  quidquam  possit,  se  ipse  nobis  per  aliquam  revelationem  voluerit  multo  uberius 
praebere  cognoscendum,  colendum,  amandum ;  quo  ubi  devenerimus,  quae  inter  tarn 
multas  falso  jactatas  absurdissimas  revelationes  unica  vera  sit  perspiciemus  utique  admodum 
facile.  Sed  ea  jam  Philosophise  Naturalis  fines  excedunt,  cujus  in  hoc  opere  Theoriam 
meam  exposui,  &  ex  qua  uberes  hosce,  &  solidos  demum  fructus  percepi. 


APPENDIX  391 

any  perfection  created  ;    &  the  latter  can  therefore  offer  no  determinative  merit  to  the 
freedom  of  God  in  favour  of  its  own  creation. 

557.  With  this  infinite  Divine  liberty  is  bound  up  all  that  relates  to  wisdom;    for,  Fit  means,  however, 
God,  to  those  purposes  which  he  of  His  own  unfettered  will  had  designed,  was  always  be^ei^cted^^the 
bound  to  select  suitable  means,  such  as  would  not  allow  these  purposes  to  be  frustrated.  Author  of  Nature 
Further,  He  has  selected  these  means  for  the  most  part  suitable  for  our  welfare,  whilst  he  ^J^the  purposes 
founded  the  whole  of  Nature  ;   &  this  demands  from  us  a  remembrance  of  His  favours  &  a  He    has    designed 
thankful  heart,  nay,  even  a  love  that  shall  correspond  to  such  great  beneficence  together 

with  a  mighty  wonder  &  admiration,  as  every  one  will  see.  Him. 

558.  It  now  remains  but  to  mention  that  there  is  no  man  of  sound  mind  who  could  we  are  thus  led  to 
possibly  doubt  that  One,  Who  has  shown  such  great  foresight  in  the  arrangement  of  Nature,  revelation,     which 

,       '         .  .  ,    «      i      i  •  r         i       i  i  however    does    not 

such  great  beneficence  towards  us  in  selecting  us,  &  in  looking  after  both  our  needs  &  our  come    within   the 
comforts,  would  not  also  wish  to  accomplish  this  also  ;   namely  that,  since  our  mind  is  so  scop.e    °.f,  .such.  * 

',,,,.  ,         ,    .       f.  /.  _..      work  as  this,  which 

weak  &  dull  that  it  can  scarcely  of  itself  arrive  at  any  sort  of  knowledge  about  Him,  He  is    purely    phiio- 

would  have  wished  to  present  Himself  to  us  through  some  kind  of  revelation  much  more  fully 

to  be  known,  honoured  &  loved.     This  being  done,  we  should  indeed  quite  easily  perceive 

which  was  the  only  true  one,  from  amongst  so  many  of  those  absurdities  falsely  brought 

forward  as  revelations.     But  such  things  as  this  already  exceed  the  scope  of  a  Natural 

Philosophy,  of  which  in  this  work  I  have  explained  my  Theory,  &  from  which  I  have  finally 

gathered  such  ripe  &  solid  fruit. 


[264]   SUPPLEMENT  A 

§1 
De    Spatio,  ac  Tempore 

Argumentum:  quae  i.  Ego   materiae  extensionem   prorsus   continuam   non   admitto,   sed   earn   constituo 

spatu  attributa.  punctis  prorsus  indivisibilibus,  &  inextensis  a  se  invicem  disjunctis  aliquo  intervallo,  & 
connexis  per  vires  quasdam  jam  attractivas,  jam  repulsivas  pendentes  a  mutuis  ipsorum 
distantiis.  Videndum  hie,  quid  mihi  sit  in  hac  sententia  spatium,  ac  tempus,  quomodo 
utrumque  dici  possit  continuum,  divisibile  in  infinitum,  aeternum,  immensum,  immobile, 
necessarium,  licet  neutrum,  ut  in  ipsa  adnotatione  ostendi,  suam  habeat  naturam  realem 
ejusmodi  proprietatibus  prasditam. 


2-  InPrimis  i^ud  mmi  videtur  evidens,  tarn  eos,  qui  spatium  admittunt  absolutum, 
reaies  modos  exist-  natura  sua  reali,  continuum,  sternum,  immensum,  tarn  eos,  qui  cum  Leibnitianis,  & 
tem"  Cartesianis  ponunt  spatium  ipsum  in  ordine,  quern  habent  inter  se  res,  quae  existunt,  praeter 
ipsas  res,  quae  existunt,  debere  admittere  modum  aliquem  non  pure  imaginarium,  sed  realem 
existendi,  per  quem  ibi  sint,  ubi  sunt,  &  qui  existat  turn,  cum  ibi  sunt,  pereat  cum  ibi  esse 
desierint,  ubi  erant.  Nam  admissso  etiam  in  prima  sententia  spatio  illo,  si  hoc,  quod  est 
esse  rem  aliquam  in  ea  parte  spatii,  haberetur  tantummodo  per  rem,  &  spatium  ;  quoties- 
cunque  existeret  res,  &  spatium,  haberetur  hoc,  quod  est  rem  illam  in  ea  spatii  parte  collocari. 
Rursus  si  in  posteriore  sententia  ordo  ille,  qui  locum  constituit,  haberetur  per  ipsas  tantum- 
modo res,  quae  ordinem  ilium  habent,  quotiescunque  res  illae  existerent,  eodem  semper 
existerent  ordine  illo,  nee  proinde  unquam  locum  mutarent.  Atque  id,  quod  de  loco 
dixi,  dicendum  pariter  de  tempore. 

Quocunque  is  modus  3.  Necessario  igitur  admittendus  est  realis  aliquis  existendi  modus,  per  quem  res  est 

nomine  appeiietur.    jj^  u^-  es^  £  tunij  cum  esti     gjve  js  mO(jus  dicatur  res,  sive  modus  rei,  sive  aliquid,  sive 

nonnihil  ;  is  extra  nostram  imaginationem  esse  debet,  &  res  ipsum  mutare  potest,  habens 
jam  alium  ejusmodi  existendi  modum,  jam  alium. 

Modi  reaies,  qui  sint          ^  ]?gO  igitur  pro  singulis  materise  punctis,  ut  de  his  [265]  loquar,  e  quibus  ad  res  etiam 

tempus?  a       m'       immateriales  eadem  omnia  facile  transferri  possunt,  admitto  bina  realia  modorum  existendi 

genera,  quorum  alii  ad  locum  pertineant,  alii  ad  tempus,  &  illi  locales,  hi  dicantur  tem- 

porarii.     Quodlibet  punctum  habet  modum  realem  existendi,  per  quem  est  ibi,  ubi    est, 

&  alium,  per  quem  est  turn,  cum  est.     Hi  reaies  existendi  modi  sunt  mihi  reale  tempus, 

&  spatium  :    horum  possibilitas  a  nobis  indefinite  cognita  est  mihi  spatium  vacuum,  & 

tempus    itidem,  ut    ita  dicam,    vacuum,    sive    etiam    spatium    imaginarium,   &    tempus 

imaginarium. 

Eorum  natura,  &  c    Modi  illi  reaies  singuli  &  oriuntur,  ac  pereunt,  &  indivisibiles  prorsus  mihi  sunt,  ac 

relationes.  .•>...  °.  ,.         .  T.    „  ,  «• 

inextensi,  &  immobiles,  ac  in  suo  ordine  immutabiles.  Ii  &  sua  ipsorum  loca  sunt  realia, 
ac  tempora,  &  punctorum,  ad  quae  pertinent.  Fundamentum  praebent  realis  relationis 
distantiae,  sive  localis  inter  duo  puncta,  sive  temporariae  inter  duos  eventus.  Nee  aliud 
est  in  se,  quod  illam  determinatam  distantiam  habeant  ilia  duo  materiae  puncta,  quam  quod 
illos  determinatos  habeant  existendi  modos,  quos  necessario  mutent,  ubi  earn  mutent 
distantiam.  Eos  modos,  qui  in  ordine  ad  locum  sunt,  dico  puncta  loci  realia,  qui  in  ordine 
ad  tempus,  momenta,  quae  partibus  carent  singula,  ac  omni  ilia  quidem  extensione,  haec 
duratione,  utraque  divisibilitate  destituuntur. 

Contiguitas  puncto-  6.  pOrro  punctum  materiae  prorsus  indivisibile,  &  inextensum,  alteri  puncto  materiae 

Spa  "  contiguum  esse  non  potest  :   si  nullam  habent  distantiam  ;   prorsus  coeunt  :   si  non  coeunt 

penitus  ;    distantiam  aliquam  habent.     Neque  enim,  cum  nullum  habeant  partium  genus, 

(a)  Hie,  fcsf  sequens  paragraphus  habentur  in  Suppkmtntis  tomi  I.  Philosophic  Retentions  Benedicti  Stay,  §  6,  W  7. 

392 


SUPPLEMENTS 

§1 

Of  Space  and  Time  (a) 

1.  I  do  not  admit  perfectly  continuous  extension  of  matter  ;   I  consider  it  to  be  made  The  theme;  what 
up  of  perfectly  indivisible  points,  which  are  non-extended,  set  apart  from  one  another  by  oiespa«:  ?attrlbutes 
a  certain  interval,  &  connected  together  by  certain  forces  that  are  at  one  time  attractive  & 

at  another  time  repulsive,  depending  on  their  mutual  distances.  Here  it  is  to  be  seen, 
with  this  theory,  what  is  my  idea  of  space,  &  of  time,  how  each  of  them  may  be  said  to  be 
continuous,  infinitely  divisible,  eternal,  immense,  immovable,  necessary,  although  neither 
of  them,  as  I  have  shown  in  a  note,  have  a  real  nature  of  their  own  that  is  possessed  of  these 
properties. 

2.  First  of  all  it  seems  clear  to  me  that  not  only  those  who  admit  absolute  space,  which  Real     local     and 
is  of  its  own  real  nature  continuous,  eternal  &  immense,  but  also  those  who,  following  Leibniz  existence  "m^tf  °f 
&  Descartes,  consider  space  itself  to  be  the   relative  arrangement   which   exists  amongst  necessity    be    ad- 
things  that  exist,  over  and  above  these  existent  things ;  it  seems  to  me,  I  say,  that  all  must  ^"ed    by    eVWy 
admit  some  mode  of  existence  that  is  real  &  not  purely  imaginary ;   through  which  they  are 

where  they  are,  &  this  mode  exists  when  they  are  there,  &  perishes  when  they  cease  to  be 
where  they  were.  For,  such  a  space  being  admitted  in  the  first  theory,  if  the  fact  that 
there  Is  some  thing  in  that  part  of  space  depends  on  the  thing  &  space  alone  ;  then,  as  often 
as  the  thing  existed,  &  space,  we  should  have  the  fact  that  that  thing  was  situated  in  that 
part  of  space.  Again,  if,  in  the  second  theory,  the  arrangement,  which  constitutes  position, 
depended  only  on  the  things  themselves  that  have  that  arrangement ;  then,  as  often  as 
these  things  should  exist,  they  would  exist  in  the  same  arrangement,  &  could  never  change 
their  position.  What  I  have  said  with  regard  to  space  applies  equally  to  time. 

3.  Therefore  it  needs  must  be  admitted  that  there  is  some  real  mode  of  existence,  due  The  name  by  which 
to  which  a  thing  is  where  it  is,  &  exists  then,  when  it  does  exist.     Whether  this  mode  is  is  immaterial, 
called  the  thing,  or  the  mode  of  the  thing,  or  something  or  nothing,  it  is  bound  to  be 

beyond  our  imagination  ;  &  the  thing  may  change  this  kind  of  mode,  having  one  mode  at 
one  time  &  another  at  another  time. 

4.  Hence,  for  each  of  the  points  of  matter  (to  consider  these,  from  which  all  I  say  Real  modes ;  what 
can  be  easily  transferred  to  immaterial  things),  I  admit  two  real  kinds  of  modes  of  existence,  [fme  ^ay  be. 

of  which  some  pertain  to  space  &  others  to  time  ;  &  these  will  be  called  local  &  temporal 
modes  respectively.  Any  point  has  a  real  mode  of  existence,  through  which  it  is  where  it 
is ;  &  another,  due  to  which  it  exists  at  the  time  when  it  does  exist.  These  real  modes 
of  existence  are  to  me  real  time  &  space  ;  the  possibility  of  these  modes,  hazily  apprehended 
by  us,  is,  to  my  mind,  empty  space  &  again  empty  time,  so  to  speak  ;  in  other  words,  imagin- 
ary space  &  imaginary  time. 

5.  These  several  real  modes  are  produced  &  perish,  and  are   in  my  opinion    quite  Their  nature   & 
indivisible,  non-extended,  immovable  &  unvarying  in  their  order.     They,  as  well  as  the  relatlons- 
positions  &  times    of  them,    &   of   the    points   to  which   they  belong,   are  real.     They 

afford  the  foundation  of  a  real  relation  of  distance,  which  is  either  a  local  relation  between 
two  points,  or  a  temporal  relation  between  two  events.  Nor  is  the  fact  that  those  two 
points  of  matter  have  that  determinated  distance  anything  essentially  different  from  the 
fact  that  they  have  those  determinated  modes  of  existence,  which  necessarily  alter  when 
they  change  the  distance.  Those  mode?  which  are  descriptive  of  position  I  call  real 
points  of  position  ;  &  those  that  are  descriptive  of  time  I  call  instants ;  &  they  are  without 
parts,  &  the  former  lack  any  kind  of  extension,  while  the  latter  lack  duration  ;  both  are 
indivisible. 

6.  Further,  a  point  of  matter  that  is  perfectly  indivisible  &  non-extended  cannot  be  Contiguity    of 
contiguous  to  any  other  point  of  matter  ;  if  they  have  no  distance  from  one  another,  they 

coincide  completely ;  if  they  do  not  coincide  completely,  they  have  some  distance  between 

(a)  This  W  the  fallowing  section  are  to  be  found  in  the  Philosophise  Recentior,  by  Benedict  Stay,  Vol.  I,  §  6,  7. 

393 


394  PHILOSOPHIC  NATURALIS   THEORIA 

possunt  ex  parte  coire  tantummodo,  &  ex  parte  altera  se  contingere,  ex  altera  mutuo 
aversari.  Praejudicium  est  quoddam  ab  infantia,  &  ideis  ortum  per  sensus  acquisitis,  a 
debita  reflexione  destitutis,  qui  nimirum  nobis  massas  semper  ex  partibus  a  se  invicem 
distantibus  compositas  exhibuerunt,  cum  videmur  nobis  puncta  etiam  invisibilia,  &  inextensa 
posse  punctis  adjungere  ita,  ut  se  contingant,  &  oblongam  quandam  seriem  constituant. 
Globules  re  ipsa  nobis  confingimus,  nee  abstrahimus  animum  ab  extensione  ilia,  &  partibus, 
quas  voce,  &  ore  secludimus. 

Posse  binis  punctis  7.  Porro  ubi  bina  materiae  puncta  a  se  invicem  distant,  semper  aliud  materiae  punctum 

aaai  alia  in  directum  n          •   •       j-  i  i  IT- 

ad  distantias  squa-  potest  collocan  m  dnectum  ultra  utrumque  ad  eandem  distantiam,  &  alterum  ultra  hoc, 
^  &  *ta  Porro>  ut  Patet>  sme  ullo  fine.  Potest  itidem  inter  utrumque  collocari  in  medio  aliud 
punctum,  quod  neutrum  continget  :  si  enim  alterum  contigeret,  utrumque  contingeret, 
adeoque  cum  utroque  congrueret,  &  ilia  etiam  congruerent,  non  distarent,  contra  hypo- 
thesim.  Dividi  igitur  poterit  illud  intervallum  in  partes  duas,  ac  eodem  argumento  ilia 
itidem  duo  in  alias  quatuor,  &  ita  porro  sine  ullo  fine.  Quamobrem,  utcunque  ingens 
fuerit  binorum  punctorum  intervallum,  semper  [266]  aliud  haberi  poterit  majus,  utcunque 
id  fuerit  parvum,  semper  aliud  haberi  poterit  minus,  sine  ullo  limite,  &  fine. 


8>  Hmc  u^tra'  &  inter  kma  *oc*  Puncta  realia  quaecunque  alia  loci  puncta  realia  possibilia 
finita.  numero,  &  in  sunt,  quas  ab  iis  recedant,  vel  ad  ipsa  accedant  sine  ullo  limite  determinato,  &  divisibilitas  realis 
pOTsibii^u^nuUum  ^nterva^i  inter  duo  puncta  in  infinitum  est,  ut  ita  dicam,  interseribilitas  punctorum  realium 
finem.  sine  ullo  fine.  Quotiescunque  ilia  puncta  loci  realia  inteiposita  fuerint,  interpositis  punctis 

materiae  realibus,  finitus  erit  eorum  numerus,  finitus  intervalloium  numerus  illo  priore 
interceptorum,  &  ipsi  simul  aequalium  :  at  numerus  ejusmodi  partium  possibilium  finem 
habebit  nullum.  Illorum  singulorum  magnitudo  certa  erit,  ac  finita  :  horum  magnitude 
minuetur  ultra  quoscunque  limites,  sine  ullo  determinato  hiatu,  qui  adjectis  novis  inter- 
mediis  punctis  imminui  adhuc  non  possit;  licet  nee  possit  actuali  divisione,  sive  inter- 
positione  exhauriri. 


con-  9'  ^mc  vero  dum  concipimus  possibilia  haec  loci  puncta,  spatii  infinitatem,  & 
tinuum.necessarium  continuitatem  habemus,  cum  divisibilitate  in  infinitum.  In  existentibus  limes  est  semper 
sternum^iminobite  certus,  certus  punctorum  numerus,  certus  intervallorum  :  in  possibilibus  nullus  est  finis. 
praecisivam.  Possibilium  abstracta  cognitio,  excludens  limitem  a  possibili  augmento  intervalli,  & 

diminutione,  ac  hiatu,  infinitatem  lineae  imaginarae,  &  continuitatem  constituit,  quae 
partes  actu  existentes  non  habet,  sed  tantummodo  possibiles.  Cumque  ea  possibilitas  & 
aeterna  sit,  &  necessaria,  ab  seterno  enim,  &  necessario  verum  fuit,  posse  ilia  puncta  cum 
illis  modis  existere  ;  spatium  hujusmodi  imaginarium  continuum,  infinitum,  simul  etiam 
aeternum  fuit,  &  necessariurn,  sed  non  est  aliquid  existens,  sed  aliquid  tantummodo  potens 
existere,  &  a  nobis  indefinite  conceptum  :  immobilitas  autem  ipsius  spatii  a  singulorum 
punctorum  immobilitate  orietur. 

in  momentis  eadem,  IQ.  Atque  haec  omnia,  quae  hucusque  de  loci  punctis  sunt  dicta,  ad  temporis  momenta 

quae     in     punctis  :         j  iiir-i  •  •  111  i      • 

post  primum  nullum  e°dem  modo  admodum  facile  transferuntur,  inter  quae  ingens  qusedam  habetur  analogia. 
secundum,  aut  ulti-  Nam  &  punctum  a  puncto,  &  momentum  a  momento  quo  vis  determinato  certain  distantiam 

mum:    sed  in  tern-   i     i  •  •  •  •««••«*  ...... 

pore  unica  dimensio,  habet,  nisi  coeunt,  qua  major,  &  minor  haben  alia  potest  sine  ullo  limite.  In  quo  vis 
in  spatio  triplex.  intervallo  spatii  imaginarii,  ac  temporis  adest  primum  punctum,  vel  momentum,  &  ultimum, 
secundum  vero,  &  penultimum  habetur  nullum  :  quovis  enim  assumpto  pro  secundo,  vel 
penultimo,  cum  non  coeat  cum  primo,  vel  ultimo,  debet  ab  eo  distare,  &  in  eo  intervallo 
alia  itidem  possibilia  puncta  vel  momenta  interjacent.  Nee  punctum  continuae  lineae, 
nee  momentum  continui  temporis,  pars  est,  sed  limes  &  terminus.  Linea  continua,  & 
tempus  continuum  generari  intelligentur  non  repetitione  puncti,  vel  momenti,  sed  ductu 
continue,  in  quo  intervalla  alia  aliorum  sint  partes,  non  ipsa  puncta,  vel  momenta,  quae 
continue  ducuntur.  Illud  unicum  erit  [267]  discrimen,  quod  hie  ductus  in  spatio  fieri 
poterit,  non  in  unica  directione  tantum  per  lineam,  sed  in  infinitis  per  planum,  quod 
concipietur  ductu  continue  in  latus  lineae  jam  conceptae,  &  iterum  in  infinitis  per  solidum, 
quod  concipietur  ductu  continue  plani  jam  concept!,  in  tempore  autem  unicus  ductus 
durationis  habebitur,  quod  idcirco  soli  lineae  erit  analogum,  &  dum  spatii  imaginarii  extensio 


SUPPLEMENT  I  395 

them.  For,  since  they  have  no  kind  of  parts,  they  cannot  coincide  partly  only  ;  that  is, 
they  cannot  touch  one  another  on  one  side,  &  on  the  other  side  be  separated.  It  is  but 
a  prejudice  acquired  from  infancy,  &  born  of  ideas  obtained  through  the  senses,  which 
have  not  been  considered  with  proper  care  ;  £  these  ideas  picture  masses  to  us  as  always 
being  composed  of  parts  at  a  distance  from  one  another.  It  is  owing  to  this  prejudice 
that  we  seem  to  ourselves  to  be  able  to  bring  even  indivisible  and  non-extended  points  so 
close  to  other  points  that  they  touch  them  &  constitute  a  sort  of  lengthy  series.  We  imagine 
a  series  of  little  spheres,  in  fact ;  &  we  do  not  put  out  of  mind  that  extension,  &  the  parts, 
which  we  verbally  exclude. 

7.  Again,  where  two  points  of  matter  are  at  a  distance  from  one  another,  another  Given  two  points, 
point  of  matter  can  always  be  placed  in  the  same  straight  line  with  them,  on  the  far  side  ^the^Tn^tne^ame 
of  either,  at  an  equal  distance  ;  &  another  beyond  that,  &  so  on  without  end,  as  is  evident,  straight    line     at 
Also  another  point  can  be  pkced  halfway  between  the  two  points,  so  as  to  touch  neither  apart*  *•    &lStitnC<is 
of  them  ;  for,  if  it  touched  either  of  them  it  would  touch  them  both,  &  thus  would  coincide  possible  to  insert 
with  both  ;  hence  the  two  points  would  coincide  with  one  another  &  could  not  be  separate  them"  t^Tny 
points,  which  is  contrary  to  the  hypothesis.     Therefore  that  interval  can  be  divided  into  extent '  in     either 
two  parts  ;  &  therefore,  by  the  same  argument,  those  two  can  be  divided  into  four  others,  case' 

&  so  on  without  any  end.  Hence  it  follows  that,  however  great  the  interval  between  two 
points,  we  could  always  obtain  another  that  is  greater  ;  &,  however  small  the  interval 
might  be,  we  could  always  obtain  another  that  is  smaller  ;  '&,  in  either  case,  without  any 
limit  or  end. 

8.  Hence  beyond  &  between  two  real  points  of  position  of  any  sort  there  are  other  The   number   of 
real  points  of  position  possible  ;    &  these  recede  from  them  &  approach  them  respectively,  ^^  wUi^aiways 
without  any  determinate  limit.     There  will  be  a  real  divisibility  to  an  infinite  extent  of  be  finite,    &   the 
the  interval  between  two  points,  or,  if  I  may  call  it  so,  an  endless  '  insertibility '  of  real  the^finite^there 
points.     However  often  such  real  points  of  position  are   interpolated,  by  real  points  of  is  no  end  to  the 
matter  being  interposed,  their  number  will  always  be  finite,  the  number  of  intervals  possit 
intercepted  on  the  first  interval,  &  at  the  same  time  constituting  that  interval,  will  be  finite ; 

but  the  number  of  possible  parts  of  this  sort  will  be  endless.  The  magnitude  of  each 
of  the  former  will  be  definite  &  finite  ;  the  magnitude  of  the  latter  will  be  diminished 
without  any  limit  whatever  ;  &  there  will  be  no  gap  that  cannot  be  diminished  by  adding 
fresh  points  in  between  ;  although  it  cannot  be  completely  removed  either  by  division  or 
by  interposition  of  points. 

9.  In  this  way,  so  long  as  we  conceive  as  possibles  these  points  of  position,  we  have  Hence,  the  manner 
infinity  of  space,  &  continuity,  together  with  infinite  divisibility.     With  existing  things  a°  ^ace^hat^is 
there  is  always  a  definite  limit,  a  definite  number  of  points,  a  definite  number  of  intervals ;  finite,    continuous, 
with  possibles,  there  is  none  that  is  finite.     The  abstract  concept  of  possibles,  excluding  ^immovabi^"^ 
as  it  does  a  limit  due  to  a  possible  increase  of  the  interval,  a  decrease  or  a  gap,  gives  us  the  means  of  an  abstract 
infinity  of  an  imaginary  line,  &  continuity  ;    such  a  line  has  not  actually  any  existing  parts,  concePt- 

but  only  possible  ones.  Also,  since  this  possibility  is  eternal,  in  that  it  was  true  from 
eternity  &  of  necessity  that  such  points  might  exist  in  conjunction  with  such  modes,  space 
of  this  kind,  imaginary,  continuous  &  infinite,  was  also  at  the  same  time  eternal  &  necessary  ; 
but  it  is  not  anything  that  exists,  but  something  that  is  merely  capable  of  existing,  &  an 
indefinite  concept  of  our  minds.  Moreover,  immobility  of  this  space  will  come  from 
immobility  of  the  several  points  of  position. 

10.  Everything,  that  has  so  far  been  said  with  regard  to  points  of  position,  can  quite  The    same  things 
easily  in  the  same  way  be  applied  to  instants  of  time  ;  &  indeed  there  is  a  very  great  analogy  j^  ;f°L7or  points* 
of  a  sort  between  the  two.     For,  a  point  from  a  given  point,  or  an  instant  from  a  given  after'   the      first 
instant,  has  a  definite  distance,  unless  they  coincide  ;    &  another  distance  can  be  found  or^as't ;  n<inSetimed 
either  greater  or  less  than  the  first,  without  any  limit  whatever.     In  any  interval  of  imaginary  however,  there  is 
space  or  time,  there  is  a  first  point  or  instant,  &  a  last ;   but  there  is  no  second,  or  last  but  ^iie "In 

one.  For,  if  any  particular  one  is  supposed  to  be  the  second,  then,  since  it  does  not  coincide  there  are  three, 
with  the  first,  it  must  be  at  some  distance  from  it ;  &  in  the  interval  between,  other  possible 
points  or  instants  intervene.  Again,  a  point  is  not  a  part  of  a  continuous  line,  or  an  instant 
a  part  of  a  continuous  time  ;  but  a  limit  &  a  boundary.  A  continuous  line,  or  a  continuous 
time  is  understood  to  be  generated,  not  by  repetition  of  points  or  instants,  but  by  a  continuous 
progressive  motion,  in  which  some  intervals  are  parts  of  other  intervals ;  the  points  them- 
selves, or  the  instants,  which  are  continually  progressing,  are  not  parts  of  the  intervals. 
There  is  but  one  difference,  namely,  that  this  progressive  motion  can  be  accomplished 
in  space,  not  only  in  a  single  direction  along  a  line,  but  in  infinite  directions  over  a  plane 
which  is  conceived  from  the  continuous  motion  of  the  line  already  conceived  in  the  direction 
of  its  breadth  ;  &  further,  in  infinite  directions  throughout  a  solid,  which  is  conceived 
from  the  continuous  motion  of  the  plane  already  conceived.  Whereas,  in  time  there  will 
be  had  but  one  progressive  motion,  that  of  duration  ;  &  therefore  this  will  be  analogous 


396  PHILOSOPHIC  NATURALIS  THEORIA 

habetur  triplex  in  longum,  latum,  &  protundum,  temporis  habetur  unica  in  longum,  vel 
diuturnum  tantummodo.  In  triplici  tamen  spatii,  &  unico  temporis  geneie,  punctum, 
ac  momentum  erit  principium  quoddam,  a  quo  ductu  illo  suo  haec  ipsa  generata  intelligentur. 

Quodyis    punctum  n.  Illud  jam  hie  diligenter  notandum  :  non  solum  ubi  duo  puncta  materiae  existunt, 

tegnmf  spatium,  "ac"  &  aliquam  distantiam  habent,  existere  duos  modos,  qui  relationis  illius  distantiae  funda- 
tempus imaginanum  mentum  prsebeant,  &  sint  bina  diversa  puncta  loci  realia,  quorum    possibilitas  a  nobis 

suum :  quidsitcom-  i_-i.  u-  ...*...        j  •    £    •   •  •t-'VL 

penetratio.  concepta,  exhibeat   bma   puncta  spatii  imaginarn,  adeoque  mfimtis  numero  possibmbus 

materiae  punctis  respondere  infinites  numero  possibiles  existendi  modos,  sed  cuivis  puncto 
materiae  respondere  itidem  infinites  possibiles  existendi  modos,  qui  sint  omnia  ipsius  puncti 
possibilia  loca.  Haec  omnia  satis  sunt  ad  totum  spatium  imaginarium  habendum,  &  quodvis 
materiae  punctum  habet  suum  spatium  imaginarium  immobile,  infinitum,  continuum, 
quae  tamen  ornnia  spatia  pertinentia  ad  omnia  puncta  sibi  invicem  congruunt,  &  habentur 
pro  unico.  Nam  si  assumatur  unum  punctum  reale  loci  ad  unum  materiae  punctum 
pertinens,  &  conferatur  cum  omnibus  punctis  realibus  loci  pertinentibus  ad  aliud  punctum 
materiae ;  est  unum  inter  haec  posteriora,  quod  si  cum  illo  priore  coexistat,  relationem 
inducet  distantiae  nullius,  quam  compenetrationem  appellamus.  Unde  patet  punctorum, 
quae  existunt,  distantiam  nullam  non  esse  nihil,  sed  relationem  inductam  a  binis  quibusdam 
existendi  modis.  Reliquorum  quivis  cum  illo  eodem  priore  induceret  relationem  aliam, 
quam  dicimus  cujusdam  determinatae  distantiae,  &  positionis.  Porro  ilia  loci  puncta,  quae 
nullius  distantiae  relationem  inducunt,  pro  eodem  accipimus,  &  quenvis  ex  infinitis  hujus- 
modi  punctis  ad  infinita  puncta  materiae  pertinentibus  pro  eodem  accipimus,  ac  ejusdem 
loci  nomine  intelligimus.  Ea  autem  haberi  debere  pro  quovis  punctorum  binario,  sic 
patet.  Si  tertium  punctum  ubicunque  collocetur,  habebit  aliquam  distantiam,  &  positionem 
respectu  primi.  Summoto  prime,  poterit  secundum  collocari  ita,  ut  habeat  eandem  illam 
distantiam,  &  positionem,  respectu  tertii,  quam  habebat  primum.  Igitur  modus  hie,  quo 
existit,  pro  eodem  habetur,  ac  modus,  quo  existebat  illud  primum,  &  si  hi  bini  modi  simul 
existerent,  nullius  distantiae  relationem  inducerent  inter  primum,  ac  secundum  :  &  haec 
pariter,  quae  hie  de  spatii  punctis  dicta  sunt,  aeque  temporis  momentis  conveniunt. 


piura  momenta  [268]  12.  An  autem  possint  simul  existere,  id  vero  pertinet  ad  relationem,  quam 
n  nabent  puncta  loci  cum  momentis  temporis,  sive  spectetur  unicum  materiae  punctum, 
sive  plura.  Inprimis  plura  momenta  ejusdem  puncti  materiae  coexistere  non  possunt,  sed 
alia  necessario  post  alia,  sic  itidem  bina  puncta  localia  ejusdem  puncti  materiae  conjungi 
non  possunt,  sed  alia  jacere  debent  extra  alia,  atque  id  ipsum  ex  eorum  natura,  &  ut 
ajunt,  essentia. 


J3'  Deinde  considerentur  conjunctiones  variae  punctorum  loci,  &  momentorum. 
poris  pro  unico  Quodvis  punctum  materiae,  si  existit,  conjungit  aliquod  punctum  spatii  cum  aliquo 
Puatuor  "ro*  binis  momento  temporis.  Nam  necessario  alicubi  existit,  &  aliquando  existit  ;  ac  si  solum 
notabiies  :  existat,  semper  suum  habet,  &  localem,  &  temporarium  existendi  modum,  per  quod,  si 
5  osrtf  a^uc*  °iuodpiam  existat,  quod  suos  itidem  habebit  modus,  distantiae  &  localis,  &  temporis 
relationem  ad  ipsum  acquiret.  Id  saltern  omnino  accidet,  si  omnium,  quae  existunt,  vel 
existere  possunt,  commune  est  spatium,  ut  puncta  localia  unius,  punctis  localibus  alterius 
perfecte  congruant,  singula  singulis.  Quid  enim,  si  alia  sunt  rerum  genera,  vel  a  nostris 
dissimilium,  vel  nostris  etiam  prorsus  similium,  quae  aliud,  ut  ita  dicam,  infinitum  spatium 
habeant,  quod  a  nostro  itidem  infinite  non  per  intervallum  quoddam  finitum,  vel  infinitum 
distet,  sed  ita  alienum  sit,  ita,  ut  ita  dicam,  alibi  positum,  ut  nullum  cum  hoc  nostro  com- 
mercium  habeat,  nullam  relationem  distantiae  inducat.  Atque  id  ipsum  de  tempore  etiam 
dici  posset  extra  omne  nostrum  aeternum  tempus  collocate.  At  id  menti,  ipsum  conanti 
concipere,  vim  summam  infert,  ac  a  cogitatione  directa  admitti  vel  nullo  modo  potest, 
vel  saltern  vix  potest.  Quamobrem  iis  rebus,  vel  rerum  spatiis,  &  temporibus,  quae  ad  nos 
nihil  pertinere  possent,  prorsus  omissis,  agamus  de  notris  hisce.  Si  igitur  primo  idem 
punctum  materiae  conjungat  idem  punctum  spatii,  cum  pluribus  momentis  temporis  aliquo 
a  se  invicem  intervallo  disjunctis  ;  habebitur  regressus  ad  eundem  locum.  Si  secundo  id 
conjungat  cum  serie  continua  momentorum  temporis  continui  ;  habebitur  quies,  quae 
requirit  tempus  aliquod  continuum  cum  eodem  loci  puncto,  sine  qua  conjunctione  habetur 
continuus  motus,  succedentibus  sibi  aliis,  atque  aliis  loci  punctis,  pro  aliis,  atque  aliis 


SUPPLEMENT   I  397 

to  a  single  line.  Thus,  while  for  imaginary  space  there  is  extension  in  three  dimensions, 
length,  breadth  &  depth,  there  is  only  one  for  time,  namely  length  or  duration  only. 
Nevertheless,  in  the  threefold  class  of  space,  &  in  the  onefold  class  of  time,  the  point  & 
the  instant  will  be  respectively  the  element,  from  which,  by  its  progression,  motion,  space 
&  time  will  be  understood  to  be  generated. 

11.  Now  here  there  is  one  thing  that  must  be  carefully  noted.     Not  only  when  two  Every    point    of 
points  of  matter  exist,  &  have  a  distance  from  one  another,  do  two  modes  exist  which  give  ^fhe *t h°o lefTi 
the  foundation  of  the  relation  of  this  distance  ;    &  there  are  two  different  real  points  of  imaginary     space, 
position,  the  possibility  of  which,  as  conceived  by  us,  will  yield  two  points  of  imaginary  naturelmof;   con> 
space  ;   &  thus,  to  the  infinite  number  of  possible  points  of  matter  there  will  correspond  penetration. 

an  infinite  number  of  possible  modes  of  existence.  But  also  to  any  one  point  of  matter 
there  will  correspond  the  infinite  possible  modes  of  existing,  which  are  all  the  possible 
positions  of  that  point.  All  of  these  taken  together  are  sufficient  for  the  possession  of 
the  whole  of  imaginary  space  ;  &  any  point  of  matter  has  its  own  imaginary  space,  immovable, 
infinite  &  continuous ;  nevertheless,  all  these  spaces,  belonging  to  all  points  coincide  with 
one  another,  &  are  considered  to  be  one  &  the  same.  For  if  we  take  one  real  point  of 
position  belonging  to  one  point  of  matter,  &  associate  it  with  all  the  real  points  of  position 
belonging  to  another  point  of  matter,  there  is  one  among  the  latter,  which,  if  it  coexist 
with  the  former,  will  induce  a  relation  of  no-distance,  which  we  call  compenetration. 
From  this  it  is  clear  that,  for  points  which  exist,  no-distance  is  not  nothing,  but  a  relation 
induced  by  some  two  modes  of  existence.  Any  of  the  others  would  induce,  with  that 
same  former  point  of  position,  another  relation  of  some  determinate  distance  &  position, 
as  we  say.  Further,  those  points  of  position,  which  induce  a  relation  of  no-distance,  we 
consider  to  be  the  same  ;  &  we  consider  any  of  the  infinite  number  of  such  points  belonging 
to  the  infinite  number  of  points  of  matter  to  be  the  same  ;  &  mean  them  when  we  speak 
of  the  '  same  position.'  Moreover  this  is  evidently  bound  to  be  true  for  any  pair  of  points. 
If  now  a  third  point  is  situated  anywhere,  it  will  have  some  distance  &  position  with  respect 
to  the  first.  If  the  first  is  removed,  the  second  can  be  so  situated  that  it  has  the  same 
distance  &  position  with  respect  to  the  third  as  the  first  had.  Hence  the  mode,  in  which 
it  exists,  will  be  taken  to  be  the  same  in  this  case  as  the  mode  in  which  the  first  point  was 
existing  ;  &  if  these  two  modes  were  existing  together,  they  would  induce  a  relation  of 
no-distance  between  the  first  point  &  the  second.  All  that  has  been  said  above  with  regard 
to  points  of  space  applies  equally  well  to  instants  of  time. 

12.  Now,  whether  they  can  coexist  is  a  question  that  pertains  to  the  relation  between  Several  instants 
points  of  position  &    instants  of  time,  whether  we  consider  a  single  point  of  matter  or  ^"gp"fnt  Cannot 
several  of  them.     In  the  first  place,  several  instants  of  time  belonging  to  the  same  point  coexist. 

of  matter  cannot  coexist ;  but  they  must  necessarily  come  one  after  .the  other  ;  &  similarly, 
two  points  of  position  belonging  to  the  same  point  of  matter  cannot  be  conjoined,  but 
must  lie  one  outside  the  other  ;  &  this  comes  from  the  nature  of  points  of  this  kind,  & 
is  essential  to  them,  to  use  a  common  phrase. 

13.  Next,  we  have  to  consider  the  different  kinds  of  combinations  of  points  of  space  Four  combinations 
&  instants  of  time.     Any  point  of  matter,  if  it  exists,  connects  together  some  point  of  ^  sTn^ie  VoSt  *  of 
space  &  some  instant  of  time  ;    for  it  is  bound  to  exist  somewhere  &  sometime.     Even  if  matter  ;     four 
it  exists  alone,  it  always  has  its  own  mode  of  existence,  both  local  &  temporal ;   &  by  this  fo^twtr'pofa^ 
fact,  if  any  other  point  of  matter  exists,  having  its  own  modes  also,  it  will  acquire  a  relation  extraordinary  idea 
of  distance,  both  local  &  temporal,  with  respect  to  the  first.     This  at  least  will  certainly  s 

be  the  case,  if  the  space  belonging  to  all  that  exist,  or  can  possibly  exist,  is  common  ;  so 
that  the  points  of  position  belonging  to  the  one  coincide  perfectly  with  those  belonging 
to  the  other,  each  to  each.  But,  what  if  there  are  other  kinds  of  things,  either  different  from 
those  about  us,  or  even  exactly  similar  to  ours,  which  have,  so  to  speak,  another  infinite 
space,  which  is  distant  from  this  our  infinite  space  by  no  interval  either  finite  or  infinite, 
but  is  so  foreign  to  it,  situated,  so  to  speak,  elsewhere  in  such  a  way  that  it  has  no  com- 
munication with  this  space  of  ours ;  &  thus  will  induce  no  relation  of  distance.  The 
same  remark  can  be  made  with  regard  to  a  time  situated  outside  the  whole  of  our  eternity. 
But  such  an  idea  requires  an  intellect  of  the  greatest  power  to  try  to  grasp  it ;  &  it  cannot 
be  admitted  by  direct  consideration,  in  any  way,  or  at  least  with  difficulty.  Hence, 
omitting  altogether  such  things,  or  the  spaces  &  times  of  such  things  which  are  no  concern 
of  ours,  let  us  consider  the  things  that  have  to  do  with  us.  If  therefore,  firstly,  the  same, 
point  of  matter  connects  the  same  point  of  space  with  several  instants  of  time  separated 
from  one  another  by  any  interval,  there  will  be  return  to  the  same  place.  If,  secondly, 
it  connects  the  point  of  space  to  a  continuous  series  of  instants  of  continuous  time,  there 
will  be  rest,  which  requires  a  certain  continuous  time  to  be  connected  with  the  same  point 
of  position  ;  without  this  connection  there  will  be  continuous  motion,  points  of  position 
succeeding  one  another  corresponding  to  instants  of  time,  one  after  the  other.  Thirdly, 


398  PHILOSOPHIC  NATURALIS  THEORIA 

momentis  temporis.  Si  tertio  idem  punctum  materiae  conjungat  idem  momentum  tem- 
poris  cum  pluribus  punctis  loci  a  se  invicem  distantibus  aliquo  intervallo  ;  habebitur  ilia, 
quam  dicimus  replicationem.  Si  quarto  id  conjungat  cum  serie  continua  punctorum  loci 
aliquo  intervallo  continue  contentorum,  habebitur  quaedam  quam  plures  Peripatetici 
admiserunt,  virtualem  appellantes  extensionem,  qua  indivisibilis,  &  partibus  omnino 
destituta  materiae  particula  spatium  divisibile  occuparet.  Sunt  alias  quatuor  combina- 
tiones,  ubi  plura  materiae  pun-[269J-cta  considerentur.  Nimirum  quinto  si  conjungant 
idem  momentum  temporis  cum  pluribus  punctis  loci,  in  quo  sita  est  coexistentia.  Sexto 
si  conjungant  idem  punctum  spatii  cum  diversis  momentis  temporis,  quod  fieret  in  successive 
appulsu  diversorum  punctorum  materiae  ad  eundem  locum.  Septimo  si  conjungant  idem 
momentum  temporis  cum  eodem  puncto  spatii,  in  quo  sita  esset  compenetratio.  Octavo 
si  nee  momentum  ullum,  nee  punctum  spatii  commune  habeant,  quod  haberetur,  si  nee 
coexisterent,  nee  ea  loca  occuparent,  quae  ab  aliis  occupata  fuissent  aliquando. 

Relation**  earum  j,    gx  }j,isce  octo  casibus  primo  respondet  tertius,  secundo  quartus,  quinto  sextus, 

ad  se  invicem :  quae,  ,  T  __,.  ..*«..  T- 

&  quomodo  possi-  septimo  octavus.  Tertium  casum,  mmirum  replicationem,  commumtur  censent  naturahter 
haberi  non  posse.  Quartum  censent  multi  habere  animam  rationalem,  quam  putant  esse 
in  spatio  aliquo  divisibili,  ut  plures  Peripatetici  in  toto  corpore,  alii  Philosophi  in  quadam 
cerebri  parte,  vel  in  aliquo  nervorum  succo  ita,  ut  cum  indivisibilis  sit,  tota  sit  in  toto 
spatio,  &  tota  in  quavis  spatii  parte,  quemadmodum  eadem  indivisibilis  Divina  Natura 
est  in  toto  spatio,  &  tota  in  qualibet  spatii  parte,  ubique  necessario  praesens,  &  omnibus 
creatarum  rerum  realibus  locis  coexistens,  ac  adstans.  Eundem  alii  casum  in  materia 
admittunt,  cujus  particulas  eodem  pacto  extendi  putant,  ut  diximus  ;  licet  simplices  sint, 
licet  partibus  expertes,  non  modo  actu  separatis,  sed  etiam  distinctis,  ac  tantummodo 
separabilibus.  Earn  sententiam  amplectendam  esse  non  censeo  idcirco,  quod  ubicunque 
materiam  loca  distincta  occupantem  sensu  percipimus,  separabilem  etiam,  ingenti  saltern 
adhibita  vi,  videmus ;  sejunctis  partibus,  quae  distabant  :  nee  vero  alio  ullo  argumento 
excludimus  a  Natura  replicationem,  nisi  quia  nullam  materiae  partem,  quantum  sensu 
percipere  possumus,  videmus,  bina  simul  occupare  loca.  Virtualis  ilia  extensio  materiae 
infinities  ulterius  progreditur  ultra  simplicem  replicationem. 


Quietem,  &  regres-  jr    Si  secundus  casus  quietis,  &  primus  casus  regressus  ad  eundem  locum  naturaliter 

sum     ad     eundem    ,     ,       .  .A  .  ,  ,  ...  A 

locum    in    Natura  haberi  possent,  esset  is  quidem  defectus  quidam  analogiae  inter  spatium,  &  tempus.     At 
esse  in   mfinitum  j^j^j  v}deor  probare  illud  posse,  neutrum  unquam  in  Natura  contingere,  adeoque  naturaliter 

improbabiles,  &,,.  .  ,  c. 

indeingensanaiogia.  haberi  non  posse.  Id  autem  evinco  hoc  argumento.  bit  punctum  matenas  quodam 
momento  in  quodam  spatii  puncto,  &  pro  quovis  alio  momento  ignorantes,  ubi  sit,  quaeramus, 
quanto  probabilius  sit,  ipsum  alibi  esses,  quam  ibidem.  Tanto  erit  probabilius  illud, 
quam  hoc,  quanto  plura  sunt  alia  spatii  puncta,  quam  illud  unicum.  Hsec  in  quavis  linea 
sunt  infinita,  infinitus  in  quovis  piano  linearum  numerus,  infinitus  in  toto  spatio  planorum 
numerus.  Quare  numerus  aliorum  punctorum  est  infinitus  tertii  generis,  adeoque  ilia 
probabilitas  major  infinities  tertii  generis  infinitate,  ubi  de  quovis  alio  determinato  momento 
agitur.  Agatur  jam  inde-[269]-finite  de  omnibus  momentis  temporis  infiniti,  decrescet 
prior  probabilitas  in  ea  ratione,  qua  momenta  crescunt,  in  quorum  aliquo  saltern  posset 
ibidem  esse  punctum.  Sunt  autem  momenta  numero  infinita  infinitate  ejusdem  generis, 
cujus  puncta  possibilia  in  linea  infinita.  Igitur  adhuc  agendo  de  omnibus  momentis 
infiniti  temporis  indefinite,  est  infinities  infinite  improbabilius,  quod  punctum  in  eodem 
illo  priore  sit  loco,  quam  quod  sit  alibi.  Consideretur  jam  non  unicum  punctum  loci 
determinato  unico  momento  occupatum,  sed  quodvis  punctum  loci,  quovis  indefinite 
momento  occupatum,  &  adhuc  probabilitas  regressus  ad  aliquod  ex  iis  crescet,  ut  crescit 
horum  loci  punctorum  numerus,  qui  infinite  etiam  tempore  est  infinitus  ejusdem  ordinis, 
cujus  est  numerus  linearum,  in  quovis  piano.  Quare  improbabilitas  casus,  quo  determin- 
atum  quodpiam  materiae  punctum  redeat,  quovis  indefinite  momento  temporis,  ad  quodvis 
indefinite  punctum  loci,  in  quo  alio  quovis  fuit  momento  temporis  indefinite  sumpto, 
remanet  infinita  primi  ordinis.  Eadem  autem  pro  omnibus  materiae  punctis,  quae  numero 
finita  sunt,  decrescit  in  ratione  finita  ejus  numeri  ad  unitatem  (quod  secus  accidit  in  com- 
muni  sententia,  in  qua  punctorum  materise  numerus  est  infinitus  ordinis  tertii).  Quare 


SUPPLEMENT  I  399 

if  the  same  point  of  matter  connects  the  same  instant  of  time  with  several  points  of  position 
distant  from  one  another  by  some  interval,  then  we  shall  have  replication.  Fourthly,  if 
it  connects  the  instant  with  a  continuous  series  of  points  of  position  contained  within  some 
continuous  interval,  we  shall  have  something  which  several  of  the  Peripatetics  admitted, 
calling  it  virtual  extension  ;  by  virtue  of  which  an  indivisible  particle  of  matter,  quite 
without  parts,  could  occupy  divisible  space.  There  are  four  other  combinations,  when 
several  points  are  considered.  That  is  to  say,  fifthly,  if  several  points  connect  the  same 
instant  of  time  with  several  points  of  position  ;  in  this  is  involved  coexistence.  Sixthly, 
if  they  connect  the  same  point  of  space  with  several  instants  of  time  ;  as  would  be  the 
case  when  different  points  of  matter  were  forced  successively  into  the  same  position. 
Seventhly,  if  they  connect  the  same  point  of  space  with  the  same  instant  of  time  ;  in 
this  is  involved  compenetration.  Eighthly,  if  they  have  no  instant  of  time,  &  no  point 
of  space,  common  to  them  ;  as  would  be  the  case,  if  they  did  not  coexist,  nor,  any  of  them, 
occupied  the  positions  that  had  been  occupied  by  any  of  the  others  at  any  time. 

14.  Out  of  these  eight  cases,  the  third  corresponds  to  the  first,  the  fourth  to  the  second,  The    relations    of 
the  sixth  to  the  fifth,  the  eighth  to  the  seventh.     The  third  case,  namely  replication,  is  anotherT^Mch0"" 
usually  considered  to  be  naturally  impossible.     Many  think  that  the  fourth  case  holds  them  are  possible, 
good  for  the  rational  soul,  which  they  consider  to  have  its  seat  in  some  divisible  space  ; 

for  instance,  the  Peripatetics  think  that  it  pervades  the  whole  of  the  body,  other  philosophers 
think  it  is  situated  in  a  certain  part  of  the  brain,  or  in  some  juice  of  the  nerves  ;  so  that, 
since  it  is  indivisible,  the  whole  of  it  must  be  in  the  whole  of  the  space,  &  the  whole  of  it  in 
any  part  of  the  space.  Just  in  the  same  way  as  the  same  indivisible  Divine  Nature  is  as 
a  whole  in  the  whole  of  space,  &  as  a  whole  in  any  part  of  space,  being  necessarily  present 
everywhere,  &  coexisting  with  &  accompanying  created  things  wherever  created  things  are. 
Others  admit  this  same  case  for  matter,  &  consider  that  particles  of  matter  can  be  extended  in 
a  similar  manner,  as  we  have  said  ;  although  they  are  simple,  &  although  they  are  devoid 
of  parts,  not  only  parts  that  are  really  separated,  but  also  such  as  are  distinct  &  only  -separable. 
I  do  not  consider  that  this  supposition  can  be  entertained,  for  the  reason  that,  whenever 
we  perceive  with  our  senses  matter  occupying  positions  distinct  from  one  another,  we  see 
that  it  is  also  separable,  although  we  may  have  to  use  a  very  great  force  ;  here,  parts  are 
separated  which  were  at  a  distance  from  one  another.  Indeed,  by  no  other  argument 
can  we  exclude  replication  from  Nature,  than  that  we  never  see  any  portion  of 
matter,  as  far  as  can  be  perceived  by  the  senses,  occupying  two  positions  at  the  same 
time.  The  idea  of  Virtual  extension  of  matter  goes  infinitely  further  beyond  the  idea 
of  simple  replication. 

15.  If  the  second  case  of  rest,  &  the  first  case  of  return  to  the  same  position  could  ^est  &  "turn  to 
be  obtained  naturally,  then  indeed  there  would  be  a  certain  defect  in  the  analogy  between 


space  &  time.     But  it  seems  to  me  that  I  can  prove  that  neither  ever  happens  in  Nature  ;  £able   in  Nature  ; 

&  so  they  cannot  be  obtained  naturally  ;    this  is  my  argument.     If  a  point  of  matter  at  great6  analogy 

any  instant  of  time  is  at  a  certain  point  of  space,  &  we  do  not  know  where  it  is  at  some  tween  them- 

other  instant,  let  us  inquire  how  much  more  probable  it  is  that  it  should  be  somewhere 

else  than  at  the  same  point  as  before.     The  former  will  be  more  probable  than  the  latter 

in  the  proportion  of  the  number  of  all  the  other  points  of  space  to  that  single  point.     There 

are  an  infinite  number  of  these  points  in  any  straight  line,  the  number  of  lines  in  any  plane 

is  infinite,  &  the  number  of  planes  in  the  whole  of  space  is  infinite.     Hence,  the  number 

of  other  points  of  space  is  an  infinity  of  the  third  order  ;  &  thus  the  probability  is  infinitely 

greater  with  an  infinity  of  the  third  order,  when  we  are  concerned  with  any  other  particular 

instant  of  time.     Now  let  us  deal  indefinitely  with  all  the  instants  of  infinite  time  ;   then 

the  first  probability  will  decrease  in  proportion  as  the  number  of  instants  increases,  at  any 

of  which  it  might  at  least  be  possible  that  the  point  was  in  the  same  place  as  before.    Moreover, 

there  are  an  infinite  number  of  instants,  the  infinity  being  of  the  same  order  as  that  of  the 

number  of  possible  points  in  an  infinite  line.  Hence,  still  considering  indefinitely  all  the  instants 

of  infinite  time,  it  is  infinitely  more  improbable  that  the  point  should  be  in  the  same  position 

as  before,  than  that  it  should  be  somewhere  else.     Now  consider,  not  a  single  point  of  position 

occupied  at  a  single  particular  instant,  but  any  point  of  position  occupied  at  any  indefinite 

instant  ;    then  still  the  probability  of  return  to  any  one  of  these  points  of  position  will 

increase  as  the  number  of  them  increases  ;   &  this  number,  in  a  time  that  is  also  infinite, 

is  an  infinity  of  the  same  order  as  the  number  of  lines  in  any  plane.     Hence  the  improbability 

of  this  case,  in  which  any  particular  point  of  matter  returns  at  some  indefinite  instant  of 

time  to  some  indefinite  point  of  position,  in  which  it  was  assumed  to  be  at  some  other 

indefinite  instant  of  time,  remains  an  infinity  of  the  first  order.     Moreover,  this,  for  all 

points  of  matter,  which  are  finite  in  number,  will  decrease  in  the  finite  ratio  of  this  number 

to  infinity  (which  would  not  be  the  case  with  the  usual  theory,  in  which  the  number  of 

points  of  matter  is  taken  to  be  an  infinity  of  the  third  order).     Hence  we  are  still  left  with 


400  PHILOSOPHIC  NATURALIS  THEORIA 

adhuc  remanet  infinita  improbabilitas  regressus  puncti  materiae  cujusvis  indefinite,  ad 
punctum  loci  quodvis,  occupatum  quovis  momento  praecedenti  indefinite,  regressus  inquam, 
habendi  quovis  indefinite  momento  sequent!  temporis,  qui  regressus  idcirco  sine  ullo  erroris 
metu  debet  excludi,  cum  infinitam  improbabilitatem  in  relativam  quandam  impossibilitatem 
migrare  censendum  sit  :  quae  quidem  Theoria  communi  sententiae  applicari  non  potest. 
Quamobrem  eo  pacto,  patet,  in  mea  materiae  punctorum  Theoria  e  Natura  tolli  &  quietem, 
quam  etiam  supra  exclusimus,  &  vero  etiam  regressum  ad  idem  loci  punctum,  in  quo 
semel  ipsum  punctum  materias  extitit  :  unde  fit,  ut  omnes  illi  primi  quatuor  casus  exclud- 
antur  ex  Natura,  &  in  iis  accurata  temporis,  &  spatii  servetur  analogia. 

Nuiium  punctum  jg.  Quin  imo  si  quaeratur,  an  aliquod  materiae  punctum  occupare  debeat  quopiam 

materue  advenire  ,      -1  ,     ,.  ^          ,.  ,.     ,r  .  .  T."wr*' 

ad  uiium  punctum  momento  punctum  loci,  quod  alio  momento  aliquo  aliud  materiae  punctum  occupavit ;  adhuc 
spatii    in   quo  improbabilitas  erit  infinities  infinita.     Nam  numerus  punctorum  materiae  existentium  est 

aliquando     fuerit    c    f  .  .          .         .         ,         r  .      . 

aliud  punctum  fimtus,  adeoque  si  pro  regressu  puncti  cujusvis  ad  puncta  loci  a  se  occupata  adhibeatur 
quodvis.    in    sola  regressus  ad  puncta  occupata  a  quovis  alio,  numerus  casuum  crescit  in  ratione  unitatis  ad 

coexistentia  respon-  cr.  *    •  .      .  .  .  r    .  TT. 

dente  huic  adventui  numerum  punctorum  nnitum  utique,  nimirum  in  ratione  finita  tantummodo.  Hmc 
analogiam.  improbabilitas  appulsus  alicujus  puncti  materiae  indefinite  sumpti  ad  punctum  spatii 
aliquando  ab  alio  quovis  puncto  occupati  adhuc  est  infinita,  &  ipse  appulsus  habendus  pro 
impossibili,  quo  quidem  pacto  excluditur  &  sextus  casus,  qui  in  eo  ipso  situs  erat  regressu, 
&  multo  magis  septimus,  qui  binorum  punctorum  mate-[27i]-riae  simultaneum  appulsum 
continet  ad  idem  aliquod  loci  punctum,  sive  compenetrationem.  Octavus  autem  pro 
materia  excluditur,  cum  tota  simul  creata  perpetuo  duret  tota,  adeoque  semper  idem 
momentum  habeat  commune.^)  Solus  quintus  casus,  quo  plura  materiae  puncta  idem 
momentum  temporis  cum  diversis  punctis  loci  conjungant,  non  modo  possibilis  est,  sed 
etiam  necessarius  pro  omnibus  materiae  punctis,  coexistentibus  nimirum  :  fieri  enim  non 
potest,  ut  septimus,  &  octavus  excludantur  ;  nisi  continue  ob  id  ipsum  includatur  quintus 
ille,  ut  consideranti  patebit  facile.  Quamobrem  in  eo  analogia  deficit,  quod  possint  plura 
materiae  puncta  conjungere  diversa  puncta  spatii  cum  eodem  momento  temporis,  qui  est 
hie  casus  quintus,  non  autem  possit  idem  punctum  spatii,  cum  pluribus  momentis  temporis, 
qui  est  casus  tertius,  quern  defectum  necessario  inducit  exclusio  septimi,  &  octavi,  quorum 
altero  incluso,  excludi  posset  hie  quintus,  ut  si  possent  materiae  puncta,  quae  simul  creata 
sunt,  nee  pereunt,  non  coexistere,  turn  enim  idem  momentum  cum  diversis  loci  punctis 
nequaquam  conjungeretur. 


^"'  ibilUS  Si"r  17.  Ex  illis    7  casibus  videntur   omnino  6  per    Divinam  Omnipotentiam    possibiles, 

Divinam   Omnipo-  dempta  nimirum  virtuali  ilia  materiae  extensione,  de  qua  dubium  esse  poterit,  quia  deberet 
tentiam:    usus  simui  existere  numerus  absolute  infinitus  punctorum  illorum  loci  realium,  quod  impossibile 

supenons      theore-  ..,,.  .  .  ,.../•»«* 

matis  in  impenetra-  est ;   si  infimtum  numero  actu  existens  repugnat  in  modis  ipsis.     (juoniam  autem  possunt 
bihtate.  omnia  existere  alia  post  alia  puncta  loci  in  qua  vis  linea  constituta,  in  motu  nimirum  con- 

tinuo,  &  possunt  itidem  momenta  omnia  temporis  continui,  alia  itidem  post  alia  in  rei 
cujusvis  duratione  ;  ambigi  poterit,  an  possint  &  omnia  simul  ipsa  loci  puncta,  quam  quaes- 
tionem  definire  non  ausim.  Illud  unum  moneo,  sententiam  hanc  meam  de  spatii  natura, 
&  continuitate  praecipuas  omnes  difficultates,  quibus  premuntur  reliquae,  peni-[272]-tus 
evitare,  &  ad  omnia,  quae  hue  pertinent,  explicanda  commodissimam  esse.  Turn  illud 
addo,  excluso  appulsu  puncti  cujusvis  materiae  ad  punctum  loci,  ad  quod  punctum  quodvis 
materiae  quovis  momento  appellit,  &  inde  compenetratione,  veram  impenetrabilitatem 
materiae  necessario  consequi,  quod  in  decimo  nobis  libro  (')  plurimum  proderit.  Nimirum 


(b)  Hie  casus  nusquam  itidem  haberetur  ;   si  duratio  non  esset  quid  continenter  permanent,  sed  loco  ipsius  admit- 
Uretur  qutedam  existentia,  ut  ita  dicam,  saltitans,  nimirum  si  quodvis  materits  punctum  (Eif  idem  potest  transferri  ad 
qutsvis  creata   entia)  existeret  tantum  in  momentis  indivisililibus  a  se  invicem  remotis,  in  omnibus  vero  intermediis 
possibilibus  omnino  non  existeret.     Eo  casu  coexistentia  esset  infinite  improbabilis  eodem  fere  argumento,  quo  adventus 
unius  puncti  materite  ad  punctum  spatii,  in  quo  aliud  quodvis  punctum  unquam  fuerit.     In  eodem  nullum  haberetur 
reale  continuum  ne  in  motu  quidem  ;    diversis  celeritates  multo  melius  explicarentur :    multo  magis   pateret,  quomodo 
vita  insecti  brevissima  possit  tequivalere  vitce  cuivis  longissimce,  per  eundem  nimirum  numerum  existentiarum    inter 
extrema  momenta.     Verum  y  exclusio  cujusvis  coexistentid  abriperet  secum  omnes  prorsus  influxus  pbysicos  immediatos, 
ac  determinations,  y  deberet  haberi  continua  reproductio,  immo  creatio  nova  perpetua,  y  alia  ejusmodi,  quts  admitti, 
non  possunt,  haberentur. 

(c)  Stay  ants  nimirum  philosophies,  in  quo  Auctor  elegantissimus,  &  doctissimus  hanc  meam  Philosophiam   exponit. 
Hunc  ejus  theorematis  fructum  jam  cepimus  hie  supra,  ubi  in  ipso  opere  de  impenetrabilitate  egimus,  W  de  apparenti 
compenetratione,  qute  sine  viribus  mutuis  haberetur  a  num.  360. 


SUPPLEMENT  1  401 

an  infinite  improbability  of  the  return  of  any  indefinitely  chosen  point  of  matter  to  any 
point  of  position,  occupied  at  any  previous  instant  of  time  indefinitely,  of  a  return,  I  say, 
taking  place  at  any  indefinite  instant  of  subsequent  time  ;  hence,  such  a  return  must  be 
excluded,  without  any  fear  as  to  error,  since  it  must  be  considered  that  an  infinite 
improbability  merges  into  a  sort  of  relative  impossibility.  This  Theory  indeed  cannot 
be  applied  to  the  ordinary  view.  Hence,  in  this  way  it  is  clear,  in  my  Theory  of  points 
of  matter,  there  must  be  excluded  from  Nature  both  rest,  which  also  we  excluded  above, 
&  even  return  to  the  same  point  of  position  in  which  that  point  of  matter  once  was  situated. 
Therefore  it  comes  about  that  all  those  first  four  cases  will  be  excluded  from  Nature,  & 
in  them  the  analogy  of  time  &  space  will  be  preserved  accurately. 

1 6.  Finally,  if  we  seek  to  find  whether  any  point  of  matter  is  bound  to  occupy  at  some  NO  point  of  matter 
instant  a  point  of  position  which  was  occupied  by  some  other  point  of  matter  at  some  p^nH>TspTce  that 
other  instant,  still  the  improbability  will  be  infinitely  infinite.     For  the  number  of  existing  wa»  once  occupied 
points  of  matter  is  finite  ;  &  thus,  if  instead  of  the  return  of  any  point  to  points  of  position  it^1^0  onfymin 
occupied  by  itself  we  consider  the  return  to  points  that  have  been  occupied  by  another,  coexistence,  which 
the  number  of  cases  increases  in  the  ratio  of  unity  to  a  number  of  points  that  is  in  every  that^the  " 

case  finite,  that  is  to  say,  in  a  finite  ratio  only.  Hence,  the  improbability  of  the  arrival  is  broken, 
of  any  point  of  matter  indefinitely  taken  at  a  point  of  space  that  has  been  occupied  at  some 
time  by  any  other  point  is  still  infinite  ;  &  this  arrival  must  therefore  be  taken  to  be  impossible. 
In  this  way,  indeed,  the  sixth  case,  which  depended  on  this  return,  is  excluded  ;  &  much 
more  so  the  seventh  case,  which  involves  the  simultaneous  arrival  of  a  pair  of  points  of  matter 
at  any  the  same  point  of  position,  that  is  to  say,  compenetration.  The  eighth  case  also 
is  excluded  for  matter  ;  for  all  things  created  together  as  a  whole  will  continually  last  as 
a  whole,  &  so  will  always  have  a  common  instant  of  time.(*)  Only  the  fifth  case,  in  which 
several  points  of  matter  connect  the  same  instant  of  time  with  different  points  of  position 
remains  ;  &  this  is  not  only  possible,  but  also  necessary  for  all  points  of  matter,  seeing 
that  they  coexist.  For  it  cannot  be  the  case  that  the  seventh  &  the  eighth  are  excluded, 
unless  straightway,  on  that  very  account,  the  fifth  is  included,  as  will  be  easily  seen  on 
consideration.  Therefore  in  this  point  the  analogy  fails,  namely,  in  that  several  points 
of  matter  can  connect  different  points  of  space  with  the  same  instant  of  time,  which  is 
the  fifth  case  ;  whereas  it  is  impossible  for  the  same  point  of  space  to  be  connected  with 
several  instants  of  time,  which  is  the  third  case.  This  defect  is  necessarily  induced  by 
the  exclusion  of  the  seventh  &  eighth  cases ;  for  if  either  of  the  latter  is  included,  the 
fifth  might  be  excluded  ;  just  as  if  it  were  possible  for  points  of  matter,  which  had  been 
created  together,  &  do  not  perish,  not  to  coexist ;  for  then  the  same  instant  of  time  would 
in  no  way  be  connected  with  different  points  of  position. 

17.  At  least  six  of  the  seven  cases  seem  to  be  possible  through  Divine  Omnipotence,  Which  of  the  cases 
that  is  to  say,  omitting  the  virtual  extension  of  matter,  about  which  there  may  be  possibly  through0  oYvine 
some  doubt ;  for  in  this  case  there  must  exist  at  the  same  time  an  absolutely  infinite  number  Omnipotence;   use 
of  those  real  points  of  position  ;    &  this  is  impossible,  if  an  existing  thing  that  is  infinite  above  " 
in  number  is  contradictory  in  the  modes.     Moreover,  since  all  points  of  position  can  exist  trabiiity. 

one  after  another,  arranged  along  any  line,  for  instance,  in  continuous  motion,  &  so  can 
also  all  instants  of  continuous  time,  one  after  another  in  the  duration  of  any  thing,  there 
will  be  reason  for  doubt  as  to  whether  all  those  points  of  position  can  also  exist  at  the  same 
time.  This  is  a  matter  upon  which  I  dare  not  make  a  definite  statement.  All  I  say  is 
that  this  theory  of  mine  with  regard  to  the  nature  of  space  &  continuity  completely  avoids 
all  the  chief  difficulties  that  are  obstacles  in  other  theories ;  &  that  it  is  very  suitable  for 
the  explanation  of  everything  in  connection  with  this  matter.  I  will  also  add  the  remark 
that,  if  the  arrival  of  any  point  of  matter  at  a  point  of  position,  at  which  any  point  of 
matter  has  arrived  at  any  instant,  is  excluded,  &  along  with  it  compenetration  is  thus 
excluded,  then  real  impenetrability  of  matter  must  necessarily  follow,  which  will  be  of  great 
service  to  us  in  our  tenth  book  (^).  That  is,  unless  repulsive  forces  prevent  such  a  thing,  any 

(b)  This  case  also  would  never  happen,  if  the  duration  were  not  something  continuously  permanent ;    in  -place  of 
it,  we  should  have  to  admit  a  kind  of,  so  to  speak,  skipping  existence  ;    that  is  to  say,  as  if  any  point  of  matter  (and 
the  same  thing  applies  to  all  created  entities)  existed  only  in  indivisible  instants  remote  from  one  another,  and  in  all 
intermediate  instants  possible  did  not  exist  at  all.     Coexistence,  in  this  case,  would  be  infinitely  improbable,  the  argu- 
ment being  nearly  the  same,  as  in  the  case  of  the  arrival  of  one  point  of  matter  at  a  point  of  space  in  which   some   other 
point  had  once  been.     In  this  case  too,  there  would  be  no  real  continuum  even  in  motion  ;    different  velocities  could  be 
explained  much  more  easily  ;    it  would  be  much  more  evident  in  what  way  the  very  short  life  of  an   insect   can   be 
equivalent  to  the  longest  of  lives,  by  means  of  the  same  number  of  existences  coming  in  between  the  first  &  last  instants. 
Indeed  the  exclusion  of  any  coexistence  would  carry  away  with  it  all  immediate  physical  influence  altogether,  13  deter- 
minations ;   indeed,  a  continually  fresh  creation,  13  other  inadmissible  things  of  that  sort,  would  be  obtained. 

(c)  The  reference  is  to  Stay's  "  Philosophy,"  in  which  that  most  refined  {3   learned  author  expounds  my  Philosophy. 
On  what  I  have  said  above,  I  have  plucked  the  fruit  of  the  theorem,  in  which,  in  Art.  360  of  this  work,  I  dealt  with 
impenetrability,  13  the  apparent  compenetration  that  would  result,  if  there  were  no  mutual  forces. 

DP 


402  PHILOSOPHISE   NATURALIS  THEORIA 

nisi  vires  repulsivse  prohiberent ;  liberrime  massa  qusevis  per  quamvis  aliam  massam 
permearet,  sine  ullo  periculo  occursus  ullius  puncti  cum  alio  quovis,  ubi  haberetur  apparens 
quaedam  compenetratio  similis  penetrationi  luminis  per  crystalla,  olei  per  ligna,  &  marmora, 
sine  ulla  reali  compenetratione  punctorum.  In  massis  crassioribus,  &  minori  celeritate 
praeditis  vires  repulsivse  motum  ulteriorem  plerumque  impediunt  sine  ullo  impactu,  & 
sensibilem  etiam  illam,  ac  apparentem  compenetrationem  excludunt  :  in  tenuissimis,  & 
celerrimis,  ut  in  luminis  radiis  per  homogeneas  substantias,  vel  per  alios  radios  propagatis, 
evitatur  per  celeritatem  ipsam,  actionum  exigua  insequalitas,  ex  circumjacentium  punctorum 
insequali  distantia  orta,  ac  liberrimus  habetur  progressus  in  omnes  plagas  sine  ullo  occursus 
periculo,  quod  summam,  &  unicam  difficultatem  propagations  luminis  per  substantiam 
emissam,  &  progredientem,  penitus  amovet.  Sed  de  his  jam  satis. 


SUPPLEMENT   I  403 

perfectly  free  mass  will  permeate  through  any  other  mass,  without  there  being  any  danger 
of  a  collision  of  one  point  with  another.  Here  there  would  be  an  apparent  compenetration 
similar  to  the  penetration  of  light  through  crystals,  oils  through  wood,  &  marble,  without 
any  real  compenetration  of  the  points.  In  denser  masses,  &  those  endowed  with  a  smaller 
velocity,  the  repulsive  forces  for  the  most  part  prevent  further  motion  without  any  impact ; 
&  this  also  excludes  sensible  as  well  as  apparent  compenetration.  In  very  tenuous  masses 
moving  with  very  great  velocities,  as  rays  of  light  propagated  through  homogeneous 
substances,  or  through  other  rays,  the  very  slight  inequality  of  the  actions,  derived  from 
the  unequal  distances  of  the  circumjacent  points,  will  be  prevented  by  the  high  velocity  ; 
&  perfectly  free  progress  will  take  place  in  all  directions  without  any  danger  of  collisions. 
This  removes  altogether  the  greatest  &  only  real  difficulty  in  the  idea  of  the  propagation 
of  light  by  means  of  a  substance  that  is  emitted  &  travels  forward.  But  I  have  now  said 
quite  enough  upon  this  matter. 


D 


[273]  §   II 
Tempore,   ut  a    nobis  cognoscuntur 


Nos  nee  modos  ig.  Diximus  in  superiore  Supplemento  de  spatio,  ac  tempore,  ut  sunt  in  se  ipsis  : 

posse60  absolute  superest,  ut  illud  attingam,  quod  pertinet  ad  ipsa,  ut  cognoscuntur.     Nos  nequaquam 
cognoscere,  nee  immediate  cognoscimus  per  sensus  illos  existendi  modos  reales,  nee  discernere  possumus 

absolute  distantias       T          i       !••          o         •  •  i  s»       •     •        •  i  •  •  • 

&  magnitudines.  a^los  a®  a^lls'  Sentimus  quidem  a  discrimme  idearum,  quae  per  sensus  excitantur  m  ammo, 
relationem  determinatam  distantiae,  &  positionis,  quae  e  binis  quibusque  localibus  existendi 
modis  exoritur,  sed  eadem  idea  oriri  potest  ex  innumeris  modorum,  sive  punctorum  realium 
loci  binariis,  quae  inducant  relationes  aequalium  distantiarum,  &  similium  positionum 
tarn  inter  se,  quam  ad  nostra  organa,  &  ad  reliqua  circumjacentia  corpora.  Nam  bina 
materiae  puncta,  quae  alicubi  datam  habent  distantiam,  &  positionem  inductam  a  binis 
quibusdam  existendi  modis,  alibi  possunt  per  alios  binos  existendi  modos  habere  relationem 
distantise  aequalis,  &  positionis  similis,  distantiis  nimirum  ipsis  existentibus  parallelis. 
Si  ilia  puncta,  &  nos,  &  omnia  circumjacentia  corpora  mutent  loca  realia,  ita  tamen,  ut 
omnes  distantise  aequales  maneant,  &  prioribus  parallelae  ;  nos  easdem  prorsus  habebimus 
ideas,  quin  imo  easdem  ideas  habebimus  ;  si  manentibus  distantiarum  magnitudinibus, 
directiones  omnes  in  sequali  angulo  converterentur,  adeoque  aeque  ad  se  invicem  inclinarentur 
ac  prius.  Et  si  minuerentur  etiam  distantiae  illas  omnes,  manentibus  angulis,  &  manente 
illarum  ratione  ad  se  invicem,  vires  autem  ex  ea  distantiarum  mutatione  non  mutarentur, 
rite  mutata  virium  scala  ilia,  nimirum  curva  ilia  linea,  per  cujus  ordinatas  ipsae  vires 
exprimuntur  ;  nullam  nos  in  nostris  ideis  mutationem  haberemus. 


Motum  communem  19.  Hinc  autem  consequitur  illud,  si  totus  hie  Mundus  nobis  conspicuus  motu  parallelo 

non1Sposse  a  nobis  promoveatur  in  plagam  quamvis,  &  simul  in  quovis  angulo  convertatur,  nos  ilium  motum, 

cognosci,  nee  s  i  &  conversionem  sentire  non  posse.     Sic  si  cubiculi,  in  quo  sumus,  &  camporum,  ac  montium 

ipse    in    quavis 


ratione      au 


'geatur,  tractus  omnis  motu  aliquo  Telluris  communi  ad  sensum  simul  convertatur  ;  motum  ejusmodi 
vei  minuatur  totus.  sentire  non  possumus  :    ideas  enim  eaedem  ad  sensum  excitantur  in  animo.     Fieri  autem 
posset,  ut  totus  itidem  Mundus  nobis  conspicuus  in  dies  contraheretur,  vel  produceretur, 
scala  virium  tantundem  contracta,  vel  producta  ;    quod  si  fieret ;    nulla  in  animo  nostro 
idearum  mutatio  haberetur,  adeoque  nullus  ejusmodi  mutationis  sensus. 


Mutata  positione 
nostra,  &  omnium, 
quae  videmus,  non 
mutari  n  o  s  t  r  a  s 
ideas,  &  idcirco  nos 
motum  nee  nobis 
adscribere,  nee 
reliquis. 


20.  Ubi  vel  objecta  externa,  vel  nostra  organa  mutant  illos  suos  existendi  modos  ita, 
ut  prior  ilia  aequalitas,  [274]  vel  similitudo  non  maneat,  turn  vero  mutantur  ideae,  & 
mutationis  habetur  sensus,  sed  ideas  eaedem  omnino  sunt,  sive  objecta  externa  mutationem 
subeant,  sive  nostra  organa,  sive  utrumque  inaequaliter.  Semper  ideae  nostrae  difrerentiam 
novi  status  a  priore  referent,  non  absolutam  mutationem,  quae  sub  sensus  non  cadit.  Sic 
sive  astra  circa  Terram  moveantur,  sive  Terra  motu  contrario  circa  se  ipsam  nobiscum  ; 
eaedem  sunt  ideae,  idem  sensus.  Mutationes  absolutas  nunquam  sentire  possumus,  discrimen 
a  priori  forma  sentimus.  Cum  autem  nihil  adest,  quod  nos  de  nostrorum  organorum 
mutatione  commoneat ;  turn  vero  nos  ipsos  pro  immotis  habemus  communi  praejudicio 
habendi  pro  nullis  in  se,  quae  nulla  sunt  in  nostra  mente,  cum  non  cognoscantur,  &  muta- 
tionem omnem  objectis  extra  nos  sitis  tribuimus.  Sic  errat,  qui  in  navi  clausus  se  immotum 
censet,  littora  autem,  &  monies,  ac  ipsam  undam  moveri  arbitratur. 


404 


§  II 

Of  Space  ^f  Time,  as  we  know  them 

1 8.  We  have  spoken,  in  the  preceding  Supplement,  of  Space  &  Time,  as  they  are  in  We  cannot  obtain 
themselves ;   it  remains  for  us  to  say  a  few  words  on  matters  that  pertain  to  them,  in  so  fe^geof°iocai  modes 
far  as  they  come  within  our  knowledge.     We  can  in  no  direct  way  obtain  a  knowledge  of    existence  ;  nor 
through  the  senses  of  those  real  modes  of  existence,  nor  can  we  discern  one  of  them  from  tances  ^or^agni- 
another.     We  do  indeed  perceive,  by  a  difference  of  ideas  excited  in  the  mind  by  means  tudes. 

of  the  senses,  a  determinate  relation  of  distance  &  position,  such  as  arises  from  any  two 
local  modes  of  existence  ;  but  the  same  idea  may  be  produced  by  innumerable  pairs  of 
modes  or  real  points  of  position  ;  these  induce  the  relations  of  equal  distances  &  like  positions, 
both  amongst  themselves  &  with  regard  to  our  organs,  &  to  the  rest  of  the  circumjacent 
bodies.  For,  two  points  of  matter,  which  anywhere  have  a  given  distance  &  position  induced 
by  some  two  modes  of  existence,  may  somewhere  else  on  account. of  two  other  modes  of 
existence  have  a  relation  of  equal  distance  &  like  position,  for  instance  if  the  distances  exist 
parallel  to  one  another.  If  those  points,  we,  &  all  the  circumjacent  bodies  change  their 
real  positions,  &  yet  do  so  in  such  a  manner  that  all  the  distances  remain  equal  &  parallel 
to  what  they  were  at  the  start,  we  shall  get  exactly  the  same  ideas.  Nay,  we  shall  get  the 
same  ideas,  if,  while  the  magnitudes  of  the  distances  remain  the  same,  all  their  directions 
are  turned  through  any  the  same  angle,  &  thus  make  the  same  angles  with  one  another  as 
before.  Even  if  all  these  distances  were  diminished,  while  the  angles  remained  constant, 
&  the  ratio  of  the  distances  to  one  another  also  remained  constant,  but  the  forces  did  not 
change  owing  to  that  change  of  distance  ;  then  if  the  scale  of  forces  is  correctly  altered, 
that  is  to  say,  that  curved  line,  whose  ordinates  express  the  forces ;  then  there  would  be 
no  change  in  our  ideas. 

19.  Hence  it  follows  that,  if  the  whole  Universe  within  our  sight  were  moved  by  a  The  motion,  if  any, 
parallel  motion  in  any  direction,  &  at  the  same  time  rotated  through  any  angle,  we  could  thTuniverse  could 
never  be  aware  of  the  motion  or  the  rotation.     Similarly,  if  the  whole  region  containing  not   come    within 
the  room  in  which  we  are,  the  plains  &  the  hills,  were  simultaneously  turned  round  by  some  ncV^ould  winnow 
approximately  common  motion  of  the  Earth,  .we  should  not  be  aware  of  such  a  motion ;  it.  if  it    were  in- 
for  practically  the  same  ideas  would  be  excited  in  the  mind.     Moreover,  it  might  be  the  ratfofor diminished^ 
case  that  the  whole  Universe  within  our  sight  should  daily  contract  or  expand,  while  the  as  a  whole, 
scale  of  forces  contracted  or  expanded  in  the  same  ratio  ;  if  such  a  thing  did  happen,  there 

would  be  no  change  of  ideas  in  our  mind,  &  so  we  should  have  no  feeling  that  such  a  change 
was  taking  place. 

20.  When  either  objects  external  to  us,  or  our  organs  change  their  modes  of  existence  s.mce-  '£  °!?r  P°si- 
in  such  a  way  that  that  first  equality  or  similitude  does  not  remain  constant,  then  indeed  everything  we  see 
the  ideas  are  altered,  &  there  is  a  feeling  of  change  ;    but  the  ideas  are  the  same  exactly,  is  changed,  our  ideas 
whether  the  external  objects  suffer  the  change,  or  our  organs,  or  both  of  them  unequally,  therefore  Cwen8can 
In  every  case  our  ideas  refer  to  the  difference  between  the  new  state  &  the  old,  &  not  to  ascribe  no  motion 
the  absolute  change,  which  does  not  come  within  the  scope  of  our  senses.     Thus,  whether  anything  eise.° 
the  stars  move  round  the  Earth,  or  the  Earth  &  ourselves  'move  in  the  opposite  direction 

round  them,  the  ideas  are  the  same,  &  there  is  the  same  sensation.  We  can  never  perceive 
absolute  changes ;  we  can  only  perceive  the  difference  from  the  former  configuration  that 
has  arisen.  Further,  when  there  is  nothing  at  hand  to  warn  us  as  to  the  change  of  our 
organs,  then  indeed  we  shall  count  ourselves  to  have  been  unmoved,  owing  to  a  general 
prejudice  for  counting  as  nothing  those  things  that  are  nothing  in  our  mind  ;  for  we  cannot 
know  of  this  change,  &  we  attribute  the  whole  of  the  change  to  objects  situated  outside 
of  ourselves.  In  such  manner  any  one  would  be  mistaken  in  thinking,  when  on  board  ship, 
that  he  himself  was  motionless,  while  the  shore,  the  hills  &  even  the  sea  were  in 
motion. 


405 


406 


PHILOSOPHISE  NATURALIS  THEORIA 


Quomodo  judice- 
mus  de  aequalitate 
duorum,  ex  aequal- 
itate cum  tertio  : 
nunquam  h  a  b  e  r  i 
congrue  n  t  i  a  m  in 
longitudine,  ut  nee 
in  tempore,  sed  in- 
ferri  a  causis. 


21.  Illud  autem  notandum  inprimis  ex  hoc  principio  immutabilitatis  eorum,  quorum 
mutationem  per  sensum  non  cognoscimus,  oriri  etiam  methodum,  quam  adhibemus  in 
comparandis  intervallorum  magnitudinibus  inter  se,  ubi  id,  quod  pro  mensura  assumimus, 
habemus  pro  immutabili.      Utimur  autem   hoc    principio,  quce  sunt   cequalia  eidem,  sunt 
cequalia  inter  se,  ex  quo  deducitur  hoc  aliud,  ad  ipsum  pertinens,  quce  sunt  ceque  multipla, 
vel  submuti-pla  alterius,  sunt  itidem  inter  se  cequalia,  &  hoc  alio,  quce  congruant,  cequalia  sunt. 
Assumimus   ligneam,  vel  ferream  decempedam,   quam  uni  intervallo  semel,  vel  centies 
applicatam   si    inveniamus  congruentem,  turn  alteri  intervallo  applicatam   itidem  semel, 
vel  centies  itidem  congruentem,  ilia  intervalla  aequalia  dicimus.     Porro  illam  ligneam,  vel 
ferream  decempedam  habemus  pro  eodem  comparationis  termino  post  translationem.     Si 
ea  constaret  ex  materia  prorsus  continua,  &  solida,  haberi  posset  pro  eodem  comparationis 
termino  ;  at  in  mea  punctorum  a  se  invicem  distantium  sententia,  omnia  illius  decempedse 
puncta,  dum   transferuntur,  perpetuo  distantiam  revera  mutant.     Distantia   enim  con- 
stituitur  per  illos  reales  existendi  modos,  qui  mutantur  perpetuo.     Si  mutentur  ita,  ut 
qui  modi  succedunt,  fundent  reales  sequalium  distantiarum  relationes ;   terminus  compara- 
tionis non  erit  idem,  adhuc  tamen  Eequalis  erit,  &  aequalitas  mensuratorum  intervallorum 
rite  colligetur.     Longitudinem  decempedae  in  priore  situ  per  illos  priores  reales  modos 
constitute,  cum  longitudine  in  posteriore  situ  constituta  per  hosce  posteriores,  immediate 
inter  se  conferre  nihilo  magis  possumus,  quam  ilia  ipsa  intervalla,  quae  mensurando  conferi- 
mus.     Sed  quia  nullam  in  translatione  mutationem  sentimus,  quae  longitudinis  relationem 
nobis  ostendat,  idcirco    pro  eadem  habemus  longitudinem  ipsam.     At  ea  revera  semper 
in  ipsa  translatione  non  nihil  mutabitur.     Fieri  posset,  ut  ingentem  etiam  mutationem 
aliquam  subiret    [275]  &  ipsa,  &  nostri  sensus,  quam   nos  non  sentiremus,  &  ad  priorem 
restituta  locum  ad  priori  sequalem,  vel  similem  statum  rediret.     Exigua  tamen  aliqua 
mutatio  habetur  omnino  idcirco,  quod  vires,  quse  ilia  materiae  puncta  inter  se  nectunt, 
mutata  positione  ad  omnia  reliquarum  Mundi  partium  puncta,  non  nihil  immutantur. 
Idem  autem  &  in  communi  sententia  accidit.     Nullum  enim  corpus  spatiolis  vacat  inter jectis, 
&    omnis    penitus    compressionis,  ac  dilatationis  est  incapax,  quae    quidem  dilatatio,   & 
compressio  saltern  exigua  in  omni  translatione  omnino  habetur.     Nos    tamen    mensuram 
illam  pro  eadem  habemus,  cum,  ut  monui,  nullam  mutationem  sentiamus. 


Conciusio 


discri-  22.  Ex  his  omnibus  consequitur,  nos  absolutas  distantias  nee  immediate    cognoscere 

omnino  posse,  nee  per  terminum  communem  inter  se  comparare,  sed  sestimare  magnitudines 
ab  ideis,  per  quas  eas  cognoscimus,  &  mensuras  habere  pro  communibus  terminis,  in  quibus 
nullam  mutationem  factam  esse  vulgus  censet.  Philosophi  autem  mutationem  quidem 
debent  agnoscere,  sed  cum  nullam  violatae  notabili  mutatione  sequalitatis  causam  agnoscant, 
mutationem  ipsam  pro  aequaliter  facta  habent. 


Licet  translata  de- 
cempeda,  mutentur 
modi,  qui  intervalli 
relationem  consti- 
tuunt  ;  tamen  inter- 
valla aequalia  haberi 
pro  eodem  ex 
causis. 


23.  Porro  licet,  ubi  puncta  materiae  locum  mutant,  ut  in  decempeda  translata,  mutetur 
revera  distantia,  mutatis  iis  modis  realibus,  quae  ipsam  constituunt  ;  tamen  si  mutatio  ita 
fiat,  ut  posterior  ilia  distantia  aequalis  prorsus  priori  sit,  ipsam  appellabimus  eandem,  &  nihil 
mutatam  ita,  ut  eorundem  terminorum  aequales  distantiae  dicantur  distantia  eadem,  & 
magnitude  dicatur  eadem,  quae  per  eas  aequales  distantias  definitur,  ut  itidem  ejusdem 
directionis  nomine  intelligantur  binae  etiam  directiones  paralleiae  ;  nee  mutari  distantiam, 
vel  directionem  dicemus  in  sequentibus,  nisi  distantiae  magnitudo,  vel  parallelismus  mutetur. 


Eadem  ad  tempus 
transferenda,  sed 
in  eo  etiam  vulgo 
notum  esse,  inter- 
vallum  t  e  m  p  o  r- 
arium  no  n  posse 
transferri  idem  pro 
comparatione  duo- 
rum  :  errari  ab  eo 
circa  spatium. 


24.  Quae  de  spatii  mensura  diximus,  haud  difficulter  ad  tempus  transferentur,  in  quo 
itidem  nullam  habemus  certam,  &  constantem  mensuram.  Desuminus  a  motu  illam, 
quam  possumus,  sed  nullum  habemus  motum  prorsus  aequabilem.  Multa,  quae  hue  perti- 
nent, &  quae  ad  idearum  ipsarum  naturam,  &  successionem  spectant,  diximus  in  notis. 
Unum  hie  addo,  in  mensura  temporis,  ne  vulgus  quidem  censere  ab  uno  tempore  ad  aliud 
tempus  eandem  temporis  mensuram  transierri.  Videt  aliam  esse,  sed  aequalem  supponit  ob 
motum  suppositum  aequalem.  In  mensura  locali  aeque  in  mea  sententia,  ac  in  mensura 
temporaria  impossibile  est  certam  longitudinem,  ut  certam  durationem  e  sua  sede  abducere 
in  alterius  sedem,  ut  binorum  comparatio  habeatur  per  tertium.  Utrobique  alia  longi- 
tude, ut  alia  duratio  substituitur,  quae  priori  illi  aequalis  censetur,  nimirum  nova  realia 


SUPPLEMENT   II  407 

21.  Again,  it  is  to  be  observed  first  of  all  that  from  this  principle  of  the  unchangeability  The    man 
of  those  things,  of  which  we  cannot  perceive  the  change  through  our  senses,  there  comes  ^g^ofTh 
forth  the  method  that  we  use  for  comparing  the  magnitudes  of  intervals  with  one  another  ;  ity  of  two  things 
here,  that,  which  is  taken  as  a  measure,  is  assumed  to  be  unchangeable.     Also  we  make  wrt^tnird^there 
use  of  the  axiom,  things  that  are  equal  to  the  same  thing  are  equal  to  one  another ;  &  from  this  never  can  be  con- 
is  deduced  another  one  pertaining  to  the  same  thing,  namely,  things  that  are  equal  multiples,  fJ^moreThanthe-e 
or  submulti-ples,  of  each,  are  also  equal  to  one  another;   &  also  this,  things  that  coincide  are  can  be  in  time;  the 
equal.     We  take  a  wooden  or  iron  ten-foot  rod  ;   &  if  we  find  that  this  is  congruent  with  j^f  ^rr  ed  tfrom 
one  given  interval  when  applied  to  it  either  once  or  a  hundred  times,  &  also  congruent  to  causes, 
another  interval  when  applied  to  it  either  once  or  a  hundred  times,  then  we  say  that  these 

intervals  are  equal.  Further,  we  consider  the  wooden  or  iron  ten-foot  rod  to  be  the  same 
standard  of  comparison  after  translation.  Now,  if  it  consisted  of  perfectly  continuous  & 
solid  matter,  we  might  hold  it  to  be  exactly  the  same  standard  of  comparison  ;  but  in 
my  theory  of  points  at  a  distance  from  one  another,  all  the  points  of  the  ten-foot  rod,  while 
they  are  being  transferred,  really  change  the  distance  continually.  For  the  distance  is 
constituted  by  those  real  modes  of  existence,  &  these  are  continually  changing.  But  if  they 
are  changed  in  such  a  manner  that  the  modes  which  follow  establish  real  relations  of  equal 
distances,  the  standard  of  comparison  will  not  be  identically  the  same  ;  &  yet  it  will  still 
be  an  equal  one,  &  the  equality  of  the  measured  intervals  will  be  correctly  determined. 
We  can  no  more  transfer  the  length  of  the  ten-foot  rod,  constituted  in  its  first  position 
by  the  first  real  modes,  to  the  place  of  the  length  constituted  in  its  second  position  by  the 
second  real  modes,  than  we  are  able  to  do  so  for  intervals  themselves,  which  we  compare 
by  measurement.  But,  because  we  perceive  none  of  this  change  during  the  translation, 
such  as  may  demonstrate  to  us  a  relation  of  length,  therefore  we  take  that  length  to  be 
the  same.  But  really  in  this  translation  it  will  always  suffer  some  slight  change.  It  might 
happen  that  it  underwent  even  some  very  great  change,  common  to  it  &  our  senses,  so  that 
we  should  not  perceive  the  change  ;  &  that,  when  restored  to  its  former  position,  it  would 
return  to  a  state  equal  &  similar  to  that  which  it  had  at  first.  However,  there  always  is 
some  slight  change,  owing  to  the  fact  that  the  forces  which  connect  the  points  of  matter, 
will  be  changed  to  some  slight  extent,  if  its  position  is  altered  with  respect  to  all  the  rest 
of  the  Universe.  Indeed,  the  same  is  the  case  in  the  ordinary  theory.  For  no  body  is 
quite  without  little  spaces  interspersed  within  it,  altogether  incapable  of  being  compressed 
or  dilated  ;  &  this  dilatation  &  compression  undoubtedly  occurs  in  every  case  of  translation, 
at  least  to  a  slight  extent.  We,  however,  consider  the  measure  to  be  the  same  so  long 
as  we  do  not  perceive  any  alteration,  as  I  have  already  remarked. 

22.  The  consequence  of  all  this  is  that  we  are  quite  unable  to  obtain  a  direct  knowledge  Conclusion  reached; 
of  absolute  distances ;  &  we  cannot  compare  them  with  one  another  by  a  common  standard.  *he   difference.  be- 

•r     .  ,1-1  }  •  tween       ordinary 

We  have  to  estimate  magnitudes  by  the  ideas  through  which  we  recognize  them  ;    &  to  people     &    phiio- 
take  as  common  standards  those  measures  which  ordinary  people  think  suffer  no  change,  sophers     in    the 

...  .  .  '    r      r  .  6       matter  of  judgment. 

.but  philosophers  should  recognize  that  there  is  a  change  ;  but,  since  they  know  of  no  case 
in  which  the  equality  is  destroyed  by  a  perceptible  change,  they  consider  that  the  change 
is  made  equally. 

23.  Further,  although  the  distance  is  really  changed  when,  as  in  the  case  of  the  translation  Although,     when 
of  the  ten-foot  rod,  the  position  of  the  points  of  matter  is  altered,  those  real  modes  which  the  t^11:*00*  r°d  1S 

T        v  i     •  i  i     1         -r  i  •  moved  in    position, 

constitute  the  distance  being  altered  ;  nevertheless  it  the  change  takes  place  in  such  a  way  those   modes   that 
that  the  second  distance  is  exactly  equal  to  the  first,  we  shall  call  it  the  same,  &  say  that  it  is  constitute  thereia- 

,    .  '     i1..  -11    i  •  i  i        tions  of  the  interval 

altered  in  no  way,  so  that  the  equal  distances  between  the  same  ends  will  be  said  to  be  the  are  also  altered,  yet 
same  distance  &  the  magnitude  will  be  said  to  be  the  same  :  &  this  is  defined  by  means  of  eq"al  intervals  are 

,    ,.  .     °  HIT-  -11  i         i       •      i  i  reckoned    as    same 

these  equal  distances,  just  as  also  two  parallel  directions  will  be  also  included  under  the  name  for     the     reasons 
of  the  same  direction.      In  what  follows  we  shall  say  that  the  distance  is  not  changed,  or  stated- 
the  direction,  unless  the  magnitude  of  the  distance,  or  the  parallelism,  is  altered. 

24.  What  has  been  said  with  regard  to  the  measurement  of  space,  without  difficulty  The    same    obser- 
can  be  applied  to  time  ;    in  this  also  we  have  no  definite  &  constant  measurement.     We  equally11  to ^TimJ; 
obtain  all  that  is  possible  from  motion  ;  but  we  cannot  get  a  motion  that  is  perfectly  uniform.  but  in  it.  *  is  wel1 
We  have  remarked  on  many  things  that  belong  to  this  subject,  &  bear  upon  the  nature  &  ordinary  people! 
succession  of  these  ideas,  in  our  notes.     I  will  but  add  here,  that,  in  the  measurement  of  that    the.    sam<j 
time,  not  even  ordinary  people  think  that  the  same  standard  measure  of  time  can  be  translated  c^m^bWansfated 
from  one  time  to  another  time.     They  see  that  it  is  another,  consider  that  it  is  an  equal,  for  the  purpose  of 
on  account  of  some  assumed  uniform  motion.     Just  as  with  the  measurement  of  time,  m^en/a^V^is  be- 
so  in  my  theory  with  the  measurement  of  space  it  is  impossible  to  transfer  a  fixed  length  cause  of  this  that 

f  •    '     i  ,.  ...r.11  .  -,.  i       r     •  they  fall  into   error 

irom  its  place  to  some  other,  just  as  it  is  impossible  to  transfer  a  fixed  interval  ot  time,  ^th  regard  to 
so  that  it  can  be  used  for  the  purpose  of  comparing  two  of  them  by  means  of  a  third.     In  space, 
both  cases,  a  second  length,  or  a  second  duration  is  substituted,  which  is  supposed  to  be 
equal  to  the  first ;    that  is  to  say,  fresh  real  positions  of  the  points  of  the  same  ten-foot 


4o8  PHILOSOPHIC  NATURALIS  THEORIA 

punctorum  ejusdem  decempedae  loca  novam  distantiam  constituentia,  ut  [276]  novus 
ejusdem  styli  circuitus,  sive  nova  temporaria  distantia  inter  bina  initia,  &  binos  fines.  In 
mea  Theoria  eadem  prorsus  utrobique  habetur  analogia  spatii,  &  temporis.  Vulgus  tan- 
tummodo  in  mensura  locali  eundem  haberi  putat  comparationis  terminum  :  Philosophi 
ceteri  fere  omnes  eundem  saltern  haberi  posse  per  mensuram  perfecte  solidam,  &  continuam, 
in  tempore  tantummodo  aequalem  :  ego  vero  utrobique  aequalem  tantum  agnosco,  nuspiam 
eandem. 


SUPPLEMENT  II  409 

rod  which  constitute  a  new  distance,  such  as  a  new  circuit  made  by  the  same  rod,  or  a 
fresh  temporal  distance  between  two  beginnings  &  two  ends.  In  my  Theory,  there  is  in  each 
case  exactly  the  same  analogy  between  space  &  time.  Ordinary  people  think  that  it  is 
only  for  measurement  of  space  that  the  standard  of  measurement  is  the  same  ;  almost  all 
other  philosophers  except  myself  hold  that  it  can  at  least  be  considered  to  be  the  same 
from  the  idea  that  the  measure  is  perfectly  solid  &  continuous,  but  that  in  time  there  is  only 
equality.  But  I,  for  my  part,  only  admit  in  either  case  the  equality,  &  never  the  identity. 


[277]   §   HI 

Solutio  analytica   'Problematis  determinants  naturam   Legis 

Virium   (d) 

Denomin  a  t  i  o,  ac  25.  Ut  hasce  conditiones  impleamus,  formulam  inveniemus  algebraicam,  quae  ipsam 

continebit  legem  nostram,  sed  hie  elementa  communia  vulgaris  Cartesianae  algebras  suppone- 
mus  ut  nota,  sine  quibus  res  omnino  confici  nequaquam  potest.  Dicatur  autem  ordinata 
y,  abscissa  x,  ac  ponatur  xx  =  z.  Capiantur  omnium  AE,  AG,  AI  &c.  valores  cum  signo 
negative,  &  summa  quadratorum  omnium  ejusmodi  valorum  dicatur  a,  summa  productorum 
e  binis  quibusque  quadratis  b,  summa  productorum  e  ternis  c,  &  ita  porro  ;  productum, 
autem  ex  omnibus  dicatur  /.  Numerus  eorundem  valorum  dicatur  m.  His  positis  ponatur 
zm  +  azm~*  +  bzmz  +  czm~3  &c  ...+/=  P.  Si  ponatur  P  =  o,  patet  aequationis  ejus 
omnes  radices  fore  reales,  &  positivas,  nimirum  sola  ilia  quadrata  quantitatum  AE,  AG, 
AI  &c,  qui  erunt  valores  ipsius  z  ;  adeoque  cum  ob  xx  =  z,  sit  x  =  ±  Vz,  patet,  valores 
x  fore  tarn  AE,  AG,  AI  positivas,  quam  AE',  AG',  &c  negativas. 

Assumptio    cujus-  26.  Deinde  sumatur  qusecunque  quantitas  data  per  z,  &  constantes  quomodocunque, 

1  dummodo  non  habeat  ullum  divisorem  communem  cum  P,  ne  evanescente  z,  eadem  evan- 
escat,  ac  facta  x  infinitesima  ordinis  primi,  evadat  infinitesima  ordinis  ejusdem,  vel  inferioris, 
ut  erit  quaecunque  formula  z'  +  gz'"1  +  hzr~z  &c  +  /,  quae  posita  =  o  habeat  radices 
quotcunque  imaginarias,  &  quotcunque,  &  quascunque  reales  (dummodo  earum  nulla  sit 
ex  iis  AE,  AG,  AI  &c,  sive  positiva,  sive  negativa),  si  deinde  tota  multiplicetur  per  z.  Ea 
dicatur  Q. 

Formula  continens  2y.  Si  jam  fiat  P  —  Qy  =  o  ;    dico,   hanc    aequationem   satisfacere  reliquis    omnibus 

stain.10  hujus  curvae  conditionibus,  &  rite  determinate  valore  Q,  posse  infmitis  modis  satisfied 

etiam  postremae  condition!  expositae  sexto  loco. 
Aequationem    fore  [278]  28.  Nam  inprimis,  quoniam  valores  P,  &    Q  positi  =  o,  nullam    habent  radicem 

simplicem    non    re-    *•    '     J  n    *     i     i     i  T    •  TT-         i     ' 

soiubiiem  in  piures.  communem,  nullum  habebunt  divisorem  communem.  Hinc  hsec  aequatio  non  potest  per 
divisionem  reduci  ad  binas,  adeoque  non  est  composita  ex  binis  aequationibus,  sed  simplex, 
&  proinde  simplicem  quandam  curvam  continuam  exhibet,  quae  ex  aliis  non  componitur. 
Quod  erat  primum. 

Exhibituram     da-  29.  Deinde  curva  hujusmodi  secabit  axem  C'AC  in  iis  omnibus,  &  solis  punctis,  E, 

ter^ctiomim^curvae]  G,  I,  &c,  E',  G',  &c.      Nam  ea  secabit  axem  C'AC  solum  in  iis  punctis,  in  quibus  y  =  o, 

in  datis  punctis.    '  &  secabit  in  omnibus.     Porro  ubi  fuerit  y  =  o,  erit  &  Qy  =  o,  adeoque  ob  P  —  Qy  =  o  ; 

erit  P  =  o.      Id  autem  continget  solum  in  iis  punctis,  in  quibus  z  fuerit  una  e  radicibus 

aequationis  P  =  o,  nimirum,  ut  supra  vidimus,  in  punctis  E,  G,  I,  vel  E',  G',  &c.     Quare 

solum  in  his  punctis  evanescet  y,  &  curva  axem  secabit.     Secaturam  autem  in  his  omnibus 

patet  ex  eo,  quod  in  his  omnibus  punctis  erit  P  =  o.      Quare  erit  etiam  Q  y  =  o.      Non 

erit  autem  Q  =  o  ;    cum  nulla  sit  radix  communis   aequationum  P  =  o,  &  Q  =  o.      Quare 

erit  y  =  o,  &  curva  axem  secabit.     Quod  erat  secundum. 

p 
Singuias  ordinatas  *o>  Praeterea  cum  sit  P  —  Qx  =  o,  erit  y  =  -^- ;  determinata  autem  utcunque  abscissa 

responsuras    singu-  (J 

x,  habebitur  determinata  quaedam  z,  adeoque  &  P,  Q  erunt  unicse,  &  determinatae.  Erit 
igitur  etiam  y  unica,  &  determinata  ;  ac  proinde  respondebunt  singulis  abscissis  z  singulas 
tantum  ordinatae  y.  Quod  erat  tertium. 

(d)  Heec  solutio  excerpta  est  ex  dissertatione  De  Lege  Virium  in  Natura  existentium.  Acced.it  iis,  quce  Me 
sunt  eruta,  scholium  3  primo  adjectum  in  hue  editions  Veneta  prima.  Ipsum  problema  hie  solvendum  habetur  in  ipso 
hoc  Optre  parte  i  num.  117,  ac  ejus  conditiones  num.  118. 

410 


§111 

Analytical  Solution  of  the  Problem    to   determine  the   nature  of  the 

Law   of  Forces 


25.  To  fulfil  these  conditions,  we  will  find  an  algebraical  formula,  such  as  will  represent  statement,  &  pre. 
our  law  ;   to  do  so,  we  shall  take  it  that  the  first  principles  of  the  ordinary  Cartesian  algebra  Potion, 
are  known  ;    for,  without  that,  the  thing  can  in  no  way  be  accomplished.      Suppose  that 
y  is  the  ordinate,  x  the  abscissa,  &  let  x*  =  z.     Take  the  values  of  AE,  AG,  AI,  &c.,  all  with 
a  negative  sign,  &  let  a  be  the  sum  of  the  squares  of  all  such  values,  b  the  sum  of  the  products 
of  all  these   squares  two  at  a  time,  c  the  sum  of  the  products  three  at  a  time,  &  so  on  ;   & 
let  the  product  of  them  all  together  be  called  / ;   suppose  that  the  number  of  these  values 
is  m.     Then  suppose  P  to  stand  for 


If  P  is  put  equal  to  zero,  it  is  plain  that  all  the  roots  of  this  equation  will  be  real  &  positive, 
namely,  only  the  squares  of  the  quantities  AE,  AG,  AI,  &c.  ;  &  these  will  be  the  values  of  z. 
Hence,  since  x*  =  z,  &  therefore  x  =  ±  V  z,  it  is  evident  that  the  values  of  x  will  be 
AE,  AG,  AI,  positive,  &  AE',  AG',  &c.,  negative.  [See  Fig.  i.] 

26.  Next,  assume  some  quantity  that  is  given  by  z,  &  constants,  in  any  manner,  so  Assumption   of 
long  as  it  has  not  got  any  common  measure  with  P,  nor  vanishes  when  z  vanishes  ;   also,   to'the^aSer11^'6 
if  x  is  made  an  infinitesimal  of  the  first  order,  let  the  quantity  become  an  infinitesimal  of 

the  same  order,  or  of  a  lower  order.     Such  a  formula  will  be  any  one  such  as 

zf  +  gz'~~l  -f-  hz'~2  -\- -j-  / 

(if  this  is  put  equal  to  zero,  it  will  have  a  number  of  imaginary,  &  a  number  of  real  roots  of 
ome  kind  ;  but  none  of  them  will  be  equal  to  AE,  AG,  AI,  &c.,  whether  positive  or  negative) 
-f  we  multiply  the  whole  by  z.     Call  the  product  Q. 

27.  If  now  we  put  P  —  Qy  =•  o,  I  say  that  this  equation  will  satisfy  all  the  remaining  Formula  containing 
conditions  of  the  curve  ;  &  if  Q  is  correctly  determined,  it  can  satisfy  in  an  infinite  number  quired*1113*10 

of  ways  the  last  condition  also,  given  as  sixthly. 

28.  For,  first  of  all,  since  the  values,  P  &Q,  when  separately  put  equal  to  zero,  have  no  The  equation  will 
common  root,  they  cannot  have  a  common  divisor.     Hence  this  equation  cannot  by  division  jj.e  ^j^'  ^l*  re'- 
be  reduced  to  two  ;  &  therefore  it  is  not  a  composite  equation  formed  from  two  equations,  solved  into  several 
but  is  simple.     Hence,  it  will  represent  some  simple  continuous  curve,  which  is  not  made  c 

up  of  others.     This  was  the  first  condition. 

29.  Next,  this  curve  will  cut  the  axis  C'AC  in  all  those  points,  &  in  them  only,  such  it    will   represent 
as  E,  G,  I,  &c.,E',  G',  &c.     For  it  will  cut  the  axis  C'AC  in  those  points  only,  for  which  »f  {££5ecto£bS 
y  =  o,  &  it  will  cut  it  in  all  of  them.     Further,  when  y  =  O,  we  have  also  Qy  =  0  ;  &  there-  the  curve  at  given 
fore,  since  P  —  Qy  =  o,  we  have  P  =  o.     Now  this  happens  only  at  those  points  for  which  P°mts- 

z  would  be  one  of  the  roots  of  the  equation  P  =  o  ;  that  is  to  say,  as  we  saw  above,  at 
the  points  E,  G,  I,  &c.,  E',  G',  &c.  Hence  it  is  only  at  these  points  that  y  will  vanish, 
&  the  curve  will  cut  the  axis.  It  is  clear  that  it  will  cut  the  axis  at  all  these  points,  from 
the  fact  that  at  all  these  points  we  have  P  =  0.  Hence  also  Qy  =  o.  But  Q  is  not 
equal  to  zero,  since  there  is  no  root  common  to  the  equations  P  =  o,  Q  =  o.  Hence 
y  =  o,  &  the  curve  will  cut  the  axis.  This  was  the  second  condition. 

30.  Further,  since  P  —  Qy  =  o,  it  follows  that  y  =  P/Q  ;  hence,  for  any  determinate  To  each.  abscissa 
abscissa  x,  there  will  be  a  determinate  z  ;    &  thus  P  &  Q  will  be  uniquely  determinate,  sporfd  one  ordinate 
Therefore  also  y  will  be  uniquely  determinate  ;  hence,  to  each  abscissa  x  there  will  correspond  &  °ne  only. 

one  ordinate,  y,  &  only  one.     This  was  the  third  condition. 

(d)  This  solution  is  abstracted  from  my  dissertation  De  Lege  Virium  in  Natura  existentium.  In  addition  to 
these  things  that  have  been  taken  from  that  dissertation,  there  has  been  added  a  third  scholium,  which  appears  for  the 
first  time  in  this  Venetian  edition.  The  problem  here  set  for  solution  will  be  found  in  Art.  117  of  the  first  part  of 
this  work,  y  the  conditions  in  Art.  118. 

411 


412  PHILOSOPHIC  NATURALIS  THEORIA 

Abscissis  hinc  inde  31.  Rursus  sive  x  assumatur  positiva,  sive  negativa,  dummodo  ejusdem  longitudinis 

^raf  aqualerord"*-  s'lt>  semPer  val°r  z  =  xx  er^  idem  ;   ac  proinde  valores  tarn  P,  quam  Q  erunt  semper  iidem. 
natas.    '  Quare  semper  eadem  y.     Sumptis  igitur  abscissis  z  aequalibus  hinc,  &  inde  ab  A,  altera 

positiva,  altera  negativa,  respondebunt  ordinatse  aequales.     Quod  erat   quartum. 

Primum     arcum  32.  Si  autem  x  minuatur  in  infinitum,  sive  ea  positiva  sit,  sive  negativa;    semper  z 

toticumUcumSyarea  minuetur  in  infinitum,  &  evadet  infinitesima  ordinis  secundi.     Quare  in  valore  P  decrescent 
infinita.  in  infinitum  omnes  termini  praeter  /,  quia  omnes  praeter  eum  multiplicantur  per  z,  adeoque 

valor  P  erit  adhuc  finitus.     Valor  autem  Q,  qui   habet    formulam  ductam  in  z   totam, 

p 
minuetur  in  infinitum,  eritque  infinitesimus  ordinis  secundi.     Igitur  ^-=  y  augebitur  in 

infinitum  ita,  ut  evadat  infinita  ordinis  secundi.  Quare  curva  habebit  pro  asymptoto 
rectam  AB,  &  area  BAED  excrescet  in  infinitum,  &  si  ordinatae  y  assumantur  ad  partes  AB, 
&  exprimant  vires  repulsivas,  arcus  asymptoticus  ED  jacebit  ad  partes  ipsas  AB.  Quod 
erat  quintum. 

Post  eas  condi-  [279]  33.  Patet  igitur,  utcunque  assumpto  Q  cum  datis  conditionibus,  satisfieri  primis 
indeterSmina«onem  quinque  conditionibus  curvae.  Jam  vero  potest  valor  Q  variari  infinitis  modis  ita,  ut 
parem  cuicunque  adhuc  impleat  semper  conditiones,  cum  quibus  assumptus  est.  Ac  proinde  arcus  curvae 
curva-Tin1  punctis  mtercepti  intersectionibus  poterunt  infinitis  modis  variari  ita,  ut  primae  quinque  ipsius 
datis  quibusvis.  curvae  conditiones  impleantur  ;  unde  fit,  ut  possint  etiam  variari  ita,  ut  sextam  conditionem 
impleant. 

Quid  requirature  34.  Si  enim  dentur  quotcunque,  &  quicunque  arcus,  quarumcunque  curvarum,  modo 

s^nt  ejusmodi,  ut  ab  asymptoto  AB  perpetuo  recedant,  adeoque  nulla  recta  ipsi  asymptoto 
parallela  eos  arcus  secet  in  pluribus,  quam  in  unico  puncto,  &  in  iis  assumantur  puncta 
quotcunque,  utcunque  inter  se  proxima  ;  poterit  admodum  facile  assumi  valor  P  ita,  ut 
curva  per  omnia  ejusmodi  puncta  transeat,  &  idem  poterit  infinitis  modis  variari  ita,  ut 
adhuc  semper  curva  transeat  per  eadem  ilia  puncta. 


standum°  "*  pra"  35'  ^  enim  numerus  punctorum  assumptorum  quicunque  =  r,  &  a  singulis  ejusmodi 

punctis  demittantur  rectae  parallelae  AB  usque  ad  axem  C'AC,  quae  debent  esse  ordinatae 
curvae  quaesitae,  &  singulae  abscissae  ab  A  usque  ad  ejusmodi  ordinatas  dicantur  Mi.  Mz, 
M3,  &c,  singulae  autem  ordinatae  N'l,  N'z,  N'3,  &c.  Assumatur  autem  quaedam  quantitas 
Azr  +  Bz™  +  Cz'"2  +  ....  -f"  Gz,  quae  ponatur  =  R.  Turn  alia  assumatur 
quantitas  T  ejusmodi,  ut  evanescente  z  evanescat  quivis  ejus  terminus,  &  ut  nullus  sit 
divisor  communis  valoris  P,  &  valoris  R  -f-  T,  quod  facile  fiet,  cum  innotescant  omnes 
divisores  quantitatis  P.  Ponatur  autem  Q  =  R  -f-  T,  &  jam  aequatio  ad  curvam  erit 
P  —  R  y  —  Ty  =  o.  Ponantur  in  hac  sequatione  successive  Mi,  M2,  M3,  &c,  pro  x,  & 
N'l,  N'z,  N'3,  &c  pro  y.  Habebuntur  aequationes  numero  r,  quae  singulae  continebunt 
valores  A,  B,  C,  .  .  .  G,  unius  tantum  dimensionis  singulos,  numero  pariter  r,  &  praeterea 
datos  valores  Mi,  Mz,  M3,  &c,  Ni,  Nz,  N3,  &c,  ac  valores  arbitrarios,  qui  in  T  sunt 
coefficientes  ipsius  z. 

Progressus  ulterior.  36.  per  illas  aequationes  numero  r  admodum  tacile  determinabuntur  illi  valores  A,  B, 

C,  .  .  .  G,  qui  sunt  pariter  numero  r,  assumendo  in  prima  aequatione,  juxta  methodos 
notissimas,  &  elementares  valorem  A,  &  eum  substituendo  in  sequationibus  omnibus  sequen- 
tibus,  quo  pacto  habebuntur  aequationes  r  —  I.  Hae  autem  ejecto  valore  B  reducentur 
ad  r  —  z,  &  ita  porro,  donee  ad  unicam  ventum  fuerit,  in  qua  determinato  valore  G,  per 
ipsum  ordine  retrograde  determinabuntur  valores  omnes  praecedentes,  singuli  in  singulis 
aequationibus. 

Conciusio,  &  cohaer.  37.  Determinatis  hoc  pacto  valoribus  A,  B,  C,  .  .  .  G  [280]  in  aequatione  P  —  Ry 

pr^dTtirTslon!  --Ty  =  o,  siveP-Qy  =  o,  patet   positis  successive  pro  x  valorib™    Mi,  Mz,  M3,  &c, 

ditionibus.  debere  valores  ordinatae  y  esse  successive    Ni,   Nz,   N3,  &c  ;    ac  proinde   debere  curvam 

transire   per  data   ilia  puncta  in  datis  illis    curvis  :    &   tamen    valor    Q    adhuc    habebit 

omnes    conditiones  prsecedentes.     Nam  imminuta  z  ultra  quoscunque  limites,  minuentur 

singuli  ejus  termini  ultra  quoscunque  limites,  cum  minuantur  termini  singuli  valoris  T, 

qui  ita  assumpti  sunt,  &  minuantur  pariter  termini  valoris  R,  qui  omnes  sunt  ducti  in  z, 

&  praeterea  nullus  erit  communis  divisor  quantitatum  P,  &  Q,  cum  nullus  sit  quantitatum 

P,  &  R  +  T. 

inde     contactus,  38.  Porro  si  bina  proxima  ex  punctis  assumptis  in  arcubus  curvarum  ad  eandem  axis 

partem  concipiantur  accedere  ad  se  invicem  ultra  quoscunque  limites,  &  tandem  congruere, 
factis  nimirum  binis  M  aequalibus,  &  pariter  asqualibus  binis  N  ;  jam  curva  quaesita  ibidem 


aCC<      5 


SUPPLEMENT  III  413 

31.  Again,  whether  x  is  taken  positive  or  negative,  so  long  as  its  length  is  the  same,  T?    equal   at>- 

i  i          r  -11  i        i  TT  i  i    '        t  i       i    T«  a    r\      -11  i  scissae,      therefore, 

the  value  of  z,  or  x*,  will  be  the  same.     Hence  the  values  of  both  r  &  Q  will  be  the  same,  there    will   corre- 

Hence  y  will  always  be  the  same  for  either.     Hence,  if  equal  abscissae  x  are  taken  one  on  sP°nd    equal    or- 

either  side  of  A,  the  one  positive  &  the  other  negative,  the  corresponding  ordinates  will  side1  of'  the  origin! 
be  equal.     This  was  the  fourth  condition. 

32.  Now,  if  x  is  diminished  indefinitely,  whether  it  is  positive  or  negative,  z  will  be  The  first  arc   wil1 
also  diminished  indefinitely,  &  will  become  an  infinitesimal  of  the  second  order.     Hence,  branch  aSwith>t°an 
every  term  in  the  value  of  P,  except  /,  will  diminish  indefinitely  ;   for  each  of  them  except  infinite  area, 
this  one  has  a  factor  z.     Thus  the  value  of  P  will  remain  finite.     But  the  value  of  Q,  in 

which  the  whole  expression  was  multiplied  by  z,  will  diminish  indefinitely  ;  &  it  will  become 
an  infinitesimal  of  the  second  order.  Hence  y,  which  is  equal  to  P/Q,  will  be  increased 
indefinitely,  so  that  it  becomes  an  infinity  of  the  second  order.  Therefore,  the  curve  will 
have  the  straight  line  AB  as  an  asymptote,  &  the  area  BAED  will  become  infinite  ;  also, 
if  AB  is  taken  to  be  the  positive  direction  for  the  ordinates  y,  these  will  represent  repulsive 
forces,  &  the  asymptotic  arc  ED  will  fall  in  the  direction  given  by  AB.  This  was  the  fifth 
condition. 

33.  Hence,  it  is  clear  that,  however  Q  is  chosen  subject  to  the  given  conditions,  the  After    these    con- 
first  five  conditions  for  our  curve  will  be  satisfied.     Now,  the  value  of  Q  can  be  varied  ^ifMed  htherebetm 
in  an  infinite  number  of  ways,  such  that  it  will  still  fulfil  the  conditions  under  which  it  remains  an  equal 
was  assumed.     Then  the  arcs  of  the  curves  intercepted  between  the  intersections  with  indeterminat1°n 
the  axis  could  be  varied  in  an  infinite  number  of  ways,  such  that  the  first  five  conditions  any    given  curves 
for  the  curve  are  satisfied.     Hence  it  follows  that  they  can  be  varied  also,  in  such  a  way  at  any  given  P°ints- 
that  the  sixth  condition  is  satisfied. 

34.  Now,  if  any  number  of  arcs  of  any  kind,  belonging  to  any  curves,  are  given;   so  The  conditions 
long  as  these  are  such  that  they  continually  recede  from  the  asymptote  AB,  &  therefore  th^o'ug^g^vTn 
such  that  no  straight  line  parallel  to  this  asymptote  will  cut  any  of  them  in  more  than  one  points     of    these 
point ;   &  if  in  these  arcs  there  are  taken  any  number  of  points,  no  matter  how  close  they  c 

are  together,  a  value  of  P  can  be  obtained  quite  easily,  such  that  the  curve  will  pass  through 
all  these  points.  Moreover,  this  can  be  done  in  an  infinite  number  of  ways,  such  that 
the  curve  will  still  pass  through  all  these  points  in  every  case. 

35.  For,  let  the  number  of  points  taken  be  any  number  r.     From  each  of  these  points,  How  this  can  be 
let  a  straight  line  be  drawn  parallel  to  AB,  to  meet  the  axis  C'AC  ;  these  must  be  ordinates  mana8ed- 

of  the  curve  required.  Let  the  several  abscissae  measured  from  A  to  these  ordinates  be 
MI,  M»,  Ms,  &c.  ;  &  let  the  corresponding  ordinates  be  NI,  Nz,  Na,  &c.  Then  assume 
some  quantity  Azf  +  B  z^1  +  Cz'1"2  +  .  .  .  +  Gz,  &  suppose  that  this  is  R.  Next, 
take  another  quantity,  T,  of  such  a  kind  that,  when  z  vanishes,  each  term  of  T  vanishes, 
&  there  is  no  common  divisor  of  P  &  R  +  T.  This  can  easily  be  done,  since  the  divisors 
of  the  quantity  P  are  known.  Now,  suppose  that  Q  =  R+  T ;  the  equation  to  the  curve, 
will  then  be  P  —  Ry  --  Ty  =  O.  In  this  equation,  substitute  in  succession  MI,  Ma,  M3 
&c.  for  x,  &  NI,  NS,  Ns  &c.  for  y.  Then  we  shall  have  r  equations,  each  of  which  will 

contain  the  values  A,  B,  C, ,  G,  which  are  also  r  in  number  ;  &  these  will  all  appear 

linearly.  The  equations  will  also  contain,  in  addition,  the  given  values  MI,  M2,  M3,  &c., 
NI,  Nj,  NS,  &c.,  &  the  arbitrary  values  which  aopear  as  the  coefficients  of  z  in  the  expression 
T. 

36.  From  these  equations,  r  in  number,  the  values  of  A,  B,  C,  .  .  .  ,  G,  which  are  also  Further  progress. 
r  in  number,  can  quite  easily  be  determined.     Thus,  from  the  first  equation,  according 

to  well-known  elementary  methods,  obtain  the  value  of  A  in  terms  of  the  rest,  &  substitute 
this  value  in  each  of  the  other  equations.  In  this  way  we  shall  obtain  r  —  I  equations. 
Eliminating  B  from  these,  we  shall  get  r  —  ^  equations ;  &  so  on,  until  at  last  we  shall  come 
to  a  single  equation.  Having  determined  from  this  the  value  of  G,  we  can  determine,  by 
retracing  our  steps,  the  preceding  values  in  succession,  one  value  from  each  set  of  equations. 

37.  The    values    of   A,    B,    C,  .  .  .  .  ,  G,    in    the   equation  P  —  Ry  —  Ty  =  o,  or  Conclusion ;  agree- 
P  — Qy  =  o,  having  been  thus  found,  it  is  clear  that,  if  the  values  Ml5  M,,  M3,  &c.,  are  ^nt preceding 
substituted  for  x  in  succession,  the  values  of  y  will  be  NI,  N»,  N3,  &c.     Hence,  the  curve  conditions, 
must  pass  through  the  given  points  on  the  given  arcs ;   &  still  the  value  of  Q  will  satisfy 

all  the  preceding  conditions.  For,  if  z  is  diminished  beyond  all  limits,  each  of  its  terms 
will  be  diminished  beyond  all  limits ;  since  each  of  the  terms  of  the  value  of  T,  according 
to  the  supposition  made,  will  be  so  diminished,  &  likewise  each  of  the  terms  of  R,  which 
all  contain  a  factor  z.  In  addition,  there  will  be  no  common  divisor  of  P  &  Q,  since  there 
is  none  for  the  quantities  P  &  R  +  T. 

38.  Again,  if  two  of  the  chosen  points,  next  to  one  another  in  the  arcs  of  the  curves,  Hence  contacts, 
are  supposed  to  approach  one  another  on  the  same  side  of  the  axis  beyond  all  limits,  &  approach' °ofS' any 
finally  to  coincide  with  one  another,  namely,  by  making  two  values  of  M  equal  to  one 

another,  &  therefore  also  the  corresponding  values  of  N,  then  also  the  required  curve  will 


4H  PHILOSOPHIC  NATURALIS   THEORIA 

tanget  arcum  curvae  datae  :  &  si  tria  ejusmodi  puncta  congruant,  earn  osculabitur  :  quin 
immo  illud  prasstari  poterit,  ut  coeant  quot  libuerit  puncta,  ubi  libuerit,  &  habeantur 
oscula  ordinis  cujus  libuerit,  &  ut  libuerit  sibi  invicem  proxima,  arcu  curvae  datse  accedente, 
ut  libuerit,  &  in  quibus  libuerit  distantiis  ad  arcus,  quos  libuerit  curvarum,  quarum  libuerit, 
&  tamen  ipsa  curva  servante  omnes  illas  sex  conditiones  requisitas  ad  exponendam  legem 
illam  virium  repulsivarum,  ac  attractivarum,  &  datos  limites. 

Adhuc  indetermina-  »Q    Cum  vero  adhuc  infinitis   modis  variari  possit    valor    T;    infinitis   modis   idem 

tio  relicta  pro  infmi-  J*  *  .  i     •    r-    •   •  T    •  ••  •  •        i         i      •  v   •       -i 

tis  modis.  praestan  poterit  :  ac  promde  infinitis  modis  mvenin  poterit  curva  simplex  datis  conditiombus 

satisfaciens.     Q.E.F. 

Posse    &    axem  ^O-  Coroll.    I.     Curva   poterit   contingere   axem   C'AC   in   quot   libuerit    punctis,   & 

con  mgere.  os  a.n,  contmgere  gimu^  ac  secare  in  iisdem,  ac  proinde  eum  osculari  quocunque  osculi  genere. 
Nam  si  binse  quaevis  e  distantiis  limitum  fiant  aequales  ;  curva  continget  rectam  C'A, 
evanescente  arcu  inter  binos  limites  ;  ut  si  punctum  I  abiret  in  L,  evanescente  arcu  IKL  ; 
haberetur  contactus  in  L,  repulsio  per  arcum  HI  perpetuo  decresceret,  &  in  ipso  contactu 
IL  evanesceret,  turn  non  transiret  in  attractionem,  sed  iterum  cresceret  repulsio  ipsa  per 
arcum  LM.  Idem  autem  accideret  attractioni,  si  coeuntibus  punctis  LN,  evanesceret 
arcus  repulsivus  LMN. 

Posse  contingere  ^j.  Si  autem  tria  puncta  coirent,  ut  LNP  ;  curva  contingeret  simul  axem  C'AC,  & 

ab  eodem  simul  secaretur,  ac  promde  haberet  in  eodem  puncto  contactus  flexum 
contrarium.  Haberetur  autem  ibidem  transitus  ab  attractione  ad  repulsionem,  vel  vice 
versa,  adeoque  verus  limes. 

Quid  congruentia  4.2.  Eodem  pacto  possunt  congruere  puncta  quatuor,  quinque,  quotcunque  :  &  si 

congruat  numerus  punctorum  par;  habebitur  contactus  :  si  impar  ;  contactus  simul,  & 
sectio.  Sed  quo  plura  puncta  coibunt ;  eo  magis  curva  accedet  ad  [281]  axem  C'AC  in 
ipso  limite,  eumque  osculabitur  osculo  arctiore. 

Posse  axem  secari  43.  Coroll.  2.     In  iis  limitibus,  in  quibus  curva  secat  axem  C'AC,  potest  ipsa  curva 

anguiis,U&aSCquavis  secare  eundem  in  quibuscunque  angulis  ita  tamen,  ut  angulus,  quern  efficit  ad  partes  A 
g  nit  u  d  i  n  e  arcus  curvae  in  perpetuo  recessu  ab  asymptote  appellens  ad  axem  C'AC  non  sit  major  recto, 
&  ibidem  potest  aut  axem,  aut  rectam  axi  perpendicularem  contingere,  aut  osculari,  quo- 
cunque contactus,  aut  osculi  genere,  nimirum  habendo  in  utrolibet  casu  radium  osculi 
magnitudinis  cujuscunque,  &  vel  utcunque  evanescentem,  vel  utcunque  abeuntem  in 
innnitum. 

Demonstratio :  A  A.  Nam  pro  illis  punctis  datis  in  arcubus  curvarum    quarumcunque,   quas    curva 

umitatio  necessana.  jnventa  pOtest  vel  contingere,  vel  osculari  quocunque  osculi  genere,  ex  quibus  definitus 
est  valor  R,  possunt  assumi  arcus  curvarum  quarumcunque  secantium  axem  C'AC,  in 
angulis  quibuscunque  :  solum  quoniam  semper  arcus  curvae,  ut  *Ny  debet  ab  asymptote 
recedere,  non  poterit  punctum  ullum  t  praecedens  limitem  N  jacere  ultra  rectam  axi  perpen- 
dicularem erectam  ex  N,  vel  punctum  y  sequens  ipsum  N  jacere  citra  ;  ac  proinde  non 
poterit  angulus  AN*,  quern  efficit  ad  partes  A  arcus  *N  in  perpetuo  recessu  ab  asymptote 
appellens  ad  axem  C'AC,  esse  major  recto. 

Quid  possint  arcus  ^  Possunt  autem  arcus  curvarum  assumptarum  in    iisdem   punctis  aut    axem,  aut 

taVum™  omTiPa  rectam  axi  perpendicularem  contingere,    aut    osculari,  quocunque    contactus,  aut    osculi 

posse  &  inventam.    genere,  ut  nimirum  sit  radius  osculi  magnitudinis  cujuscunque,  &  vel  utcunque  evanescens, 

vel  utcunque  abiens  in  innnitum.     Quare  idem  accidere  poterit  ut  innuimus,  &    arcui 

curvae  inventae,  quae  ad  eos  arcus  potest  accedere,  quantum  libuerit,  &  eos  contingere,  vel 

osculari  quocunque  osculi  genere  in  iis  ipsis  punctis. 

Conditio  necessana,  ^g.  Solum  si  curva  inventa  tetigerit  in  ipso  limite  rectam  axi  C'AC  perpendicularem, 

naturaUJUS  CUrV8e  debebit  simul  ibidem  eandem  secare  ;  cum  debeat  semper  recedere  ab  asymptote,  adeoque 
debebit  ibidem  habere  flexum  contrarium. 

Coroi.  i  includi  in  ^7.  Scholium  I.     Corollarium  I  est  casus  particularis  hujus  corollarii  secundi,  ut  patet  : 

sed  libuit  ipsum  seorsum  diversa  methodo,  &  faciliore  prius  eruere. 

Quid  ubivis  etiam  ^g.  Coroll.  3.     Arcus  curvae  etiam  extra  limites  potest  habere  tangentem  in  quovis 

angulo  inclinatam  ad  axem,  vel  ei  parallelam,  vel  perpendicularem  cum  iisdem  contactuum, 

&  osculorum  conditionibus,  quae  habentur  in  corollario  2. 
Demonstratio    ea-  ^  Demonstratio  est    prorsus  eadem  :    nam  arcus   curvarum    dati,  ad   quos    arcus 

curvse  inventae  potest  accedere    ubicunque,  quantum  libuerit,    possunt  habere    ejusmodi 

conditiones. 


m  a 
arcuum. 


SUPPLEMENT  III  415 

touch  the  arc  of  the  given  curve  at  this  point.  If  three  such  points  coincide  with  one 
another,  it  will  osculate  the  given  curve.  Indeed,  it  can  be  brought  about  that  any  number 
of  points  desired  shall  coincide,  &  thus  osculations  of  any  order  desired  can  be  obtained. 
These  may  be  as  close  together  as  desired,  the  arc  approaching  the  given  curve  to  any  desired 
degree  of  closeness ;  or  they  may  be  at  any  distances  from  any  of  the  arcs  of  any  of  the 
curves,  as  desired.  Yet  the  curve  will  observe  all  those  six  conditions,  which  are  required 
for  representing  the  law  of  repulsive  &  attractive  forces,  as  well  as  the  limit-points. 

39.  Now,  since  the  value  of  T  can  still  be  varied  in  an  infinite  number  of  ways,  this  can  There  is  stm   left 
be  brought  about  in  an  infinite  number  of  ways.     Hence,  in  an  infinite  number  of  ways,  coun^ie^wa10^  '" 
a  simple  curve  can  be  found  satisfying  the  given  conditions.     Q  .  E  .  F  . 

40.  Cor.  i.     The  curve  may  touch  the  axis  C'AC  in  any  desired  number  of  points  ;  it  is  possible  also 
or  at  the  same  time  touch  &  cut  it  at  the  same  points ;   &  hence  it  may  osculate  the  axis  touchhtheCaxIs  o° 
with  any  kind  of  osculation.     For,  if  any  two  of  the  distances  for  the  limit-points  become  to  osculate  it,  etc. 
equal,  the  curve  will  touch  the  straight  line  C'A,  the  arc  between  these  two  limit-points 

vanishing.  Thus,  if  the  point  I  should  go  off  to  L,  the  arc  IKL  vanishing,  we  should  have 
contact  at  L,  &  repulsion  would  continually  decrease  along  the  arc  HI,  vanish  at  the  point 
of  contact  IL  ;  after  that  it  would  not  become  an  attraction,  but  the  repulsion  would 
continually  increase  along  the  arc  LM.  The  same  thing  would  also  happen  in  the  case 
of  attraction,  if,  owing  to  the  points  L,N  coinciding,  the  repulsive  arc  LMN  should  vanish. 

41.  Again,  if  three  points,  say  L,N,P,  should  coincide,  the  curve  would  at  the  same  *t  is  possible  that 
time  touch  the  axis  C'AC  &  intersect  it ;    thus,  at  that  point  of  contact  there  would  be  simultaneous7  con! 
contrary  flexure.     Also,  there  would  be  there  a  passage  from  attraction  to  repulsion,  or  tact  &  section  of 
vice  versa,  &  therefore  a  true  limit-point. 

42.  In  the  same  way,  four  points  may  coincide,  or  five,  or  any  number.     If  the  number  The  result  of  the 
of  points  that  coincide  is  even,  there  will  be  touching  contact;    if  the  number  is  odd,  there  several  ninterse°c- 
will  be  contact  &  intersection  at  the  same  time.     The  greater  the  number  of  the  points  tions. 

that  coincide,  the  more  the  curve  will  approach  to  coincidence  with  the  axis  C'AC  at  that 
limit-point ;  &  thus  the  higher  the  order  of  the  osculation. 

43.  Cor.  2.      At  these  limit-points,  where  the  curve  cuts  the  axis  C'AC,  the  curve  The  axis  may  be 
may  cut  it  at  any  angle  ;    but  in  such  a  way  that  the  angle,  which  the  arc  of  the  curve,  atlny^ngfe,  C&rby 
in  its  continuous  recession  from  the  asymptote,  makes  with  the  direction  of  A  as  it  comes  arcs  of  any  size, 
up  to  the  axis  C'AC,  is  not  greater  than  a  right  angle  ;  &  it  may  touch  either  the  axis  or 

the  straight  line  at  right  angles  to  the  axis,  or  osculate  the  axis ;  the  contact  or  the  osculation 
being  of  any  order.  That  is  to  say,  it  may  have  in  either  case  a  radius  of  osculation  of 
any  magnitude  whatever,  either  vanishing  or  becoming  infinite,  in  any  way  whatever. 

44.  For,  we  may  take  as  our  chosen  points  in  the  arcs  of  any  curves,  which  the  curve  Demonstration; 
of  forces  is  found  to  touch  or  to  osculate  with    an  osculation  of  any  order,  from  which  tion.SSa 

the  value  of  R  is  determined,  arcs  of  any  curves  cutting  the  axis  C'AC  at  any  angles.  Except 
that,  since  the  arc  of  the  curve,  such  as  tNy,  must  always  recede  from  the  asymptote,  it 
would  not  be  possible  for  any  point  such  as  t,  which  precedes  the  limit-point  N,  to  lie  on 
the  far  side  of  the  straight  line  perpendicular  to  the  axis  erected  at  N  ;  or  for  the  point  y, 
which  follows  N,  to  lie  on  the  near  side  of  this  perpendicular.  Thus,  the  angle  AN/,  which 
it  makes  with  the  direction  of  A,  as  the  arc  tN  continually  recedes  from  the  asymptote, 
as  it  comes  up  to  the  axis  C'AC,  cannot  be  greater  than  a  right  angle. 

45.  Again,  the  arcs  of  the  assumed  curves  may,  at  these  points  either  touch  the  axis  what  the  arcs  of 
or  the  straight  line  perpendicular  to  the  axis,  or  they  may  osculate,  the  contact  or  the  Say^^the^me 
osculation  being  of  any  order  ;   that  is,  the  radius  of  osculation  may  be  of  any  magnitude  properties  may  aii 
whatever,  either  vanishing  or  becoming  infinite,  in  any  way.     Hence,  as  I  said,  this  may  curve^oumi  by  the 
also  be  the  case  for  an  arc  of  the  curve  that  has  been  found  ;  for  it  can  be  made  to  approximate 

as  closely  as  desired  to  these  curves,  so  as  to  touch  them  or  osculate  them,  with  any  order 
of  osculation,  at  these  points. 

46.  Except   that,   if  the   curve   should   touch   at   the   limit-point   the   straight   line  Necessary     condi- 
perpendicular  to  the  axis  C'AC,  it  must  at  the  same  time  cut  it  at  that  point ;    for  the  the '  nature^f  The 
curve  must  always  recede  from  the  asymptote,  &  thus  is  bound  to  have  contrary  flexure  curve- 

at  the  point. 

47.  Scholium  i.     The  first  corollary  is  a  particular  case  of  the  second,  as  is  evident.  Th<r  *jrs,t  corollary 

•n  e          i  ^       •       c.  •   i_  •     j  j  t   L  J-JT  a  •        ls   included    in   the 

but  1  preferred  to  take  it   first,  with  an  independent    proof  by  a  different  &  an  easier  second, 
method. 

48.  Cor.  3.     Even  beyond  the  limit-points,  the  arc  of  the  curve  can  have  a  tangent  What  happens  also 
inclined  at  any  angle  to  the  axis,  or  parallel  to  it,  or  perpendicular  to  it ;   with  the  same  the  "mut-pointl°n 
conditions  as  to  contact  or  osculation  as  we  had  in  the  second  corollary. 

49.  The  proof  is  exactly  the  same  as  before  ;  for,  the  given  arcs  of  the  curves,  to  ^^  the  same  as 
which  the  arc  of  the  curve  that  is  found  can  be  made  to  approximate  as  closely  as  desired, 

may  have  the  conditions  stated. 


PHILOSOPHIC  NATURALIS  THEORIA 

sssposs     are  5?-  Coroll.   5.     Mutata  abscissa  per  quodcunque  intervallum  datum,  potest  ordinata 

ad  mutationem  or-  mutari  per  aliud  quodcunque  datum  utcunque  minus,  vel  majus  ipsa  mutatione  abscissae, 
quan^nque^10  *  &  ut-[282]-cunque  majus  quantitate  quacunque  data  ;  ac  si  differentia  abscissa;  sit  infini- 
tesima,  &  dicatur  ordinis  primi  ;  poterit  differentia  ordinatae  esse  ordinis  cujuscunque, 
vel  utcunque  inferioris,  vel  intermedii,  inter  quantitates  finitas,  &  quantitates  ordinis 
primi. 

prO  51-  Patet  primum  ex  eo,  quod,  ubi  determinatur  valor  R,  potest  curva  transire  per 

quotcunque,  &  quaecunque  puncta,  adeoque  per  puncta,  ex  quibus  ducts  ordinals  sint 
utcunque  inter  se  proximae,  &  utcunque  insequales. 


iSe  s'mor°uvm  ^2'  7*tQt  secundum  :    quia  in  curvis,  ad  quas  accedit  arcus  curvs  invents  vel  quas 

ordine.  osculatur  quocunque  osculi  genere,  potest  differentia  abscissae  ad  differentiam  ordinats 

esse  pro  diversa  curvarum  natura  in  datis  earum  punctis  in  quavis  ratione,  quantitatis 
infinitesimae  ordinis  cujuscunque  ad  infinitesimam  cujuscunque  alterius. 


mottpdere  a  .        53>  Scholi.um  2'     Illud  notandum,  ubicunque  fuerit  tangens  curvae  invents  inclinata 
positione  tangentis.  'm  angulo  finito  ad  axem,  fore  differentiam  abscissae  ejusdem  ordinis,  ac    est  differentia 
ordinats  :    ubi  tangens  fuerit  parallela  axi,  fore  differentiam  ordinats  ordinis  inferioris, 
quam  sit  differentia  abscissae,  &  vice  versa,  ubi  tangens  fuerit  perpendicularis  axi. 


mnr™  54-  Prseterea  notandum  :  si  abscissa  fuerit  ipsa  distantia  limitis,  quae  vel  augeatur,  vel 

termini,  tiir    in    Jim*         •  t*^v  *  t*  *       •  i  •  • 

ite.  mmuatur  utcunque  ;    differentia  ordinats  erit  ipsa  ordinata  Integra  ;    cum  nimirum  in 

limite  ordinata  sit  nihilo  aequalis. 
Posse  arcus  utcun.  55.  Coroll.    c.     Arcus    repulsionum,    vel     attractionum     intercepti     binis     limitibus 

que     rccedcre     ftD          **_  i  i  i»i»i  /•» 

axe.  quibuscunque,  possunt  recedere  ab  axe,  quantum  libuent,  adeoque  fieri  potest,  ut  alii 

propiores  asymptote  recedant  minus,  quam  alii  remotiores,  vel  ut  quodam  ordine  eo  minus 
recedant  ab  axe,  quo  sunt  remotiores  ab  asymptoto,  vel  ut  post  aliquot  arcus  minus 
recedentes  aliquis  arcus  longissime  recedat. 

Demonstrate.  ^  Omnia   manifesto  consequuntur  ex  eo,   quod  curva  possit  transire  per  quaevis 

data  puncta. 

m0uSmhcrusrirs°ySmp.  57'     CorolL  6-     Potest  curva  ipsum  axem  C'AC  habere  pro  asymptoto  ad  partes  C', 

toticum.&aiiacrura  &  C  ita,  ut  arcus  asymptoticus  sit  vel  repulsivus,  vel  attractivus  ;    &  potest  arcus  quivis 
asymptotica.  bims  limitibus   quibuscunque  interceptus   abire  in  infinitum,  ac  habere  pro  asymptoto 

rectam  axi  perpendicularem,  utcunque  proximam  utrilibet  limiti,  vel  ab  eo  remotam. 

Ratio   praestandi  eg.  Nam  si  concipiatur,  binos  postremos  limites  coire,  abeuntibus  binis  intersectionibus 

primum.  *  •••*«••  .      .     «    .  . 

in  contactum,  turn  concipiatur,  ipsam  distantiam  contactus  excrescere  in  infinitum  ;  jam 
axis  aequivalet  rectae  curvam  tangenti  in  puncto  infinite  remoto,  adeoque  evadit  asymptotus  : 
&  si  arcus  evanescens  inter  postremos  duos  limites  coeuntes  fuerit  arcus  repulsionis  ; 
postremus  arcus  asymptoticus  erit  arcus  attractionis.  Contra  vero,  si  arcus  evanescens 
fuerit  arcus  attractionis. 


&  [2^5l  59'  Eodem  pacto  si  concipiatur,  quamvis  ordinatam  respondentem  puncto  cuilibet, 
per  quod  debet  transire  curva,  abire  in  infinitum  ;  jam  arcus  curvae  abibit  in  infinitum,  & 
erit  ejus  asymptotus  in  ilia  ipsa  ordinata  in  infinitum  excrescens. 


exhfberiVpermfunc>  ^°'  ^c^°^um   3-     Ope   formulae  exhibentis   curvam   propositam  habetur  lex  virium 

tionem    distantiae,  expressa   per  functionem   quandam   distantiae  constantem   plurimis   terminis,   immo  per 

^1°z™e1r°en<d1aem  ^qu*11^0116111  commiscentem  abscissam,  &  ordinatam,  ac  utriusque  potentias  inter  se,  & 

unicam  potentiam  :  cum  rectis  datis,  non  per  solam  ipsius  distantiae  potentiam.     Sunt,  qui  censeant  expres- 

sionem  per  solam  potentiam  debere  prseferri  expressioni  per  functionem  aliam,  quia  haec  sit 

simplicior,  quam  ilia,  &  quia  in  ilia  praeter  distantias  debeant  haberi  aliquae  aliae  parametri, 

quae  non  sint  solae  distantiae  ;   dum  in  formula  -  exprimente  x  distantias,  distantiae  solae 

oc 

rem  confidant,  videatur  autem  vis  debere  pendere  a  solis  distantiis,  potissimum  si  sit 
quaedam  essentialis  proprietas  materiae  :  praeterea  addunt,  nullam  fore  rationem  suffi- 
cientem,  cur  una  potius,  quam  alia  parameter  expressionem  virium  deberet  ingredi,  si 
parametri  sint  admiscendae. 

Qua  occasione  haec  6i.  Hsc  agitata  sunt  potissimum  ante  hos  aliquot  annos    in  Academia  Parisiensi, 

a^tetaY^parfeiensi  cum  censeretur,  motum  Apogei  Lunaris  observatum  non  cohaerere  cum  gravitate  decrescente 
Academia.  ^  ratione  reciproca  duplicata  distantiarum,  &  ad  ipsum  exhibendum  adhiberetur  gravitas 


SUPPLEMENT  III  417 

50.  Cor.  4.     If  the  abscissa  is  changed  by  any  given  interval,  the  ordinate  can  be  The  change  of  the 
changed  by  any  other  given  interval  however  much  the  latter  may  be  smaller  or  greater  an^^atio^to  ^he 
than  the  change  of  the  abscissa,  or  however  much  greater  than  any  given  quantity  it  may  change  of  the  ordi- 
be.     Further,  if  the  difference  in  the  abscissa  is  infinitesimal,  &  we  call  it  an  infinitesimal  nate' 

of  the  first  order,  then  the  difference  in  the  ordinate  may  be  of  any  order,  either  of  any 
order  below  the  first  whatever,  or  intermediate  between  finite  quantities  &  quantities  of 
this  first  order. 

51.  The  first  part  is  evident  from  the  fact  that,  when  the  value  of  R  is  determined,  P""?0*     for    fini*e 
a  curve  can  be  made  to  pass  through  any  number  of  points  of  any  sort ;   &  thus,  through 

points,  from  which  ordinates  are  drawn  as  close  to  one  another  as  we  please,  &  unequal 
to  one  another  in  any  way. 

52.  The  second  part  is  evident,  because  in  the  curves,  to  which  the  arcs  of  the  curve  The  same  for  any 
found  approximates,  or  which  it  osculates  with  any  order  of  osculation,  the  difference  of  ^fsr  of  mfimtesi- 
the  abscissa  can  bear  any  ratio  to  the  difference  of  the  ordinate  for  a  different  nature  of 

the  curves  at  given  points  on  them  ;  this  ratio  may  be  that  of  an  infinitesimal  quantity 
of  any  order  to  an  infinitesimal  quantity  of  any  other  order. 

53.  Scholium  2.     It  is  to  be  observed  that,  whenever  the  tangent  to  the  curve  that  This    relation    de. 
has  been  found  is  inclined  at  a  finite  angle  to  the  axis,  the  difference  of  the  abscissa  is  of  tkm  ^"t 

the  same  order  as  the  difference  of  the  ordinate  ;   when  the  tangent  is  parallel  to  the  axis,  gent, 
the  difference  of  the  ordinate  will  be  of  an  inferior  order  to  the  difference  of  the  abscissa  ; 
&  the  opposite  is  the  case  when  the  tangent  is  perpendicular  to  the  axis. 

54.  In  addition,  it  is  to  be  observed  that,  if  the  abscissa  corresponds  to  a  limit-point,  The.  case 
&  this  is  either  increased  or  diminished  in  any  way,  the  difference  of  the  ordinate  will  be  mg^at 
the  whole  ordinate  itself,  for  at  the  limit-point  itself  the  ordinate  is  indeed  equal  to  zero,  limit-points. 

55.  Cor.  5.     The  arcs  of  repulsion  or  attraction,  which  are  intercepted  between  any  The  arcs  may  re. 
pair  of  limit-points,  may  recede  from  the  axis  to  any  extent ;   &  thus,  it  may  happen  that  ^  aeny  gSent6  aX1S 
some  that  are  nearer  to  the  asymptote  may  recede  less  than  others  that  are  more  remote  ; 

or  that,  to  any  order,  they  may  recede  the  less,  the  further  they  are  from  the  asymptote  ; 
or  that,  after  a  number  of  arcs  that  recede  less,  there  may  be  one  which  recedes  by  a  very 
large  amount. 

56.  Everything  clearly  follows  from  the  fact  that  the  curve  can  be  made  to  pass  through  Prooi  ot  this  state- 
any  given  points. 

1:7.  Cor.  6.     The  curve  may  have  the  axis  C'AC  as  an  asymptote  in   the  directions  ,The  .curve   can 

,  f^fi   .",    .  ,  ,        J,  .  .      .  ,  t  •  •  have  lts  last  branch 

or  C  &  C,  m  such  a  manner  that  the  asymptotic  arc  is  either  repulsive  or  attractive  ;  also  any  asymptotic,  &  also 
arc   intercepted   between  a    pair   of   limit-points   may  go  off  to  infinity,  &  have  for   an  other     asymptotic 

•    iv  ,.      ,  i  ,    '  ,      v  •  t       v     •       branches. 

asymptote  a  straight  line  perpendicular  to  the  axis,  however  near  or  tar  irom  either  limit- 
point. 

58.  For,  if  we  suppose  that  the  last  two  limit-points  coincide,  as  the  two  intersections  Jhe  Proof  of   the 

.       •  ,       „    i  r  •  f  •  i  i        first     part   of   this 

coincide  &  become  a  point  where  the  curve  touches  the  axis ;  &  then  suppose  that  the  statement, 
distance  of  this  point  of  contact  becomes  infinite  ;  then  the  axis  will  become  equivalent 
to  a  straight  line  touching  the  curve  at  a  point  infinitely  remote,  &  will  thus  be  an  asymptote. 
If  the  vanishing  arc  that  is  intercepted  between  those  two  last  coincident  limit-points 
should  be  an  arc  of  repulsion,  the  last  asymptotic  arc  will  be  an  arc  of  attraction.  But 
the  opposite  would  be  the  case  if  the  vanishing  arc  should  be  an  arc  of  attraction. 

59.  In  the  same  way,  if  it  is  supposed  that  any  ordinate  corresponding  to  any  point,  The  proof  of  the 
through  which  the  curve  has  to  pass,  should  go  off  to  infinity ;   then  the  arc  of  the  curve  statement.  °f  the 
will  also  go   off  to  infinity,  and  that  ordinate,  as  it  increases  indefinitely,  will  become  an 

asymptote  of  the  curve. 

60.  Scholium  3.     By  the  help  of   the  formula  corresponding  to  the  proposed  curve,  Th?r  law  of  foilce? 

,,  ....  9       .    ]  .        .  ,  ,-,.rr.  °    .         ..      r      r      .  ,  'is  here  represented 

the  law  of  forces  is  obtained  expressed  as  a  definite  function  of  the  distance  with  many  by  a  function  of  the 
terms ;  or  rather,  by  means  of  an  equation  involving  the  abscissa  &  the  ordinate,  &  powers  distance;  many 

,  .  i '    .  •    i       i  •  i  •       i  f     i        i  •  Vni  others  think  that  a. 

of  these,  along  with  given  straight  lines,  &  not  by  a  single  power  of  the  distance.     There  single  power  of  the 
are  some  who  think  that  representation  by  means  of  a  single  power  is  to  be  preferred  to  djstance  .1S  prefer- 

.  rr.  ,        J  ,       ,  .  °     . r     ,  ,      r ..  able ;  their  reasons. 

representation  by  another  function  ;  because  the  latter  is  simpler  than  the  former  ;  & 
because  in  it,  besides  the  distances,  there  are  bound  to  be  other  parameters  that  are  not 
merely  distances.  Whereas,  in  the  formula  i/;e»»,  where  x  represents  the  distances,  the 
distances  alone  settle  the  matter ;  &  it  is  seen  that  the  force  must  depend  on  the  distance 
alone,  especially  if  it  should  be  an  essential  property  of  matter.  Besides,  they  add,  there 
is  no  sufficient  reason  why  any  one,  rather  than  any  other,  parameter  should  enter  the 
expression  for  the  forces,  if  parameters  are  to  be  admitted. 

61.  This  question  came  in  for  a  large  amount  of  discussion  a  number  of  years  ago  in  The    occasion    on 

...  r  T>     •          T-I         •  ,  111  •  rit  11     which  this  question 

the  Academy  of  Pans.     For,  it  was  thought  that  the  motion  of  the  lunar  apogee,  as  observed,  was  discussed  in  the 
did  not  agree  with  the  idea  of  gravity  decreasing  in  the  inverse  duplicate  ratio  of  the  distances.  Academy  of  Paris. 
They  considered  that  an  expression  for  gravity  should  be  employed,  in  which  it  was  represented 

EE 


4i  8  PHILOSOPHIC  NATURALIS   THEORIA 

,  .         .          a         b  .       .  .  .... 

expressa  per  bmommm   —  -\  --     cujus   pars   prior  in  magnis,  pars  posterior  in  exiguis 

X  X 

distantiis  respectu  socise  partis  evanesceret  ad  sensum,  sed  ilia  prior  in  distantia  Lunae  a 
Terra  adhuc  turbaret  hanc  posteriorem,  quantum  satis  erat  ad  earn  praestandam  rem. 
Atque  earn  ipsam  binomii  expressionem  adhibuerant  jam  plures  Physici  ad  deducendam 
simul  ex  eadem  formula  gravitatem,  &  majores  minimarum  particularum  attractiones,  ac 
multo  validiorem  cohaesionem,  ut  innuimus  num.  121  :  atque  hae  difficultates  in  Parisiensi 
Encyclopaedia  inculcantur  ad  vocem  dttractio,  Tomo  I  turn  edito. 

Occasionem  substi-  62.  Paullo  post,  correctis  calculis  innotuit,  motum  Apogei  lunaris  ea  composita  formula 

tuendi  turn  functio-  •    j  •  .  .  , 

nem  cessasse,  sed  non  mdigere  :    at  rationes  contra  id  propositae,  quae  multo  magis  contra  meam  virium 
rationes     contra  legem  pugnarent,  meo  quidem  iudicio  nullam  habent  vim.      Nam  in  primis  quod  ad 

allatas      nullam      •        v    •  •  1-111  •  j-  ••  /• 

habere  vim  :  curvas  simplicitatem  pertmet,  hie  habent  locum  ea  omnia,  quas  dicta  sunt  in  ipso  opere  num.  no 
omnes  unifprmis  de  simplicitate  curvarum.  Formula  exprimens  solam  potentiam  quandam  distantiae 
aqvuTsimpiices!  Se  designatae  per  abscissam  exprimit  ordinatam  ad  locum  geometricum  pertinentem  ad 
familiam,  quam  exhibet  [284]  y  =  x",  qui  quidem  locus  est  Parabola  quaedam  ;  si  m 
sit  numerus  positivus,  nee  sit  unitas  :  recta  ;  si  sit  unitas,  vel  zero  :  quaedam  Hyperbola  ; 
si  sit  numerus  negativus  :  formula  autem  continens  functionem  aliam  quamvis  exprimit 
ordinatam  ad  aliam  curvam,  quae  erit  continua,  &  simplex,  si  ilia  formula  per  divisionem  non 
possit  discerpi  in  alias  plures.  Omnes  autem  ejusmodi  curvae  sunt  aeque  simplices  in  se, 
&  alia;  aliis  sunt  magis  affines,  aliae  minus.  Nobis  hominibus  recta  est  omnium  simplicissima, 
cum  ejus  naturam  intueamur,  &  evidentissime  perspiciamus,  ad  quam  idcirco  reducimus 
alias  curvas,  &  prout  sunt  ipsi  magis,  vel  minus  affines,  habemus  eas  pro  simplicioribus,  vel 
magis  compositis  ;  cum  tamen  in  se  aeque  simplices  sint  omnes  illae,  qua  ductum  uniformem 
habent,  &  naturam  ubique  constantem. 

Esse  «eque  simpli-  63.  Hinc  ipsa  ordinata  ad  quamvis  naturae  uniformis  curvam  est  quidam  terminus 

cem  relationem      •         ..  .     .  \      .       .          .       ,  .     ,  •,.  j     i_      •  •  *. 

ordinatarum  ad  simplicissimae  relatioms  cujusdam,  quam  habet  ordinata  ad  abscissam,  cui  termmo  impositum 
abscissas  :  termino-  est  generale  nomen  functionis  continens  sub  se  omnia  functionum  genera,  ut  etiam  quam- 

rum    multitudinem  ,  .  „       •    i     i  •  j       •  j  •   j-  •  j  • 

pro  ea  exprimenda  cunque   solam  potentiam,  &  si  haberemus  nomma  ad  ejusmodi  functiones  denommandas 
oriri  a  nostro  cog-  singillatim  ;    haberet  nomen  suum  quaevis  ex  ipsis,  ut  habet  quadratum,  cubus,  potestas 

noscendi  mode.  •         c-  •  •  j-        i     • 

quaevis.     Si  omnia  curvarum  genera,  omnes  ejusmodi  relationes  nostra  mens  mtueretur 

immediate  in  se  ipsis  ;   nulla  indigeremus  terminorum  farragine,  nee  multitudine  signorum 

ad  cognoscendam,  &  enuntiandam  ejusmodi  functionem,  vel  ejus  relationem  ad  abscissam. 

Origo  ejus  modi  ab  64.  Verum  nos,  quibus  uti  monui  recta  linea  est  omnium  locorum  geometricorum 

habe'muTnos  loam-  simplicissima,  omnia  referimus  ad  rectam,  &  idcirco  etiam  ad  ea,  quae  oriuntur  ex  recta, 

ines  naturae  soiius  ut  est  quadratum,  quod  fit  ducendo  perpendiculariter  rectam  super  aliam  rectam  aequalem, 

omnw  curvas  refer*  &  cubus>  qui  fit  ducendo  quadratum  eodem  pacto  per  aliam  rectam  primae  radici  aequalem, 

imus.  quibus  &  sua  signa  dedimus  ope  exponentium,  &  universalizando  exponentes  efformavimus 

nobis  ideas  jam  non  geometricas  superiorum  potentiarum,  nee  integrarum  tantummodo, 

&  positivarum,  sed  etiam  fractionariarum,  &  negativarum  :    &  vero  etiam,  abstrahendo 

semper  magis,  irrationalium.     Ad  hasce  potentias,  &  ad  producta,  quae  simili  ductu  conci- 

piuntur  genita,   reducimus  caeteras   functiones  omnes  per  relationem,   quam  habent  ad 

ejusmodi  potentias,  &  producta  earum  cum  rectis  datis,  ac  ad  earn  reductionem,  sive  ad 

expressionem  illarum  functionum  per  hasce  potentias,  &  per  haec    producta,  indigemus 

terminis  jam  paucioribus,  jam  pluribus,  &  quandoque  etiam,  ut  in  functionibus  transcendent- 

alibus,  serie  terminorum  infinita,  quae  ad  valorem,  vel  naturam  functionis  propositae  accedat 

semper  magis,  utut  in  hisce  casibus  earn  nunquam  ac-[28s]-curate  attingat  :    habemus 

autem  pro  magis,  vel  minus  compositis  eas,  quae  pluribus,  vel  paucioribus  terminis  indigent, 

sive  quae  ad  solas  potentias  relationem  habent  propiorem. 

Aiiud    mentium  65.  At  si  aliud  mentium  genus  aliam  curvam  ita  intime  cognosceret,  ut  nos  rectam; 

haberet  pro  maxime  simplici  solam  ejus  functionem,  &  ad  exprimendum  quadratum,  vel 


potentiae  necessario  aliam  potentiam,  contemplaretur  illam  eandem  relationem,  sed  inverse  assumptam  ita,  ut 
Tem!  Ve^maljorem  incipiendo  a  functione  ipsa  per  earn,  &  per  similes  ejus  functiones,  ac  functionum  citeriorum 
fan-aginem.  functiones  ulteriores,  addendo,  ac  subtrahendo  deveniret  demum  ad  quaesitam.  Relatio 

potentiae  ad  functionem,  &  nexus  mutuus  compositionem  habet,  &  multitudinem  terminorum 

inducit  :    uterque  relationis  terminus  est  in  se  aeque  simplex. 

Sola    etiam   poten- 

in'cTudi  etta^amTd6  *>6.  Quod  pertinet  ad  parametros,  quas  dicitur  includere  functio,  non  autem  potentia 

nos  homines  para-  distantiae,  non  est  verum  id  ipsum,  quod  potentia  parametros  non  includat.     Formula 

metres  plures:  para-      •.  .       .      .  ,  . 

meter   in    unitate   _  includit  unitatem  ipsam,  quae  non  est  aliquid  in  se  determmatum,  sed  potest  expnmere 

arbitraria,  &  affix-   xm 

certame£Santiamd  magnitudinem  quamcunque.     Et  quidem  ea  species  includit  omnes  species  Hyperbolarum, 


SUPPLEMENT  III  419 

by  the  formula  of  two  terms,  afxz  +  b/xz  ;  of  this,  the  first  part  at  large  distances,  &  the 
last  part  at  very  small  distances,  would  practically  become  evanescent  with  respect  to  the 
other  part  associated  with  it.  But  the  first  part,  for  the  distance  of  the  Moon  from  the 
Earth,  would  still  disturb  the  last  part  sufficiently  to  account  for  the  observed  inequality. 
Already,  several  Physicists  had  employed  such  an  expression  with  two  terms  to  deduce 
at  the  same  time  from  the  one  formula  both  gravity  &  the  greater  attractions  of  very 
small  particles,  &  much  more  so  the  still  stronger  forces  of  cohesion,  as  I  have  mentioned 
in  Art.  121.  These  difficulties  are  included  in  the  Encyclopedia  Parisiensis  under  the 
heading  Attraction,  in  Vol.  I  published  at  that  time. 

62.  Shortly  afterwards,  the  calculations  were  corrected  &  it  was  found  that  the  motion  The  reasons  for 
of  the  lunar  apogee  did  not  necessitate  this  compound  formula.  But  the  arguments  brought  formula"  *)  or  *the 
forward  against  it,  which  were  still  more  in  opposition  to  this  Theory  of  mine  with  regard  function,  which 
to  the  law  of  forces,  have  no  weight,  at  any  rate  in  my  eyes.  For,  in  the  first  place,  as  ceawd  toexitt^ 
regards  simplicity,  all  those  things  held  good  in  this  case,  which  I  stated  in  this  work,  but  the  arguments 
Art.  116,  with  regard  to  simplicity  of  curves.  A  formula  in  terms  of  a  single  power  of  agamst  it 


the  distance  represented  by  an  abscissa  expresses  the  ordinate  of  a  geometrical  locus  belonging  weight  ;  all  curves 


to  the  family,  represented  by  y  =  xm  ;  &  this  locus  is  a  Parabola,  if  m  is  any  positive  number 

except  unity  ;  a  straight  line,  if  m  is  unity  or  zero  ;  &  a  hyperbola,  if  m  is  a  negative  number,  equally  simple 

But  a  formula  containing  some  other  function  expresses  the  ordinate  of  some  other  curve  ; 

&  this  will  be  continuous  &  simple,  if  the  formula  cannot  be  separated  by  division  into 

several  others.     Further,  all  such  curves  are  equally  simple  in  themselves  ;    &  some  of 

them  are  more,  some  less,  of  the  same  nature  as  others.     To  us  men,  a  straight  line  is  the 

simplest  of  all  ;   for  we  observe  its  nature  &  understand  it  clearest  of  all.     To  it  therefore 

we  refer  all  other  curves  ;  &  according  as  they  are  more  or  less  like  it  in  nature,  we  consider 

them  to  be  the  more  or  less  simple.     However,  in  themselves,  all  curves,  which  are  composed 

of  a  continuous  line  &  have  a  constant  nature  everywhere,  are  equally  simple. 

63.  Hence,  the  ordinate  to  any  curve  of  a  uniform  nature  is  some  term  of  some  very  The    relation    be. 
simple  relation  that  the  ordinate  has  to  the  abscissa.     To  this  term  there  is  given  the  ^the  ^bscissT^s 
general  name,  function  ;    this  name  includes  every  kind  of  function,  for  instance,  even  a  equally  simple  ;  the 
single  power.     If  we  had  names  to  denote  such  functions  singly,  each  of  them  would  have  usedn't  o°f  express 
its  own  name,  just  as  a  square,  a  cube,  or  any  other  power.     If  our  minds  were  capable  of  this  relation  arises 
viewing  all  kinds  of  curves,  &  all  such  relations  in  themselves,  at  a  glance,  then  there  would  ^nowing^t^7 
be  no  need  of  a  medley  of  terms,  &  a  multitude  of  signs  in  order  to  know  &  state  such  a 

function  or  its  relation  to  the  abscissa. 

64.  But  we,  to  whom,  as  I  mentioned,  the  straight  line  is  the  simplest  of  all  geometrical  The  origin  of  this 

•      •       *«         11  •    T     T  j    i         r  i  i  •  •       f  method  comes  from 

loci,  refer  all  curves  to  a  straight  line,  and  therefore  also  to  all  those  things  that  arise  from  a  the  intuition  which 
straight  line  ;   such  as  a  square,  which  is  formed  by  moving  a  straight  line  perpendicular  we  men  have  of.the 

•    iv  >.   ,    '.  .  '  V.   ,     .      ,°          ,     ,  r  .  ,        nature  of  a  straight 

to  another  straight  line  which  is  equal  to  it  ;   &  a  cube,  which  is  formed  by  moving  the  iine  alone,  to  which 

square  in  the  same  way  all  along  another  straight  line  equal  to  its  prime  root.     To  these  we  refer  ail  curves. 

we  have  given  their  own  signs  by  the  help  of  exponents  ;    &,  generalizing  exponents,  we 

have  formed  for  ourselves  ideas,  that  are  not  now  geometrical,  of  higher  powers  ;   &  these 

not  integral  only,  &  positive,  but  also  fractional,  &  negative  ;    &  indeed,  by   continual 

abstraction,  ever  more  &  more,  ideas  of  irrational  powers.     To  these  powers,  &  to  products 

which  may  be  considered  to  arise  in  a  similar  fashion,  we  reduce  all  other  functions,  by 

means  of  the  relation  they  bear  to  such  powers  &  their  products  with  given  straight  lines. 

For  this  reduction,  or  expression  of  the  functions  by  means  of  these  powers  &  these  products 

we  require  sometimes  more,  sometimes  less,  terms  ;  even  when,  as  in  the  case  of  transcendental 

functions,  we  have  to  use  an  infinite  series  of  terms,  which  approximates  more  &  more 

closely  to  the  value  &  the  nature  of  the  given  function,  although  in  such  cases  it  never 

actually  reaches  this  value.     Moreover,  we  consider  these  to  be  more  or  less  composite, 

according  as  they  require  more  or  less  terms,  or  have  a  nearer  relation  to  single  powers. 

65.  But  if  another  type  of  mind  knew  another  curve  as  intimately  as  we  know  the  straight  Another   type    of 
line,  it  would  consider  a  single  function  of  that  curve  to  be  the  most  simple  of  all  ;   &,  to  ™en  reiatkm^of  CSa 
express  a  square  or  another  power,  it  would  consider  the  self-same  relation,  inversely  taken,  power  would  neces. 
so  that,  beginning  with  the  function,  through  it  &  like  functions  of  it,  &  of  higher  functions  an^quafor  greater 
of  these  lower  functions,  by  addition  &  subtraction,  the  mind  would  finally  arrive  at  the  medley  of  terms. 
function  required.     The  relation  of  a  power  to  a  function,  &  the  mutual  connection,  has 

a  compositeness,  &  leads  to  a  multitude  of  terms.     Each  term  of  the  relation  is  in  itself 

equally   simple.  power,     we     men 

66.  As  regards  the  introduction  of  parameters,  which  they  say  are  included  in  a  function  ^'"a  meters6  "a* 
but  not  in   a   power   of  the  distance,  it  is  not  true  that  a  power    does    not  include  a  parameter  in    the 
parameter.    The  formula   i/x™  includes  unity  itself  ;   &  this  is  not  something  that  is  the'cTmbinatfoA  ol 
self-determinate,  but     something     that    can     express    any     magnitude.     Indeed,    that  a  certain  force  with 
species  of  formula    includes   all   species   of  hyperbolas,   &,  if  the  exponent   m    is  given 


420  PHILOSOPHIC  NATURALIS  THEORIA 

ac  definite  exponente  m,  exprimit  unicam  quidem  earum  speciem,  sed  quae  continet  infinitas 
numero  individuas  Hyperbolas,  quarum  quselibet  suam  parametrum  diversam  habet  pro 
diversitate  unitatis  assumptae.  Potest  quidem  quaevis  ex  iis  Hyperbolis  ad  arbitrium 
assumi  ad  exprimendam  vim  decrescentem  in  ea  ratione  reciproca  ;  sed  adhuc  in  ipsa 
expressione  includitur  quaedam  parameter,  quae  determinet  certam  vim  a  certa  ordinata 
exprimendam,  sive  certam  vim  certae  distantiae  respondentem,  qua  semel  determinata 
remanent  determinatae  reliquse  omnes,  sed  ipsa  infinitis  modis  determinari  potest,  stante 
expressione  facta  per  ordinatas  ejusdem  curvae,  sive  per  eandem  potentise  formulam. 
Ejusmodi  primus  nexus  a  sola  distantia  utique  non  pendet. 
Parameter  in  ex-  5^^  Accedit  autem  alia  quasi  parameter  in  exponente  potentiae  :  illius  numeri  m 

ponente  potentiae.      ,  '  .  j  5-  •  j-  •  i-  • 

determmatio  utique  non  pendet  a  distantia,  nee  distantiam  aliquam  exprimit. 
Non    SS'V1  68.  Sed  nee  illud  video,  cur  etiam  si  dicatur  vis  esse  proprietas  qusedam  materise 


sola    distantia  essentialis,  ea  debeat  necessario  pendere  a  solis  distantiis.     Si  esset  quaedam  virtus,  quae  a 

etiam,    si   **  *~  materiae  puncto  quovis  egressa  progrederetur  motu  uniformi,  &  rectilineo  ad  omnes  circum 

tas  materiae.  distantias  :    turn  quidem  diffusio  ejus  virtutis  per  orbes  majores  aeque  crassos  fieret  in 

ratione  reciproca  duplicata  distantiarum,  &  a  distantiis  solis  penderet  ;   quanquam  ne  turn 

quidem  ab  iis  penitus  solis,  sed  ab  iis,  &  exponente  secundae  potentiae,  ac  primo  nexu  cum 

arbitraria  [286]  unitate.      At  cum  nulla  ejusmodi  virtus  debeat  progredi,  &  in  progressu 

ipso  ita  attenuari  ;  nihil  est,  cur  determinatio  ad  accessum  debeat  pendere  a  solis  distantiis, 

ac  proinde  solae  distantiae  ingredi  formulam  functionis  exprimentis  vim. 


69.  Verum  admisso  etiam,  quod  necessario  vis   debeat  pendere  a  solis  distantiis,  nihil 
distantiis,  "ordlna-  habetur  contra  expressioncm  factam  per  functionem  quandam.     Nam  ipsa  functio  per  se 
dataqcurvae  endere  immediate  pendet  a  distantia,  &  est  ordinata  quaedam  ad  curvam  quandam  certae  naturae, 
a  solis  abscissis.        respondens  abscissae  datae  cuilibet  sua.     Parametri  inducuntur  ex  eo,  quod  illius  relationem 
ad  abscissam  exprimere  debeamus  per  potentias  abscissae,  &  potentiarum  producta  cum 
aliis  rectis ;    sed  in  se,  uti  supra  diximus,  ejusdem  est  naturae  &  ilia  functio,  ac  potentia 
quaevis,  &  ilia,  ut  haec,  ordinatam  immediate  simplicem  exhibet  respondentem  abscissae 
ad  curvam  quandam  uniformis,  &  in  se  simplicis  curvae. 

Parametros  ipsas  70.  Praeterea  ipsae  illae  parametri,   quae  formulam  functionis   ingrediuntur,  possunt 

eTs    dfuncticmem  esse  certae  quaedam  distantiae  &  assumi  debere  ad  hoc,  ut  illis  datis  distantiis  illae  datae,  & 

esse  ingressas,  quod  non  aliae  vires  respondeant.     Sic  ubi  quaesita  est  formula,  quae  exprimeret  aequationem 

debuerit  haberrvls  a&  curvam  quaesitam,  assumpsimus   quasdam  distantias,  in  quibus  curva  secaret  axem, 

data,  vei  nulla.        nimirum  in  quibus,  evanescente  vi  haberentur  limites,  &  earum  distantiarum  valores  ingressi 

sunt   formulam   inventam,   ut   quaedam   parametri.     Possunt   igitur   ipsae  parametri   esse 

distantiae  qusedam  ;  ac  proinde  posito,  quod  omnino  debeat  vis  exprimi  per  solas  distantias, 

potest    adhuc    exprimi    per    functionem    continentem    quotcunque    parametros,   &  non 

exprimetur  necessario  per  solam  aliquam  potentiam. 

Argumentum  con-  71.  Reliquum  est,  ut  dicamus  aliquid  de  Ratione  Sufficienti,  quae  dicitur  parametros 

rationis  sufficientis!  excludcre,  cum  non  sit  ratio,  cur  aliae  prae  aliis  parametri  seligantur. 

si  vis  sit  essentialis  72.  Inprimis  si  vis  est  in  ipsa  natura  materiae  ;    nulla  ratio  ulterior  requiri    potest 

tTnum'  "param™  praeter  earn  ipsam  naturam,  quae  determinet  hanc  potius,  quam  aliam  vim  pro  hac  potius, 

trorum  esse  ipsam  quam  pro  ilia  distantia,  adeoque  hanc  potius,  quam  aliam  parametrum.     Quaeri  ad  summum 

cu"hocgenuUsma!tei  poterit,  cur  clegcrit  Naturae  Auctor  earn  potissimum  materiam,  quae  earn  legem  virium 

riae  existat,  ration-  haberet  cssentialem,  quam  aliam  :    ubi  ego  quidem,  qui  summam  in  Auctore  Naturae 

Creatoris  T^idenT  libertatem  agnosco,  censeo,  ut  in  aliis  omnibus,  nihil  aliud  requiri  pro  ratione  sufficient! 

si  ea  non  sit  essen-  electionis,   quam    ipsam   liberam   determinationem   Divinae    voluntatis,   a   cujus    arbitrio 

pendeat  turn,  quod  hanc  potius,  quam  aliam  eligat  rem,  quam  condat,  turn  quod  ea  re 

hanc  in  se  naturam  habente,  ubi  jam  condita  fuerit,  utatur  ad  hoc  potius,  quam  ad  illud 

ex  tarn  multis,  ad  quae  natura  quaevis  a  tanti  Artificis  manu  adhibita  potest  esse  idonea. 

Atque  haec  responsio  [287]  aeque  valet,  si  vis  non  est  ipsi  materiae  essentialis,  sed  libera 

Auctoris  lege  sancita,  quo  casu  ipse  pro  libero  arbitrio  suo  hanc  huic  materiae  potuit  legem 

dare  prse  aliis  electam. 

Praeter  arbitrium 
retorsio  in  poten- 
tia :  rationemutro-  73.  At  si  ratio  etiam  exhiberi  debeat,  quae  Auctorem  Naturae  potuerit  impellere  ad 

quoTsibTTpse^ro8-  seligendam  materiam  hac  potissimum  prasditam  essentiali  virium  lege,  vel  ad  seligendam 
posuerit.  qui  pos-  pro  hac  materia  hanc  legem  virium  ;  quaeri  primo  potest,  cur  hunc  potius  exponentem 
ignoti.  e  3  potentiae  elegerit,  &  hanc  parametrum  in  unitate  inclusam,  sive  in  quadam  determinata 


SUPPLEMENT  III  421 

it  represents  one  of  these  species ;  &  any  one  of  these  has  its  own  different  parameter  for 
a  difference  in  the  unity  assumed.  It  is  possible  for  any  one  of  these  hyperbolas  to  be 
arbitrarily  chosen  to  represent  a  force  which  decreases  in  that  reciprocal  ratio  ;  but  still 
there  is  included  in  the  expression  a  certain  parameter  ;  namely,  one  which  determines  a 
certain  force  to  be  represented  by  a  certain  ordinate,  or  a  certain  force  to  correspond  with 
a  certain  distance  ;  when  once  this  is  determined,  all  the  rest  are  at  the  same  time  determined. 
But  this  can  be  done  in  an  infinite  number  of  ways,  without  altering  the  generation  of  the 
expression  from  the  ordinates  of  the  self-same  curve,  or  the  same  formula  of  a  power.  A 
primary  connection  of  this  kind  certainly  does  not  depend  on  distance  alone. 

67.  Besides  there  is  another  thing,  that  is  very  like  a  parameter,  in  the  exponent  of  There   is  a  para- 
the  power  ;  the  determination  of  the  number  m  at  any  rate  does  not  depend  on  the  distance,  ™e^r ofntheepower 
nor  does  it  express  any  distance. 

68.  But,  really,  I  do  not  see  why,  if  it  is  said  that  force  is  some  property  essential  to  There  is  no  reason 
matter,  it  should  of  necessity  depend  on  distances  alone.     If  it  were  some  virtue,  which  Thy  l\  should 

i     1    r  .  i  ,  1-1  T  •         •  •   i  dependonthe 

proceeded  from  any  point  of  matter  &  progressed  with  uniform  motion  in  a  straight  line  distance   alone,  if 
to  all  distances  round ;    then  indeed  the  diffusion  of  this  virtue  through  greater  spheres  force  ls  an  e336"*^1 

.....  ,  r     i        T  property  of  matter. 

equally  thick  would  be  as  the  inverse  squares  of  the  distances ;  &  thus  would  depend  on 
distance  alone.  Although  not  even  then  would  it  depend  altogether  on  distances  alone  ; 
but  on  them  &  the  exponent  of  the  second  power,  in  addition  to  the  prime  connection 
with  an  arbitrary  unity.  But  since  no  such  virtue  is  bound  to  progress,  &  even  in  progression 
to  be  so  attenuated,  there  is  no  reason  why  determination  for  approach  should  depend 
on  distances  alone  ;  &  that  therefore  distances  alone  should  enter  the  formula  of  the  function 
that  expresses  the  force. 

69.  But  even  if  it  is  admitted  that  force  must  necessarily  depend  on  the  distances  Even  if  the  force 
alone  ;    still  there  is  nothing  against  the  expression  being  formed  of  some  function.     For  *jld,  dePend  on  the 
the  function  in  itself  depends  directly  upon  distance,  &  is  an  ordinate  to  some  curve  of  the  ordinates  also! 
known  nature,  corresponding  to  its  own  given  abscissa,  which  may  be  anything  you  please.  m  themselves,  de. 

•       l  1     l  l  r  i  i  11-  r  6i  *i.  Pend      on      the     ab- 

Parameters  are  induced  by  the  fact  that  we  have  to  express  the  relation  of  the  ordinate  scissae    alone,    for 
to  the  abscissa  by  means  of  powers  of  the  abscissa,  &  the  products  of  these  powers  with  any  siven  curve- 
other  straight  lines.     But  in  themselves,  as  I  said  above,  both  the  function  &  any  power 
are  of  the  same  nature  ;  &  the  former,  like  the  latter,  will  give  a  perfectly  simple  ordinate 
corresponding  to  the  abscissa  to  any  arc  of  a  curve  that  is  uniform  &  simple  in  itself. 

70.  Besides,  these  very  parameters,  which  come  into  the  formula,  may  be  certain  The     parameters 
known  distances ;    &  they  have  to  be  assumed  for  the  purpose  of  ensuring  that  to  these  tanc^e-lheyrehave 
given  distances  those  given  forces,  &  not  others,  correspond.      So,  when  we  seek  a  formula  come   into  the 
to  express  the  equation  to  the  curve  required,  we  assume  certain  distances  in  which  the  ^"riven  distances 
curve  shall  cut  the  axis ;  that  is  to  say,  distances  for  which,  as  the  force  vanishes,  we  shall  there   must  be   a 
obtain  limit-points ;    &  the  values  of  these  distances  have  entered  the  formula  we    have  j=jv|{| force  or  none 
found,    as    certain    parameters.     Hence    the    parameters    themselves    may    be    distances. 

Therefore,  if  it  is  stated  that  force  is  absolutely  bound  to  depend  on  distances  alone,  it  is 
still  possible  to  express  the  force  by  a  function  containing  any  number  of  parameters ; 
&  it  is  not  necessarily  expressed  by  some  single  power. 

71.  It  only  remains  to  say  a  few  words  with  regard  to  Sufficient  Reason  ;   this  being  The    argument 
said  to  exclude  parameters,  because  there  is  no  reason  why  some  parameters  should  be  jfiJ2?0f  .rfictont 
chosen  in  preference  to  others.  reason. 

72.  First  of  all,  if  force  is  an  essential  property  of  matter,  there  is  no  need  for  any  if  force    is    an 
other  reason  beside  that  of  the  very  nature  of  matter,  to  determine  that  this,  rather  esse ntiai^  prope rt y 
than  another,  force  should  correspond  to  this,  rather  than  to  another,  distance  ;  &  therefore  reason    for '  such 
this  parameter,  rather  than  any  other.     It  may  be  asked,  &  we  can  go  no  further,  why  parameters  is  the 

r  TVT  i  •  •  •      i  1111  i  •  '  i    very     nature     of 

the  Architect  of  Nature  chose  this  matter  m  particular,  such  as  should  have  this  essential  matter ;   why  such 
law  of  forces,  &  no  other.      In  that  case,  I,  who  believe  in  the  supreme   freedom  of  the  Jjf^ ^Jm* Jf  tjj* 
Architect  of  Nature,  think,  as  in  all  other  things,  that  there  is  nothing  else  required  for  the  creator;  the  same 
sufficient  reason  for  His  choice  beyond  the  free  determination  of  the  Divine  will.     Upon  thins  if  f°™e  w  not 
the  free  exercise  of  this  depends  not  only  the  fact  that  He  chose  this  thing  rather  than 
another  to  create  ;   &  also  that,  the  thing  having  this  nature  in  itself,  when  it  was  once 
created,  He  should  use  it  for  this  purpose  rather  than  for  any  other  of  the  very  many  purposes, 
to  which  any  nature  employed  by  the  hand  of  so  mighty  an  Artificer  may  be  suitable.     This 
reply  applies    just  as  well,  even  if  the  force  is  not  an  essential  property  of  matter,  but 
established  by  the  free  law  of  the  Author  ;  for,  in  that  case,  He,  of  his  own  free  will,  could 

,..'..  ,.  ..  ,  11        11  There  is  something 

give  this  law  to  this  matter,  having  chosen  it  m  preference  to  all  other  laws.  beyond  win  in  the 

73.  Now,  if  we  have  also  to  give  the  reason  which  might  have  forced  the  Author  of  limitation    of    his 
Nature  to  select  in  particular  this  matter  possessed  of  this  essential  law  of  forces,  or  to  bonfcases^s'Sfe'aim 
select  for  this  matter  this  law  of  forces  especially  ;  it  may  first  be  asked  why  He  should  have  *h.at  He  set  before 

-  ...  ,.  i  .  '  •     •      i     j    j    •        i_  •          Himself ;  &  this  we 

preference  for  this  exponent  of  the  power,  this  parameter  that  is  included  in  the  unity,  may  not  know. 


422  PHILOSOPHIC  NATURALIS  THEORIA 

distantia  quandam  determinatam  vim.  Quod  de  iis  dicitur,  applicari  poterit  parametris 
reliquis  functionis  cujusvis.  Ut  ille  exponens,  ilia  unitas,  ille  nexus  potuit  habere  aliquid, 
quod  caeteris  praestaret  ad  eos  obtinendos  fines,  quos  sibi  Naturae  Auctor  praescripsit ;  sic 
etiam  aliquid  ejusmodi  habere  poterant  reliquse  omnes  quotcunque,  &  qualescunque 
parametri. 

Evoiutio  finis   ip-  74.  Deinde   rem  ipsam  diligenter  consideranti  facile   patebit,   ad   obtinendos   fines, 

hatendi  "nfac  cluos  s*ki  Naturae  Auctor  debuit  proponere,  non  fuisse  aptam  solam  potentiam  quandam 

nexum  ab  Algebra  distantiae  pro  lege  virium,  sed  debuisse  assumi  functionem,  quae  ubi  exprimi  deberet  per 

primufiie'm1  nisi  nostram  humanum  Algebram,  alias  quoque  parametros    admisceret.     Si  ex.  gr.  voluisset 

per  functionem,  ad  per  eandem  vim  &  motum  Planetarum  ad  sensum  ellipticum  cum  Kepleriano  nexu  inter 

on'iTproWem^pro  quadrata  temporum  periodicorum,  &  cubos  distantiarum  mediarum,  &  cohsesionem  per 

hac  corporum  con  contactum,  nulla  sola  potentia  ad  utrumque  praestandum  finem  fuisset  satis,  quem  finem 

stitutionc,    &     mo-  a  L 

tuum  serie.  obtinuisset  ilia,  formula  —  +  — .     At  nee  ea  formula  potuit  ipsi  sufEcere,  si  vera  est  Theoria 

OC  X 

mea,  cum  ea  formula  nullam  habeat  in  minimis  distantiis  vim  contrariam  vi  in  maximis, 
sed  in  omnibus  distantiis  eandem,  nimirum  in  minimis  attractivam,  ut  in  maximis. 
Cohaesio  punctorum  se  invicem  repellentium  in  minimis  distantiis,  &  attrahentium  in 
majoribus  haberi  non  potuit  sine  intersectione  curvae  cum  axe,  quae  intersectio  sine  para- 
metro  aliqua  non  obtinetur.  Verum  ad  omnem  hanc  phaenomenorum  seriem  obtinendam 
multo  pluribus,  uti  ostensum  est  suo  loco,  intersectionibus  curvae,  &  flexibus  tarn  variis 
opus  erat,  quae  sine  plurimis  parametris  obtineri  non  poterant.  Consideretur  elevatissimum 
inversum  problema  affine  alteri,  cujus  mentio  est  facta  num.  547,  quo  quaeratur  numerus 
punctorum,  &  lex  virium  mutuarum  communis  omnibus  necessaria  ab  habendam  ope 
cujusdam  primae  combinationis,  hanc  omnem  tam  diuturnam,  tarn  variam  phaenomenorum 
seriem,  cujus  perquam  exiguam  particulam  nos  homines  intuemur,  &  statim  patebit  eleva- 
tissimum debere  esse,  &  respectu  habito  ad  nostros  exprimendi  modos  complicatissimum 
genus  curvae  ad  ejusmodi  problematis  solutionem  ne-[288]-cessarium  ;  quod  tamen  problema 
certas  quasdam  parametros  in  singulis  saltern  solutionibus  suis,  quae  numero  fortasse  infinite 
sunt,  involveret,  sola  unica  potentia  ad  tanti  problematis  solutionem  inepta. 


id    non    potuisse  75.^Debuit  igitur  Naturae  Auctor,  qui  hanc  sibi  potissimum  Phaenomenorum  seriem 

potentiam:  legeS  proposuit,  parametros  quasdam  seligere,  &  quidem  plures,  nee  potuit  solam  unicam  pro 
quadrati  distantia:  lege  virium  exprimenda  distantiae  potentiam  adhibere  :  ubi  &  illud  praeterea  ad  rem  eandem 
confirmandam  recolendum,  quod  a  num.  124,  dictum  est  de  ratione  reciproca  duplicata 
distantiarum,  quam  vidimus  non  esse  omnium  perfectissimam,  nee  omnino  eligendam, 
&  illud,  quod  sequenti  horum  Supplementorum  paragrapho  exhibetur  contra  vires  in 
minimis  distantiis  attractivas  &  excrescentes  in  infinitum,  ad  quas  sola  potentia  demum 
deducit. 

Conciusio     contra  76.  Atque  hoc  demum  pacto,  videtur  mihi,  dissoluta  penitus  omnis  ilia  difficultas, 

necessitate1?,     V^J  quae  proposita  fuerat,  nee  ulla  esse  ratio,  cur  sola  potentia  qusedam  distantiae  anteferri 

convementiam  sol-    >  ,      f .    r.    '      .  .  ,.r        ,  r        .     . 

ius  potentia:.  debuent  function!  utcunque,  si  nostrum  exprimendi  modum  spectemus,  complicatissimae. 


SUPPLEMENT  III 


423 


or  a  certain  determined  force  for  a  certain  determined  distance.  Now,  what  is  to  be  said 
about  these  things,  can  be  also  applied  to  all  the  other  parameters  of  any  function.  Namely, 
that  this  exponent,  this  unity,  this  connection  might  have  had  something  in  them,  which 
was  superior  to  all  other  things  for  the  purpose  of  obtaining  those  aims  which  the  Author 
of  Nature  had  set  before  Himself.  Similarly,  all  the  other  parameters  might  have  something 
of  the  same  sort,  no  matter  how  many  or  of  what  kind  they  are. 

74.  Next,  it  will  easily  be  clear  to  anyone,  who  considers  the  matter  with  care,  that, 
for  the  purpose  of  obtaining  the  aims  which  the  Author  of  Nature  was  bound  to  have  set 
Himself,  any  single  power  of  the  distance  would  not  have  been  convenient  for  the  law  of 
forces  ;  but  a  function  would  have  had  to  be  taken  ;  &  this,  as  it  was  destined  to  be  expressed 
in  our  human  algebra,  would  bring  in  other  parameters  also.  If,  for  instance,  He  had 
wished  to  make  subject  to  the  same  force,  both  the  practically  elliptic  motion  of  the  planets, 
with  the  Keplerian  connection  between  the  squares  of  the  periodic  times  &  the  cubes  of 
the  mean  distances,  &  also  cohesion  by  contact  ;  then  no  single  power  would  have  been 
sufficient  for  the  establishment  of  both  aims  ;  this  aim  would  have  been  met  by  the  formula 
a/x3  +  b/xz.  But  this  formula  even  would  not  have  been  sufficient,  if  my  Theory  is 
true  ;  for  it  has  not  the  force  at  very  small  distances  in  the  opposite  direction  to  the  force 
at  very  great  distances  ;  but  the  same  kind  of  force  at  all  distances,  that  is,  an  attractive 
force  at  very  small  distances,  just  as  at  very  great  distances.  Now,  the  cohesion  of  points 
that  repel  one  another  at  very  small  distances,  &  attract  one  another  at  very  large  distances, 
cannot  be  obtained  without  intersection  of  the  curve  &  the  axis  ;  &  this  intersection  could 
not  be  obtained  without  the  introduction  of  some  parameter.  Indeed,  to  obtain  the 
whole  series  of  phenomena,  there  was  need,  as  has  been  shown  in  the  proper  place  for  each, 
of  far  more  intersections  of  the  curve,  &  for  flexures  of  such  different  sorts  ;  &  these  could 
not  be  obtained  without  introducing  a  large  number  of  parameters.  Just  consider  for  a 
moment  this  most  intricate  problem,  akin  to  another  of  which  mention  was  made  in 
Art.  547  :  —  Required  to  find  the  number  of  points,  &  the  law  of  mutual  forces  common 
to  all  of  them,  which  would  be  necessary  to  obtain,  by  the  aid  of  a  given  initial  combination, 
the  whole  of  this  series  of  phenomena,  of  such  duration  &  variety,  of  which  we  men  behold 
but  the  very  smallest  of  small  portions.  Immediately  it  will  be  evident  that  it  is  bound 
to  be  of  the  most  intricate  character,  &,  having  regard  to  our  methods  of  expressing  things, 
that  the  kind  of  curve  necessary  for  the  solution  of  such  a  problem  must  be  very  complicated. 
This  problem,  however,  would  involve  certain  known  parameters  in  each  of  its  solutions 
at  least,  &  the  number  of  these  might  perchance  be  infinite  ;  &  a  single  power  by  itself 
would  be  ill-suited  for  the  solution  of  so  great  a  problem. 

75.  Hence,  the  Author  of  Nature,  who  decided  on  this  series  of  phenomena  in  particular, 
must  have  selected  certain  parameters,  &  indeed  a  considerable  number  of  them;    nor 
could  He  have  used  a  single  power  of  the  distance  by  itself  for  expressing  the  law  of  forces, 
In  this  connection  also,  we  must  recall  to  mind,  for  the  confirmation  of  this  matter,  what, 
from  Art.  124  onwards,  has  been  said  with  regard  to  the  inverse  ratio  of  the  squares  of  the 
distances.     We  saw  that  this  ratio  was  not  the  most  perfect  of  all,  nor  one  to  be  chosen 
in  all  circumstances.     Also,  we  must  look  at  that  which  is  shown,  in  the  next  section  of 
these  supplements,  in  opposition    to  forces  that  are  attractive  at    very  small  distances, 
increasing  indefinitely,  to  which  a  single  power  reduces  in  the  end. 

76.  Finally,  in  this  way,  it  seems  to  me  that  the  whole  of  the  difficulty  that  was  put 
forward  has  been  quite  done  away  with  ;  there  is  no  reason  why  any  single  power  of  the 
distance  should    be  preferred   to  a  function,  no  matter  how  complicated  it  may  be,  if 
regard  is  paid  to  our  methods  of  expressing  it. 


The  evolution 

the    necessity  '  for 
tnis     connection 

e^pr'essitfie  "by 
human       algebra, 


problems  of  crea- 

t»tk>&  of  bodies8  & 
series  of  motions. 


it  could    not    be 

™^  ^tneSaw^f 
the  squares  of  the 


Conclusion  against 

convenSnof  °oftha 
single  power. 


[289]  §  IV 

Contra  vires  in  minimis  distantiis  attractivas^  &  excrescentes  in 

infinitum 


,  qduodCUubi  77'  At  praeterea  contra  solam  attractionem  plures  habentur  difficultates,   quae  per 

conatus      deberet  gradus  crescunt.     Nam  inprimis  si  eae  imminutis  utcunque  distantiis  agant,  augent  veloci- 
apSmiisunaXindebeat  tatem  usque  ad  contactum,  ad  quern  ubi  deventum  est,  incrementum  velocitatis  ibi  per 
esse    nuiius,     vei  saltum  abrumpitur,  &  ubi  maxima  est,  ibi  perpetuo  incassum  nituntur  partes  ad  ulteriorem 
effectum  habendum,  &  necessario  irritos  conatus  edunt. 

Secunda,   si   ratio  yg.  Quod  si  in  infinitum  imminuta  distantia,  crescant  in  aliqua  ratione  distantiarum 

tantiae^aviabsoiute  reciproca  ;     multae   itidem    difficultates    habentur,    quae   nostrum    oppositam   sententiam 
infinita,  ad  quam  confirmant.     Inprimis  in  ea  hypothesi  virium  deveniri  potest  ad  contactum.  in  quo  vis, 

deveniri   deberet.  ,  ,  •    j-  •        j    i_  •••£.•*.  •  •         v  i-  • 

sublata  omni  distantia,  debet  augen  m  innmtum  magis,  quam  esset  in  aliqua  distantia. 
Porro  nos  putamus  accurate  demonstrari,  nullas  quantitates  existere  posse,  quae  in  se 
infinitae  sint,  aut  infinite  parvae.  Hinc  autem  statim  habemus  absurdum,  quod  nimirum 
si  vires  in  aliqua  distantia  aliquid  sunt,  in  contactu  debeant  esse  absolute  infinitae. 

Tertia    ex    eo,  79.  Augetur  difficultas,  si  debeat  ratio  reciproca  esse  major,  quam  simplex  (ut  ad  gravi- 

™ai°r  tatem  requiritur  reciproca  duplicata,  ad   cohaesionem  adhuc  major)    &  ad  bina    puncta 

quam  simplex,    ue-  _  j.  ±  .1.  .  i         i       •  i        i  •     r  •     • 

beat   in    contactu  pertmeat.     Nam  ilk  puncta  in  ipso  congressu  devement  ad  velocitatem  absolute  innmtam. 

veioSutem^infini^  Velocitas  autem  absolute  infinita  est  impossibilis,  cum  ea  requirat  spatium  finitum  percursum 

tam.  momento  temporis,  adeoque  replicationem,  sive  extensionem  simultaneam  per  spatium 

finitum  divisibile,  &  quovis  finite  tempore  requirat  spatium  infinitum,  quod  cum  inter 
bina  puncta  interjacere  non  possit,  requireret  ex  natura  sua,  ut  punctum  ejusmodi 
velocitatem  adeptum  nusquam  esset. 

Alia  absurda:  si  80.  Accedunt  plurima  absurda,  ad  quse  ejusmodi  leges  nos  deducunt.  Tendat  punctum 

'  aliquod  in  fig.  ']^  in  centrum  F  in  ratione  reciproca  duplicata  distantiarum,  &  ex  A  pro- 


regressus 

saitus  ab  acceiera.  jiciatur   directione   AB   perpendicular!   ad    AF,    cum   velocitate  satis   exigua  :    describet 

nu\"amC7nCingressu  Ellipsim  ACDE,  cujus  focus  erit  F,&  semper  regredietur  ad  A.  Decrescat  velocitas  AB  per 
insuperficiemsphae.  gradus,  donee  demum  evanescat.  Semper  magis  arctatur  Ellipsis,  &  vertex  D  accedit  ad 
focum  F,  in  quern  demum  recidit  abeunte  Ellipsi  in  rectam  AF.  Videtur  igitur  id  [290] 
punctum  sibi  relictum  debere  descendere  ad  F,  turn  post  acquisitam  ibi  infmitam  veloci- 
tatem, earn  sine  ulla  contraria  vi  convertere  in  oppositam,  &  retro  regredi.  At  si  id  punctum 
tendat  in  omnia  puncta  superficiei  sphericae,  vel  globi  EGCH  in  eadem  ilia  ratione  ; 
demonstratum  est  a  Newtono,  debere  per  AG  descendere  motu  accelerate  eodem  modo, 
quo  acceleraretur,  si  omnia  ejusmodi  puncta  superficiei,  vel  sphaerae  compenetrarentur  in 
F  :  abrupta  vero  lege  accelerationis  in  G,  debere  per  GH  ferri  motu  sequabili,  viribus 
omnibus  per  contrarias  actiones  elisis,  turn  per  HI  tantundem  procurrere  motu  retardato, 
adeoque  perpetuam  oscillationem  peragere,  velocitatis  mutatione  bis  in  singulis  oscillation- 
ibus  per  saltum  interrupta. 


snu        ^ocursus  ^Ip  In  eo  jam  absurdum  quoddam  videtur  esse  :   sed  id  quidem  multo  magis  crescit ; 

ultra'  ad   eandcm  si  consideretur,  quid  debeat  accidere,  ubi  tota  sphaerica  superficies,  vel  tota  sphaera  abeat 

distantiam,     v  e  i  •     unicum  punctum   F.     Turn  itidem  corpus  sibi  relictum,  deveniet  ad  centrum  cum 

saltus      in      tanto  .          .  J   *.  .        .  i    T      i  •  i  •   -r«iv      •  i. 

procursu,     sine  infinita  velocitate,  sed  procurret  ulterms  usque  ad  1,  dum  pnus,  ubi  Ellipsis  evanescebat, 

praeviis  minoribus.  Jebebat  redire  retro.      Nos   quidem  pluribus  in  locis    alibi    demonstravimus,  in  prima 


(e)  Hac  excer-pta  sunt  ex  eadem  dissertatione  De  Lege  Virium  in  Natura  existentium  a  num.  59. 

424 


SUPPLEMENT  IV 


425 


FIG.  72. 


426 


PHILOSOPHIC   NATURALIS  THEORIA 


§IV 

Arguments  against  forces  that  are  attractive  at  very  small 
distances  and  increase  indefinitely  ("} 

77.  Besides,  there  are  many  difficulties  in  the  way  of  attraction  alone,  which  increase  The  first  difficulty 
by  degrees.     For,  first  of  all,  if  these  act  at  diminished  distances  of  any  sort,  they  will  SiatT  ^hen*5  *the 
increase  the  velocity  right  up  to  the  moment  of  contact  :    &  when  contact  is  attained,  effort    should     be 
the  increment  of  the  velocity  will  then  be  suddenly  broken  off ;   &  when  this  is  greatest,  ^oach^it  Abound 
the  parts  will  continually  strive  in  vain  to  produce  a  further  effect,  &  the  efforts  will  to  be  either  nothing 
necessarily  turn  out  to  be  fruitless. 

78.  But  if,  when  the  distances  are  infinitely  diminished,  the  forces  increase  according  The   second   diffi. 
to  some  ratio  that  is  inversely  as  the  distances,  many  difficulties  will  again  be  had,  which  culty   arises  fro™ 

•    .  /~VT_  i_      •        r   j  -11  the    fact    that,     if 

confirm  our  opposite  opinion.  Un  that  hypothesis  ot  iorces  especially,  contact  may  be  the  ratio  is  in- 
attained,  in  which,  as  all  distance  is  taken  away,  the  force  is  bound  to  be  increased  ^erseiy  as  the 
infinitely  more  than  it  would  be  at  a  distance  of  some  amount.  Further,  I  think  that  it  come  to  Wa  force 
is  rigorously  proved  that  no  quantities  can  possibly  exist,  such  as  are  infinite  in  themselves  !hat. is  absolutely 
or  infinitely  small.  Hence,  we  immediately  have  an  absurdity ;  namely,  that  if  the  forces 
at  any  distance  are  anything,  on  contact  they  must  be  absolutely  infinite. 

79.  The  difficulty  is  increased,  if  the  inverse  ratio  is  greater  than  a  simple  ratio  (as  A  third  difficulty 
for  gravity  we  require  the  inverse  square,  &  for  cohesion  one  that  is  still  greater)  ;    &  it  J °he  Averse  ratfo 
has  to  do  with  a  pair  of  points.     For  these  points  on  collision  will  attain  a  velocity  that  is  »?  greater   than  a 
absolutely  infinite.     But  such  an  absolutely  infinite  velocity  is  impossible,  since  it  requires  bound  a°so 'tolmve 
that  a  finite  space  should  be  passed  over  in  an  instant  of  time,  that  is,  replication,  or  simul-  on  contact      an 
taneous  extension  through  finite  divisible  space  ;   &  for  any  finite  time  it  would  require  "        '  velocity- 
infinite  space,  which,  since  there  cannot  be  such  between  the  two  points,  would  require  of 

its  own  nature  that  there  should  not  be  anywhere  a  point  that  has  attained  such  a  velocity. 

80.  There  are  many  more  absurdities,  to  which  such  laws  of  forces  lead  us.     In  Fig.  other  absurdities ; 
72,  let  any  point  tend  towards  a  centre  F  in  the  inverse  ratio  of  the  squares  of  the  distances,  square  "of^th^  dis6 
&  suppose  it  to  be  projected  from  the  point  A  in  a  direction,  AB,  perpendicular  to  AF,  tance,  there  will  be 
with  a  fairly  small  velocity.     Then  it  will  describe  the  ellipse  ACDE,  of  which  F  is  the  cenutrre;  ^"suddtn 
focus ;   &  it  will  always  return  to  A.     Now  let  the  velocity  AB  decrease  by  degrees,  until  change     from    an 
finally  it  vanishes.     Then  the  ellipse  will  continually  become  more  &  more  pointed,  &  toawSnjT'to^oM 
the  vertex  D  will  approach  the  focus  F,  &  will  coincide  with  it  when  the  ellipse  becomes  tnat  is  nothing  on 
the  straight  line  AF.     It  seems  therefore  that  the  point,  if  left  to  itself  would  fall  towards  surfac"g  *  Sph< 
the  focus  F,  then,  after  acquiring  an  infinite  velocity  as  it  reaches  F,  it  would  convert  it 

into  an  equal  velocity  in  the  opposite  direction  without  the  assistance  of  any  opposing  force, 
&  return  to  its  original  position.  But  if  that  point  tended  towards  all  the  points  of  a 
spherical  surface,  or  the  sphere  EGCH,  in  that  same  ratio,  it  was  proved  by  Newton  that 
it  would  have  to  descend  along  AG  with  a  motion  accelerated  in  the  same  manner  as  it 
would  be  if  all  such  points  of  the  surface,  or  the  sphere,  were  condensed  at  F.  Now  the 
law  of  acceleration  being  broken  at  G,  it  will  have  to  go  on  along  GH  with  uniform  velocity, 
all  forces  being  counterbalanced  by  contrary  reactions ;  then  it  will  have  to  travel  along 
HI  for  the  same  interval  with  retarded  motion.  Thus,  there  would  be  a  continual 
oscillation,  with  the  change  of  velocity  suddenly  interrupted  twice  in  each  oscillation. 

81.  Here  there  is  already  seen  to  be  considerable  absurdity ;   but  there  is  still  greater  Simultaneous 
to  follow.     For,  let  us  consider  what  will  necessarily  happen  when  the  whole  of  the  spherical  ^n"™   &onmotion 
surface,  or  the  whole  of  the  sphere,  becomes  but  a  single  point  at  F.     Then  indeed,  the  beyond  it   to    an 
body  if  left  to  itself  would  arrive  at  the  centre  with  infinite  velocity ;    but  it  would  pass  a^udden^nge  °n 
through  it  &  beyond  as  far  as  I,  whereas  in  the  former  case  when  the  ellipse  vanished,  it  this  great  motion, 
had  to  return  to  its  original  position.     Indeed,  in  many  places  elsewhere,  I  have  proved  " 
. -   tions. 

(e)  These  paragraphs  are  quoted  from  the  same  dissertation  De  Lege  Virium  in  Natura  existentium,  starting  with 
Art.  59. 

427 


428  PHILOSOPHIC  NATURALIS  THEORIA 

determinatione  latere  errorem,  cum  Ellipsi  evanescente,  nullae  jam  adsint  omnes  vires, 
quae  agunt  per  arcum  situm  ultra  F  ad  partes  D,  quae  priorem  velocitatem  debebant  extin- 
guere,  &  novam  producere  ipsi  aequalem.  Verum  adhuc  habetur  saltus  quidam,  cui  & 
Natura,  &  Geometria  ubique  repugnat.  Nam  donee  utcunque  parva  est  velocitas,  habetur 
semper  regressus  ad  A  cum  procursu  FD  eo  minore,  quo  velocitas  est  minor  :  facta  autem 
velocitate  nulla,  procursus  immediate  evadit  FI,  quin  ulli  intermedii  minores  adfuerint. 
Quod  si  quis  ejus  priorem  determinationem  tueri  velit,  ut  punctum,  quod  agatur  in  centrum 
vi,  quae  sit  in  ratione  reciproca  duplicata  distantiarum,  debeat  e  centre  regredi  retro  ; 
turn  saltus  habetur  similis,  ubi  prius  in  sphaericam  superfkiem  vel  sphaeram  tendat,  quae 
paullatim  abeat  in  centrum.  Donee  enim  aderit  superficies  ilia,  vel  sphaera,  habebitur 
semper  is  procursus,  qui  abrumpetur  in  illo  appulsu  totius  superficiei  ad  centrum,  quin 
habeantur  prius  minores  procursus. 


-  quidem  in  ratione  reciproca  duplicata   distantiarum  :   in  reciproca  triplicata 

latio  puncti  in  habentur  etiam  graviora.  Nam  si  cum  debita  quadam  velocitate  projiciatur  per  rectam 
trum'SU  ad  °en"  •^••^  fy?"  73  c°ntinentem  angulum  acutum  cum  AP,  mobile,  quod  urgeatur  in  P  vi  crescente 
in  ratione  reciproca  triplicata  distantiarum  ;  demonstratur  in  Mechanica,  ipsum  debere 
percurrere  curvam  ACDEFGH,  quae  vocatur  spiralis  logarithmica,  quae  hanc  habet  pro- 
prietatem,  ut  quaevis  recta,  ut  PF,  ducta  ad  quodvis  ejus  punctum,  contineat  cum  recta 
ipsam  ibidem  tangente  angulum  aequalem  angulo  PAB,  unde  illud  consequitur,  ut  ea  quidem 
ex  una  parte  infmitis  spiris  cir-[29i]-cumvolvatur  circa  punctum  P,  nee  tamen  in  ipsum 
unquam  desinat  :  si  autem  ducatur  ex  P  recta  perpendicularis  ad  AP,  quae  tangenti  AB 
occurrat  in  B,  tota  spiralis  ACDEFGH  in  infinitum  continuata  ad  mensuram  longitudinis 
AB  accedat  ultra  quoscunque  limites,  nee  unquam  ei  sequalis  fiat  :  velocitas  autem  in 
ejusmodi  curva  in  continuo  accessu  ad  centrum  virium  P  perpetuo  crescat.  Quare  finito 
tempore,  &  sane  breviore,  quam  sit  illud,  quo  velocitate  initiali  percurreret  AB,  deberet 
id  mobile  devenire  ad  centrum  P,  in  quo  bina  gravissima  absurda  habentur.  Primo 
quidem,  quod  haberetur  tota  ilia  spiralis,  quae  in  centrum  desineret,  contra  id,  quod  ex  ejus 
natura  deducitur,  cum  nimirum  in  centrum  cadere  nequaquam  possit  :  deinde  vero,  quod 
elapso  eo  finito  tempore  mobile  illud  nusquam  esse  deberet.  Nam  ea  curva,  ubi  etiam  in 
infinitum  continuata  intelligitur,  nullum  habet  egressum  e  P.  Et  quidem  formulas  ana- 
lyticae  exhibent  ejus  locum  post  id  tempus  impossibilem,  sive,  ut  dicimus,  imaginarium ; 
quo  quidem  argumento  Eulerus  in  sua  Mechanica  afnrmavit  illud,  debere  id  mobile  in 
appulsu  ad  centrum  virium  annihilari.  Quanto  satius  fuisset  inferre,  earn  legem  virium 
impossibilem  esse  ? 


Pejus  in  potentiis  g-j.  Quanto  autem   maiora  absurda   in  ulterioribus   potentiis,   quibus  vires  alligatae 

altioribus  :    prseca-      •  *o-ii  r-  A  TIT?      «      •  •  VAT  •          • 

ratio  ad  demon-  sint,  consequentur  ?  Sit  globus  in  fig.  74  ABE,  &  intra  ipsum  alms  Abe,  qui  priorem 
strandum  absur-  contingat  in  A,  ac  in  omnia  utriusque  puncta  agant  vires  decrescentes  in  ratione  reciproca 
quadruplicata  distantiarum,  vel  majore,  &  quaeratur  ratio  vis  puncti  constituti  in  concursu 
A  utriusque  superficiei.  Concipiatur  uterque  resolutus  in  pyramides  infinite  arctas,  quae 
prodeant  ex  communi  puncto  A,  ut  BAD,  bAd.  In  singulis  autem  pyramidulis  divisis 
in  partes  totis  proportionales  sint  particulas  MN,  mn  similes,  &  similiter  positae.  Quantitas 
materiae  in  MN,  ad  quantitatem  in  mn  erit,  ut  massa  totius  globi  majoris  ad  totum  minorem, 
nimirum,  ut  cubus  radii  majoris  ad  cubum  minoris.  Cum  igitur  vis,  qua  trahitur  punctum 
A,  sit,  ut  quantitas  materiae  directe,  &  ut  quarta  potestas  distantiarum  reciproce,  quae 
itidem  distantiae  sunt,  ut  radii  sphaerarum  ;  erit  vis  in  partem  MN,  ad  vim  in  partem  mn 
directe,  ut  tertia  potestas  radii  majoris  ad  tertiam  minoris,  &  reciproce,  ut  quarta  potestas 
ipsius.  Quare  manebit  ratio  simplex  reciproca  radiorum. 


Partem  fore  majo-  g.    Minor  erit  igitur  actio  singularum  particulaium  homologarum  MN,  quam  mn, 

rem  tolo.  ~  ,.      *  \  ,  A -nT-> 

in  ipsa  ratione  radiorum,  adeoque  punctum  A  minus  trahetur  a  tota  sphaera  ABE,  quam 
a  sphaera  Abe,  quod  est  absurdum,  cum  attractio  in  earn  sphseram  minorem  debeat  esse  pars 


SUPPLEMENT  IV 


429 


FIG.  73. 


FIG.  74. 


430 


PHILOSOPHIC  NATURALIS  THEORIA 


FIG.  73. 


FIG.  74. 


SUPPLEMENT  IV  431 

that  there  is  an  error  in  the  first  determination  ;  for  when  the  ellipse  vanishes,  there  are 
no  longer  present  any  of  all  these  forces,  which  act  on  the  body  as  it  goes  along  the  arc 
situated  beyond  F  in  the  direction  of  D  ;  &  these  were  necessary  to  extinguish  the  former 
velocity  &  to  generate  a  new  velocity  equal  to  it.  But  still  there  is  a  sudden  change,  to 
which  both  Nature  &  geometry  are  in  all  cases  opposed.  For,  so  long  as  there  is  a  velocity, 
no  matter  how  small,  we  always  have  a  return  to  A  with  a  further  motion  beyond  F,  equal 
to  FD,  which  is  correspondingly  smaller  as  the  velocity  becomes  smaller  ;  &  yet,  when  the 
velocity  is  made  nothing  at  all,  the  further  motion  beyond  F  at  once  becomes  FI,  without 
there  being  present  any  intermediate  smaller  motions.  Now,  if  anyone  would  wish  to 
adhere  to  the  first  determination  of  the  problem,  so  that  a  point,  which  is  attracted  towards 
a  centre  by  a  force  in  the  inverse  ratio  of  the  squares  of  the  distances,  is  bound  to  return 
from  the  centre  to  its  original  position ;  then  there  too  there  is  a  sudden  change  of  a  like 
nature  to  that  which  took  place  in  the  first  case  when  it  tended  towards  a  spherical  surface, 
or  a  sphere,  which  gradually  dwindled  to  a  point  at  the  centre.  For,  as  long  as  the  spherical 
surface,  or  the  sphere,  is  there,  there  will  always  be  obtained  that  further  motion  ;  but 
this  is  suddenly  stopped  on  the  arrival  of  the  whole  of  the  spherical  surface,  or  the  whole 
of  the  sphere,  at  the  centre,  without  any  previous  smaller  motions  being  had. 

82.  Such  indeed  are  the  results  that  we  obtain  for  the  inverse  ratio  of  the  squares  of  if.  the  ratio  is  the 
the  distances ;    for  the  inverse  ratio  of  the  cubes,  we  have  even  more  serious  difficulties.  wo^r-feT  annihUa- 
For,  if  a  body  is  projected  along  AB,  in  Fig.  73,  making  an  acute  angle  with  AP,  with  a  tion  of  the  point  on 
certain  suitable  velocity,  &  it  is  attracted  towards  P  with  a  force  increasing  in  the  inverse  a* 
ratio  of  the  cubes  of  the  distances ;   in  that  case,  it  is  proved  in  Mechanics  that  the  motion 

will  be  along  a  curve  such  as  ACDEFGH,  which  is  called  the  logarithmic  spiral.  This 
curve  has  the  property  that  any  straight  line,  PF,  drawn  from  P  to  any  point  F  of  the  curve, 
contains  with  the  tangent  to  the  curve  at  the  point  an  angle  equal  to  the  angle  PAB.  Hence 
it  follows  that,  on  the  one  hand  indeed  it  will  rotate  through  an  infinite  number  of  con- 
volutions round  the  point  P,  but  will  never  reach  that  point ;  yet,  on  the  other  hand,  if 
a  straight  line  is  drawn  through  P  perpendicular  to  AP,  to  meet  the  tangent  AB  in  B,  then 
the  whole  length  of  the  spiral  ACDEFGH  continued  indefinitely  will  approximate  to  the 
length  of  AB  beyond  all  limits,  &  yet  never  be  equal  to  it.  Further  the  velocity  in  such 
a  curve,  as  it  continually  approaches  the  centre  of  forces  P,  continually  increases.  Hence 
in  a  finite  time,  &  that  too  one  that  is  shorter  than  that  in  which  it  would  pass  over  the 
distance  AB  with  the  given  initial  velocity,  the  moving  body  would  be  bound  to  arrive 
at  the  centre  P  ;  &  in  this  we  have  two  very  serious  absurdities.  The  first  is  that  the 
whole  of  the  spiral,  which  terminates  in  the  centre,  is  obtained,  in  opposition  to  the  principle 
deduced  from  its  nature,  since  truly  it  can  never  get  to  the  centre  ;  &  secondly,  that 
after  that  finite  time  has  elapsed  the  moving  body  would  have  to  be  nowhere  at  all.  For, 
the  curve,  even  when  it  is  understood  that  it  is  continued  to  infinity,  has  no  exit  through  & 
past  the  point  P.  Indeed  the  analytical  formulae  represent  its  position  after  the  lapse  of 
this  time  as  impossible,  or,  as  it  is  usually  called  imaginary.  By  this  very  argument,  Euler, 
in  his  Mechanics,  asserts  that  the  moving  body  on  approaching  the  centre  of  forces  is 
annihilated.  How  much  more  reasonable  would  it  be  to  infer  that  this  law  of  forces  is 
an  impossible  one  ? 

83.  How  much  greater  absurdities  are  those  that  follow  for  higher  powers,  with  which  still  worse  for 
the  forces  may  be  connected  !     In  Fig.  74,  let  ABE  be  a  sphere,  &  within  it  let  there  be  higher     P°wers; 

It  i  •  i       r  °       T  i    r  11         •  r     i  preparation  for 

another  one  Abe,  touching  the  former  at  A ;   &  suppose  that  on  all  points  of  each  of  them  demonstrating    an 

there  act  forces  which  decrease  in  the  inverse  ratio  of  the  fourth  powers  of  the  distances,  absurdlty- 

or  even  greater  ;  &  suppose  that  we  require  the  ratio  of  the  forces  due  to  a  point  situated  at 

the  point  of  contact  A  of  the  two  surfaces.     Imagine  each  of  the  spheres  to  be  divided  into 

infinitely  thin  pyramids,  proceeding  from  the  common  vertex  A,  such  as  BAD,  bAd.     In  each 

of  these  little  pyramids,  which  are  then  divided  into  parts  proportional  to  the  wholes,  let 

MN  &  mn  be  particles  that  are  similar  &  similarly  situated.     The  quantity  of  matter  in  MN 

will  be  to  the  quantity  of  matter  in  mn  as  the  mass  of  the  larger  sphere  to  the  mass  of  the 

whole  of  the  smaller  ;  i.e.,  as  the  cube  of  the  radius  of  the  larger  to  the  cube  of  the  radius 

of  the  smaller.     Hence,  since  the  force  exerted  upon  A  varies  as  the  quantity  of  matter 

directly,  &  as  the  fourth  power  of  the  distance  inversely,  &  these  distances  also  vary  as 

the  radii  of  the  spheres.     Therefore,  the  force  on  the  part  MN  is  to  the  force  on  the  part 

mn  directly  as  the  third  power  of  the  radius  of  the  larger  sphere  to  the  third  power  of  the 

radius  of  the  smaller,  &  inversely  as  the  fourth  powers  of  the  same.     That  is,  there  results 

the  simple  inverse  ratio  of  the  radii. 

84.  Hence  the  action  of  each  of  the  homologous  particles  MN  will  be  less  than  each  T*16   Part    greater 
of  the  corresponding  particles  mn,  in  the  ratio  of  the  radii ;    &  thus  the  point  A  will  be 

attracted  less  by  the  whole  sphere  ABE  than  by  the  sphere  Abe.     This  is  absurd  ;   for,  the 
attraction  on  the  smaller  sphere  must  be  a  part  of  the  attraction  on  the  greater  sphere 


432  PHILOSOPHIC  NATURALIS  THEORIA 

attractionis  in  sphaeram  majorem,  quae  continet  minorem,  cum  magna  materias  parte  sita 
extra  ipsam  usque  ad  superficiem  sphaerae  majoris,  unde  concluditur  esse  partem  majorem 
toto,  maximum  nimirum  absurdum.  Et  qui-[292]-dem  in  altioribus  potentiis  multo  major 
est  is  error  ;  nam  generaliter,  si  vis  sit  reciproce,  ut  Rm,  posito  R  pro  radio,  &  m  pro  quovis 
numero  ternarium  superante,  erit  attractio  sphserae  eodem  argumento  reciproce,  ut  Rm~3, 
quae  eo  majorem  indicat  vim  in  sphaeram  minorem  respectu  majoris  ipsam  continentis, 
quo  numerus  m  est  major. 

Omnia  absurda  gij.  Hoc  quidem  pacto  inveniuntur  plurima  absurda  in  variis  generibus  attractionum 

minimis  distantiis  quae  si  repulsiones,  in  minimis  distantiis  habeantur  pares  extinguendae  velocitati  cuilibet 
habeatur  repuisio,  utcunque  magnae,  cessant  illico  omnia,  cum  eae  repulsiones  mutuum  accessum  usque  ad 
ilnpediat. appUlSUm  concursum  penitus  impediant.  Inde  autem  manifesto  iterum  consequitur,  repulsiones 

in  minimis  distantiis  praeferendas  potius  esse  attraction!,  ex  quarum  variis  generibus  tarn 

multa  absurda  consequuntur. 


SUPPLEMENT  IV  433 

which  contains  the  smaller  one,  together  with  a  great  part  of  the  matter  situated  beyond 
it  as  far  as  the  surface  of  the  greater  sphere  ;  hence  the  conclusion  is  that  the  part  is  greater 
than  the  whole,  which  is  altogether  impossible.  Indeed,  in  still  higher  powers  the  error 
is  much  greater  ;  for,  in  general,  if  the  force  varies  inversely  as  R"*,  where  R  is  taken  as 
the  radius,  &  m  for  some  number  greater  than  three,  then  the  attraction  of  the  sphere 
will  be  inversely  as  Rw~3 ;  &  this  points  to  a  force  that  is  the  greater  on  a  smaller  sphere 
compared  with  that  on  a  larger  sphere  containing  it,  in  proportion  as  the  number  m  is 
greater. 

85.  Thus  we  find  very  many  absurdities  in  various  kinds  of  attractions ;  if  there  are  All  these  absurd- 
repulsions  at  very  small  distances,  sufficiently  great  to  destroy  any  velocity  however  large,  there^tsT  repulsion 
all  these  absurdities  would  cease  to  be  immediately,  for  these  repulsions  would  prevent  at  very  small 
mutual  approach  up  to  the  point  of  actual  contact.  Hence  it  once  again  manifestly  follows  p^e^ent* 
that  repulsions  at  very  small  distances  are  to  be  preferred  before  an  attraction ;  for  from  approach, 
the  various  kinds  of  the  latter  so  many  absurdities  follow. 


near 


FF 


[293]  §   V 

De  ALquilibrio  binarum  massarum  connexarum  invicem  per  bina 

alia  puncta   (/) 

de  squih-  ^6.  Continetur  autem,  quod  pertinet  ad  momentum  in  vecte,  &  ad  aequilibrium, 

brio  punctorum  sequentis  problematis  solutione.  Sit  in  fig.  75  quivis  numerus  punctorum  materias  in  A, 
Mnta°r'extreIma  I1"  dicatur  A,  in  D  quivis  alius,  qui  dicatur  D,  &  puncta  ea  omnia  secundum  directiones 
habeant  quas-  AZ,  DX  parallelas  rectse  datae  CF  sollicitentur  simul  viribus,  quae  sint  aequales  inter  omnia 
v^ibusexternisTibi  puncta  sita  in  A,  itidem  inter  omnia  sita  in  D,  licet  vires  in  A  sint  utcunque  diversae  a 
proportionaiitms,  &  viribus  in  D.  Sint  autem  in  C,  &  B  bina  puncta,  quae  in  se  invicem,  &  in  ilia  puncta  sita 
medus  vim  in  A,  &  D  mutuo  agant,  ac  ejusmodi  mutuis  actionibus  impediri  debeat  omnis  actio  virium 
illarum  in  A,  &  D,  &  omnis  motus  puncti  B  :  motus  autem  puncti  C  impediri  debeat 
actione  contraria  fulcri  cujusdam,  in  quod  ipsum  agat  secundum  directionem  compositam 
ex  actionibus  omnium  virium,  quas  habet  :  quaeritur  ratio,  quam  habere  debent  summae 
virium  A,  &  D  ad  hoc,  ut  habeatur  id  aequilibrium,  &  quantitas,  ac  quaeritur  directio  vis, 
qua  fulcrum  urgeri  debet  a  puncto  C. 


tremis*  in^aiterum 
e  mediis. 


^7'  Exprimant  AZ,  &  DX  vires  illas  parallelas  singulorum  punctorum  positorum 
in  A,  &  D.  Ut  ipsae  elidantur,  debebunt  in  iis  haberi  vires  AG,  DK  contrariae,  &  aequales 
ipsis  AZ,  DK.  Quoniam  eae  debent  oriri  a  solis  actionibus  punctorum  C,  &  B  agentium 
in  A  secundum  rectas  AC,  AB,  &  in  D  secundum  rectas  DC,  DB,  ductis  ex  G  rectis  GI, 
GH  parallelis  BA,  AC  usque  ad  rectas  AC,  BA,  &  ex  K  rectis  KM,  KL  parallelis  BD,  DC, 
usque  ad  rectas  DC,  BD  ;  patet,  in  A  vim  AG  debere  componi  ex  viribus  AI,  AH,  quarum 
prima  quodvis  punctum  in  A  repellat  a  C,  secunda  attrahat  ad  B,  &  in  D  vim  DK  componi 
itidem  ex  viribus  DM,  DL,  quarum  prima  quodvis  punctum  situm  in  D  repellat  a  C, 
secunda  attrahat  ad  B.  Hinc  ob  actionem  reactioni  aequalem  debebit  punctum  C  repelli 
a  quo  vis-  puncto  sito  in  A  secundum  directionem  AC  vi  aequali  IA,  &  a  quovis  puncto  sito 
in  D  secundum  directio-[294]-nem  DC  vi  aequali  MD  :  punctum  vero  B  debebit  attrahi 
a  quovis  puncto  sito  in  A  secundum  directionem  BA  vi  aequali  HA,'  &  a  quovis  puncto  sito 
in  D  vi  aequali  LD.  Habebit  igitur  punctum  C  ex  actione  punctorum  in  A,  &  D  binas 
vires,  quarum  altera  aget  secundum  directionem  AC,  &  erit  aequalis  IA  ductae  in  A,  altera 
aget  secundum  directionem  DC,  &  erit  aequalis  MD  ductas  in  D.  Punctum  vero  B  itidem 
binas,  quarum  altera  aget  secundum  directionem  BA,  &  erit  aequalis  HA  ductae  in  A,  altera 
aget  secundum  directionem  BD,  &  erit  sequalis  LD  ductae  in  D. 


vis,    quam    debet  g8.  Porro  vis  composita  ex  illis  binis,  quibus  urgetur  punctum  B,  elidi  debet  ab  actione 

mum"6  compositar  e  mutua  inter  ipsum,  &  C  ;  quare  debebit  habere  directionem  rectae  BC  in  casu,  quern  exhibet 

quatiior  :     enum-  figura,  in  quo  C  jacet  in  angulo  ABD  :   nam  si  angulus  ABD  hiatum  obverteret  ad  partes 

pertinentJm""^  oppositas,  ut  C  jaceret  extra  angulum  ;    ea  haberet    directionem  CB,  &_  reliqua  omnis 

omnia  puncta.          demonstratio  rediret  eodem.     Punctum  autem  C  ob   actionem,   &  reactionem   aequales 

debebit  habere  vim  aequalem,  &  contrarium  illi,  quam  exercet  B,  adeoque  vim  aequalem, 

&  ejusdem  directionis  cum  vi,  quam  e  prioribus  illis  binis  compositam  habet  punctum  B  : 

nempe  debebit  habere  binas  vires  aequales,  &  directionis  ejusdem  cum  viribus  illam  com- 

ponentibus,  nimirum  vim  secundum  directionem  parallelam  BA  aequalem  ipsi  HA  ductae 

in  A,  &  vim  secundum  directionem  parallelam  BD  aequalem  ipsi  LD  ductae  in  D.     Habebit 


(f)  Excerfta  btsc  sunt  ex  Synopsi  Physics  Generalis  P.  Caroli  Benvenuti  Soc.  Jesu,  num.  146,  cui  hanc  solutionem 
ibi  imprimendam  tradideram, 

434 


SUPPLEMENT  V 


435 


Kir,.    75. 


436 


PHILOSOPHIC    NATURALIS    THEORIA 


M 


FIG.  75- 


§v 

Equilibrium   of  two   masses  connected  together  by  two  other 

points  (/) 

86.  All  that    pertains  to  moment  in  the  lever,   &  to  equilibrium  is  contained  in  ^"enf""^0'  the 
the  solution  of  the  following  problem.     In  Fig.  75,  let  there  be  any  number  of  points  of  equilibrium     o  f 
matter  at  the  point  A,  &  let  the  number  be  called  A  ;    similarly,  any  other  number  at  D,  ^™ch  P^f  :t^ 
called  D  ;    &  suppose  that  all  these  are  at  the  same  time  under  the  action  of  forces  along  outside  points  have 
the  directions  AZ,  DX  parallel  to  the  given  straight  line  CF,  &  that  these  forces  are  equal  g"^^^63  fj^ 
to  one  another  for  all  the  points  situated  at  A,  &  also  for  all  the  points  situated  at  D,  proportional    to 
although  the  forces  at  A  may  be  altogether  different  from  those  at  D.     Also,  at  C  &  B,  let  ^™nne*  £™is  °s 
there  be  two  points,  which  act  mutually  upon  one  another  &  upon  the  points  situated  at  subject  to  a  force 
A  &  D.     Suppose  that  by  such  actions  the  whole  of  the  action  of  the  forces  on  A  &  D  has  from  a  fulcrum- 
to  be  prevented,  as  well  as  any  motion  of  the  point  B.     Also  suppose  that  the  motion  of 

the  point  C  is  to  be  prevented  by  the  contrary  action  of  a  fulcrum,  upon  which  the  point 
C  acts  according  to  the  direction  compounded  from  all  the  forces  that  act  upon  it.  It 
is  required  to  find- the  ratio  which  there  must  be  between  the  forces  on  A  &  D,  for  the 
purpose  of  obtaining  equilibrium  ;  also  to  find  the  quantity  &  direction  of  the  force  to 
which  the  fulcrum  must  be  subjected  by  the  point  C. 

87.  Let  AZ  &  DX  represent  the  parallel  forces  of  each  of  the  points  situated  at  A  &  D  T116  for<*  from  the 

•-n  1*1  •  i  •  r  »  /-i    „     TN  TT-     two     extremes     on 

respectively,  lo  cancel  these,  we  must  have  acting  at  these  points  forces  ACj  &  DK,  either  of  the  means, 
which  are  equal  and  opposite  to  AZ,  DX.  Now,  these  must  arise  purely  from  the  actions 
of  the  points  C  &  B,  acting  on  A  along  the  straight  lines  AC,  AB,  &  on  D  along  the  straight 
lines  DC  &  DB.  Hence,  if  we  draw  through  G  straight  lines  GI,  GH,  parallel  to  BA,  AC, 
to  meet  the  straight  lines  AC,  BA ;  &  through  K,  straight  lines  KM,  KL,  parallel  to  BD, 
DC,  to  meet  DC,  BD  ;  then  it  is  plain  that  the  force  AG  on  A  must  be  compounded  of 
the  forces  AI,  AH,  of  which  the  first  will  repel  any  one  of  the  points  at  A  away  from  C, 
&  the  second  will  attract  it  towards  B  ;  &  similarly,  the  force  DK  on  D  must  be  compounded 
of  the  two  forces  DM,  DL,  of  which  the  first  repels  any  one  of  the  points  situated  at  D 
away  from  C,  &  the  second  attracts  it  towards  B.  Hence,  on  account  of  the  equality  of 
action  &  reaction,  the  point  C  must  be  repelled  by  every  point  situated  at  A  in  the  direction 
AC  by  a  force  equal  to  IA,  &  by  every  point  situated  at  D  in  the  direction  DC  with  a  force 
equal  to  MD.  Also  the  point  B  will  be  attracted  by  every  point  situated  at  A  in  the 
direction  BA  with  a  force  equal  to  HA,  &  by  every  point  at  D  with  a  force  equal  to  LD. 
Therefore,  the  point  C  will  have,  due  to  the  actions  of  the  points  at  A  &  D,  two  forces,  of 
which  one  will  act  in  the  direction  AC,  &  be  equal  to  IA  multiplied  by  A,  &  the  other  will 
act  in  the  direction  DC  &  be  equal  to  MD  multiplied  by  D.  The  point  B  will  also  be  under 
the  action  of  two  forces,  one  of  which  will  act  in  the  direction  BA  &  be  equal  to  HA  mul- 
tiplied by  A,  &  the  other  will  act  in  the  direction  BD  &  be  equal  to  LD  multiplied  by  D. 

88.  Further  the  force  composed  from  the  two  forces,  which  act  upon  the  point  B,  The    force,  which 
must  be  cancelled  by  the  mutual  action  between  it  &  C ;   hence,  this  must  be  in  the  ^ust^na^e11*'  % 
direction  of  the  straight  line  BC,  in  the  case  given  by  the  figure,  where  C  lies  within  the  composed    out  of 
angle  ABD  ;  for,  if  the  angle  ABD  should  turn  its  opening  in  the  other  direction,  so  that  *°i"nrj£  ^forces 
C  should  lie  outside  the  angle,  then  the  force  would  have  the  direction  CB,  &  all  the  rest  pertaining   to    all 
of  the  proof  would  come  to  the  same  thing.     Now,  the  point  C,  on  account  of  the  equality  of  the  P°ints- 
action  &  reaction,  must  have  a  force  that  is  equal  &  opposite  to  that  exerted  by  B  ;  &  thus, 

a  force  that  is  equal  to,  &  in  the  same  direction  as,  the  force  which  B  has,  compounded  of 
those  first  two  forces.  That  is  to  say,  it  must  have  two  forces  that  are  equal  to,  &  in  the 
same  direction  as,  the  two  forces  that  compose  it ;  namely,  a  force  in  a  direction  parallel 
to  BA  &  equal  to  HA  multiplied  by  A,  &  a  force  in  a  direction  parallel  to  BD  &  equal  to 


(f)  These  are  quoted  from  the  Synopsis  Physicae  Generalis  of  FT.  Carolus  Senvtnuto,  S.J.,  Art.  146,  to  which 
author  I  gave  this  solution  to  print  in  that  work. 

437 


438  PHILOSOPHIC  NATURALIS  THEORIA 

igitur  quodvis  punctum  A  binas  vires  AI,  AH,  quodvis  punctum  D  binas  vires  DM,  DL, 
punctum  B  binas  vires,  quarum  altera  dirigetur  ad  A,  &  sequabitur  HA  ductae  in  A,  altera 
dirigetur  ad  D,  &  aequabitur  LD  ductae  in  D,  ex  quibus  componi  debet  vis  agens  secundum 
rectam  BC  :  &  demum  habebit  punctum  C  vires  quatuor,  quarum  prima  dirigetur  ad 
partes  AC,  &  erit  aequalis  IA  ductae  in  A,  secunda  ad  partes  DC,  &  erit  aequalis  MD  ductae 
in  D,  tertia  habebit  directionem  parallelam  BA,  &  erit  aequalis  HA  ductse  in  A ;  quarta 
habebit  directionem  BD,  &  erit  aequalis  LD  ductae  in  D  :  ac  ipsum  punctum  C  urgebit 
fulcrum  vi  composita  ex  illis  quatuor,  quae  omnia,  si  habeatur  ratio  directionis  rectarum 
secundum  ordinem,  quo  enunciantur  per  literas,  hue  reducuntur  : 

Quodvis  punctum  A  habebit  vires  binas     ....     AI,  AH 
Quodvis  punctum  D  vires  binas          .....    DM,  DL 

Punctum  B  binas A  X  HA,  D  X  DL 

Punctum  C  quatuor  .        A  X  IA,  D  X  MD,    A  X  HA,  D  X  LD 

Constructioprapar-  g^.  Exprimat  jam  recta  BC  magnitudinem  vis  compositae  e  binis  CN,  CR  parallelis 

DB,  AB  ;  expriment  BN,  BR  magnitudinem  virium  illarum  componentium,  cum  exprimant 
[295]  earum  directiones,  adeoque  RC,  NC  ipsis  aequales,  &  parallelae  expriment  vires  illas 
tertiam,  &  quartam  puncti  C.  Producantur  autem  DC,  AC  donee  occurrant  in  O,  &  T 
rectis  ex  N,  &  R  parallelis  ipsi  CF,  sive  ipsis  GAZ,  KDX,  &  demittantur  AF,  DE,  NQ,  RS 
perpendicula  in  ipsam  FC  productam,  qua  opus  est,  quae  occurrat  rectis  AB,  DB  in  V,  P. 

vires  sub  nova  9°.  Inprimis  ob  singula  latera  singulis  lateribus  parallela  erunt  similia  triangula  IAG, 

expression  inde  CTR,  &  triangula  MDK,  CON.  Quare  erit  ut  IG,  sive  AH,  ad  CR,  sive  NB,  vel  A  X  AH, 
nimirum  ut  I  ad  A,  ita  AG  ad  TR,  &  ita  AI  ad  TC.  Erit  igitur  TR  aequalis  GA,  sive  AZ 
ductae  in  A,  &  CT  aequalis  IA  ductae  in  A  ;  adeoque  ilia  exprimet  summam  omnium  virium 
AZ  omnium  punctorum  in  A,  haec  vim  illam  primam  puncti  C,  nimirum  A  X  IA.  Eodem 
prorsus  argumento,  cum  sit  MK,  sive  DL  ad  CN,  sive  RB,  vel  D  X  DL,  nimirum  I  ad  D, 
ita  DK  ad  ON,  &  ita  DM  ad  OC  ;  erit  NO  aequalis  KD,  sive  DX  ductae  in  D,  &  OC 
aequalis  MD  ductae  in  D,  adeoque  ilia  exprimet  summam  omnium  virium  DX  omnium 
punctorum  in  D,  haec  vim  illam  secundam  puncti  C,  nimirum  D  X  DM.  Quare  jam 
erunt 

Summa  virium  parallelarum  in  A  .         .         .         .  TR 

Summa  virium  parallelarum  in  D  .          .          .          .  NO 

Binse  vires  in  B BN,  BR 

Quatuor  vires  in  C CT,  OC,  RC,  NC 

vis  in  fulcrum  cui  91.  Jam  vero  patet,  ex  tertia  RC,  &  prima  CT  componi  vim  RT  aequalem  summae 

aequalls'  virium  parallelarum  A  :    &  ex  quarta  NC,  ac  secunda  OC    componi  vim  NO    aequalem 

summae  virium  parallelarum  in  D.  Quare  patet,  ab  unico  puncto  C  fulcrum  urgeri  vi, 
quae  eandem  directionem  habeat,  quam  habent  viies  parallelae  in  A,  &  D,  &  aequetur  earum 
summae,  nimirum  urgeri  eodem  modo,  quo  urgeretur,  si  omnia  ilia  puncta,  quae  sunt  in 
D,  &  A,  cum  his  viribus  essent  in  C,  &  fulcrum  per  se  ipsa  immediate  urgerent. 

Proportio       quae  g2    prseterea  °b  parallelismum  itidem  omnium  laterum  similia  erunt    triangula    if 

CNO,  DPC  :  2°  CNQ,  PDE  :  3?  CPR,  VCN  :  4?  CRS,  VNQ  :  5°  CVA,  TCR  :  6?  VAF, 
CRS.  Ea  exhibent  sequentes  sex  proportiones,  quarum  binae  singulis  versibus  continentur. 

ON  .  CP  :  :  NC  .  PD  :  :  NQ  .  DE 
CP  .  CV  :  :  CR  .  NV  :  :  RS  .  NQ 
CV  .  RT  :  :  VA  .  RC  :  :  AF  .  RS 

Porro  ex  iis  componendo  primas,  &  postremas,  ac  demendo  in  illis  CP,  CV ;  in  his  QN, 
RS  communes  tarn  antecedentibus,  quam  consequentibus,  fit  ex  aequalitate  nimirum  pertur- 
bata  ON  .  RT  :  :  AF.  DE.  Nempe  summa  omnium  virium  parallelarum  in  D,  cui 
aequatur  ON,  ad  summam  om-[296]-nium  in  A,  cui  aequatur  RT,  ut  e  contrario  distantia 
harum  perpendicularis  AF  a  recta  CF  ducta  per  fulcrum  directioni  virium  earumdem 
parallela,  ad  illarum  perpendicularem  distantiam  ab  eadem.  Quare  habetur  determinatio 
eorum  omnium  quae  quaerebantur  (8). 


(g)  Porro  applicatio  ad  vectem  est  similis  illi,  qute  habetur  hie  post  (equilibrium  trium  massarum  num.  326. 


SUPPLEMENT  V 


439 


M 


FIG.  75. 


440 


PHILOSOPHIC   NATURALIS  THEORIA 


FIG.  75. 


SUPPLEMENT  V  441 

LD  multiplied  by  D.  Hence,  any  point  at  A  will  have  two  forces,  AI,  AH  ;  any  point  at 
D  two  forces  DM,  DL  ;  the  point  B  two  forces,  of  which  one  is  directed  towards  A  &  is 
equal  to  HA  multiplied  by  A,  &  the  other  is  directed  towards  D  &  is  equal  to  LD  multi- 
plied by  D  ;  &  lastly,  the  point  C  will  have  four  forces,  of  which  the  first  is  directed  along 
AC  and  is  equal  to  IA  multiplied  by  A,  the  second  along  DC  and  equal  to  MD  multiplied  by 
D,  the  third  has  a  direction  parallel  to  BA  and  is  equal  to  HA  multiplied  by  A,  and  the  fourth 
has  a  direction  parallel  to  BD  and  is  equal  to  LD  multiplied  by  D.  The  point  C  will  exert  on 
the  fulcrum  a  force  compounded  from  all  four  forces ;  and  all  of  these,  if  the  sense  of  the 
direction  of  the  straight  lines  is  considered  to  be  that  given  by  the  order  of  the  letters  by 
which  they  are  named,  will  be  as  follows  : — 

Any  point  at  A  will  have  two  forces          .         .         .  AI,  AH 

Any  point  at  D,  two  forces       .....  DM,  DL 

The  point  B,  two A  X  HA,  D  X  LD 

The  point  C,  four  .         .        A  x  IA,  D  x  MD,  A  X  HA,  D  x  LD 

89.  Now  let  BC  represent  the  magnitude  of  the  force  compounded  from  the  two  Construction  neces 
forces  CN,  CR,  parallel  to  DB,  AB  :    then  BN,  BR  will  represent  the  magnitude  of  the  ^  £° 
component  forces,  since  they  represent  their  directions,  and  thus  RC,  NC,  which  are  equal 

and  parallel  to  them,  will  represent  the  third  and  fourth  forces  on  the  point  C.  Also 
let  DC  &  AC  be  produced,  until  they  meet  in  O  and  T  respectively  the  straight  lines  drawn 
through  N  &  R  parallel  to  CF,  i.e.,  to  GAZ  &  KDX ;  &  let  AF,  DE,  NQ,  RS  be  drawn 
perpendicular  to  CF,  produced  if  necessary ;  &  let  CF  meet  AB,  DB  in  V  &  P. 

90.  First  of  all,  on  account  of  their  corresponding  sides  being  parallel,  the  triangles  IAG,  The    forces     that 
CTR  are  similar,  &  so  also  are  the  triangles  MDK,  CON.     Hence,  as  IG,  or  AH,  is  to  CR,  new*  method  ^of 
or  NB,  i.e.,  A  X  AH,  in  other  words,  as  i  is  to  A,  so  is  AG  to  TR,  or  as  IA  to  TC.     Hence,  representation. 
TR  will  be  equal  to  GA  or  AZ  multiplied  by  A,  &  CT  will  be  equal  to  IA  multiplied  by 

A.  Therefore  the  former  will  represent  the  sum  of  all  the  forces  AZ  on  all  the  points  at  A, 
&  the  latter  that  first  force  on  the  point  C,  i.e.,  A  X  IA.  With  precisely  the  same  argu- 
ment, since  MK,  or  DL,  is  to  CN,  or  RB,  i.e.,  D  X  DL,  in  other  words,  as  I  to  D,  so  is 
DK  to  ON,  or  DM  to  OC  ;  therefore  NO  will  be  equal  to  KD  or  DX  multiplied  by  D, 
&  CO  equal  to  MD  multiplied  by  D  ;  &  therefore  the  former  will  represent  the  sum  of 
all  the  forces  DX  for  all  the  points  at  D,  &  the  latter  that  second  force  on  the  point  C 
namely,  D  X  DM.  Hence,  we  now  have  : — 

The  sum  of  the  parallel  forces  on  A          .          .          .          .  TR 

The  sum  of  the  parallel  forces  on  D          .          .          .          .  NO 

The  two  forces  on  B  ...                   ...  BN,  BR 

The  four  forces  on  C CT,  OC,  RC,  NC 

91.  And  now  it  is  plain  that,  from  the  third  force   RC,  &  the  first,  CT,  we  have  a  The  force  on    the 
resultant  force  RT  which  is  equal  to  the  sum  of  the  parallel  forces  at  A  ;  &  from  the  fourth,  ["lc™"J  :  to  what  it: 
NC,  &  the  second,  OC,  we  get  a  resultant  force  NO,  which  is  equal  to  the  sum  of  all  the 

parallel  forces  at  D.  Therefore,  it  is  evident  that  the  fulcrum  at  C  is  subject  to  but  a 
single  force,  which  has  the  same  direction  as  that  of  the  parallel  forces  on  the  points  at 
A  &  D,  &  that  its  magnitude  is  equal  to  their  sum.  In  other  words,  the  force  acting  upon 
it  is  exactly  the  same  as  if  all  those  points  which  are  at  A  &  D  were  transferred  together 
with  the  forces  acting  upon  them  to  the  point  C,  &  there  acted  upon  the  fulcrum  directly. 

92.  In  addition,  on  account  of  all  sides  being  parallel,  the  following  pairs  of  triangles  The     proportion 
are  similar  :-(i)  CNO,  DPC  ;    (2)  CNQ,  PDE  ;    (3)  CPR,  VCN  ;    (4)  CRS,  VNQ  ;    (5) 

CVA,  TCR  ;  (6)  VAF,  CRS.  These  will  give  the  following  six  proportions,  two  of  which 
are  contained  in  each  of  the  following  lines  : — 

ON  :  CP  =  NC  :  PD  =  NQ  :  DE 
CP  :  CV  =  CR  :  NV  =  RS  :  NQ 
CV  :  RT  =  VA  :  RC  =  AF  :  RS. 

Further,  by  compounding  together  the  first  &  last  of  these,  &  removing  from  the 
antecedents  the  ratio  CP  :  CV,  &  from  the  consequents  the  ratio  QN  :  RS,  we  are  left 
with  the  proportion,  ON  :  RT  =  AF  :  DE.  That  is  to  say,  the  sum  of  all  the  parallel 
forces  on  D,  to  which  ON  is  equal,  is  to  the  sum  of  all  those  on  A,  to  which  RT  is  equal, 
as  the  opposite  perpendicular  distance  AF  from  the  straight  line  CF  drawn  through  the 
fulcrum  in  a  direction  parallel  to  that  of  these  forces,  is  to  the  perpendicular  distance  of 
the  former  from  the  same  straight  line.  Hence,  we  have  obtained  a  solution  of  all  that 
was  required  (£). 


(g)  Moreover,  the  application  to  the  lever  u  similar  to  that  given  in  this  work,  after  the  equilibrium  of  three  points 
in  Art.  326. 


[297]     §VI 

EPISTOLA    AUCTORIS    AD    P.  CAROLUM    SCHERFFER 

SOCIETATIS  JESU 

Occasio,  &  argu-  93-  1°  meo  discessu  Vienna  reliqui  apud  Reverentiam  Vestram  imprimendum  opus, 
mentum  epistoiae.  cujus  conscribendi  occasionem  praebuit  Systema  trium  massarum,  quarum  vires  mutuse 
Theoremata  exhibuerunt  &  elegantia,  &  foecunda,  pertinentia  tarn  ad  directionem,  quam 
ad  rationem  virium  compositarum  e  binis  in  massis  singulis.  Ex  iis  Theorematis  evolvi 
nonnulla,  quae  in  ipso  prime  inventionis  aestu,  &  scriptionis  fervore  quodam,  atque  impetu 
se  se  obtulerunt.  Sunt  autem  &  alia,  potissimum  nonnulla  ad  centrum  percussionis 
pertinentia  ibi  attactum  potius,  quam  pertractatum,  quae  mihi  deinde  occurrerunt  &  in 
itinere,  &  hie  in  Hetruria,  ubi  me  negotia  mihi  commissa  detinuerunt  hucusque,  qua 
quidem  ad  Reverentiam  Vestram  transmittenda  censui,  ut  si  forte  satis  mature  advenerint, 
ad  calcem  operis  addi  possint  ;  pertinent  enim  ad  complementum  eorum,  quae  ibidem 
exposui,  &  ad  alias  sublimiores,  ac  utilissimas  perquisitiones  viam  sternunt. 


<^   Inprimis  ego  quidem  ibi  consideravi  directiones  virium  in  eodem  illo  piano,  in 
a  massis  jacentibus  quo  jacent  tres  massae,  &  idcirco  ubi  Theoremata  applicavi    ad    centrum    aequilibrii,  & 
ad*  iTbicuiT  ueaDUI^'  osc^ati°nis  Pro  phiribus  etiam  massis,  restrinxi  Theoriam  ad  casum,  in  quo  omnes  massae 
tas     affirmata  ^L  jaceant  in  eodem  piano  perpendiculari  ad  axem  conversionis.      In  nonnullis  Scholiis  tantum- 
°trandahlC  demon"  modo  innui,  posse  rem  transferri  ad  massas,  utcunque  dispersas,  si   eae  reducantur  ad   id 
planum  per  rectas  perpendiculares  piano  eidem  ;    sed  ejus    applicationis    per    ejusmodi 
reductionem    nullam   exhibui   demonstrationem,    &    affirmavi,    requiri    systema    quatuor 
massarum  ad  rem  generaliter  pertractandam. 

Viribus    trium  or.  At  admodum  facile  demonstratur  eiusmodi  reductionem  rite   fieri,    &  sine   nova 

massarum  in  eodem  f     •  TL  r     i.    U  •          --PL 

piano,     i  n     quo  peculiari  Theona  massarum  quatuor  generalis  habetur  applicatio  tenui  extensione  Theonae 
^?eat'  j  trans'atis  massarum  trium.     Nimirum  si  concipiatur  planum  quodvis,  &  vires  singulae  resolvantur 

ad   ahud,  rem  ob-    •      j  ,  j-      i  •      •      i  HI  •  v  i 

tinen.  in  duas,  alteram  perpendicularem  piano  ipsi,  alteram  paraiielam  ;   pnorum  summa  elidetur, 

cum  oriantur  e  viribus  mutuis  contrariis,  &  aequalibus,  quae  ad  quamcunque  datam 
directionem  redactae  aequales  itidem  remanent,  &  con-[298]-trariae,  evanescente  (£)  summa  : 
posteriores  autem  componentur  eodem  prorsus  pacto,  quo  componerentur  ;  si  massae  per 
illas  perpendiculares  vires  reducerentur  ad  illud  planum,  &  in  eo  essent,  ibique  vires  haberent 
aequales  redactas  ad  directionem  ejusdem  plani,  quarum  oppositio  &  aequalitas  redderet 
eandem  figuram,  &  eadem  Theoremata,  quae  in  opere  demonstrata  sunt  pro  viribus  jacentibus 
in  eodem  piano,  in  quo  sunt  massae.  Porro  haec  consideratio  extendet  Theoriam  aequilibrii, 
&  centri  oscillationis  ad  omnes  casus,  in  quibus  systema  quodvis  concipitur  connexum  cum 
unico  puncto  axis  rotationis,  ut  ubi  globus,  vel  systema  quotcunque  massarum  invicem 
connexarum  oscillat  suspensum  per  punctum  unicum. 


9^-  Quod  si  sint  quatuor  massae,  &  concipiatur  planum  perpendiculare  rectae  transeunti 
omnes  ad  planum  per  binas  ex  iis,  ac  fiat  resolutio  eadem,  quae  superius  ;  res  iterum  eodem  recidet  :  nam 
recta:0  Vungenti  ^^  bhiae  massae  ita  in  illud  planum  projectae,  coalescent  in  massam  unicam,  &  vires  ad 

duas  :   inde  transi- 

tus       ad       massas 

quotcunque. 

(h)  Htec  turn  quidem  in  bac  epistola.  Addi  potest  illud,  ubi  nulla  externa  vis  in  ea  directions  agens,  &  in 
contraria  applicetur  diversis  partibus  ipsius  systematis,  dtbere  vim  hujusmodi  in  singulis  etiam  ipsius  systematis  punctis 
esse  nullam.  Nam  per  mutuum  nexum  impeditur  mutatio  positionis  mututs,que  utique  induceretur,  si  in  aliquibus  tantum- 
modo  ejus  partibus  remaneret  vis  externis  viribus  non  impedita.  Porro  ubi  agitur  de  centra  oscillationis,  &  percussionis, 
ac  etiam  de  (equilibria,  nulla  supponitur  vis,  externa  agens  secundum  directionem  axis  rotationis,  sen  conversionis.  Quare 
in  iis  casibus,  pro  quibus  b<zc  tbeoria  hie  extenditur,  satis  est  considerare  reliquas  illas  vires,  qua  agunt  secundum 
directionem  plani  perpendicularis  eidem  axi,  quod  hie  pnsstatur  in  iis,  qu<z  consequuntur  . 

442 


§  VI 


transition  to 
theory  of  the 
of  oscilla- 


A  LETTER  FROM  THE  AUTHOR 

TO 
FR.  CAROLUS  SCHERFFER,  S.J. 

03.  When  I  departed  from  Vienna,  I  left  with  Your  Reverence  to  be  printed  a  work,  The  occasion  for, 

,  .   7J-,  .  .  ,  '  i  •  j  •  r  r     i  i        &  the  contents    of, 

which  I  had  written  as  an  outcome  of  the  consideration  of  a  system  of  three  masses  ;  the  the  letter, 
mutual  forces  between  these  brought  out  several  theorems  that  were  both  elegant  &  fruitful, 
with  regard  to  the  direction  &  the  ratio  of  the  forces  on  each  of  the  masses  compounded 
from  the  other  two.  From  these  theorems  I  worked  out  certain  results,  which,  in  the 
first  surge  of  discovery,  &  a  certain  fervour  &  impetus  of  writing,  had  forced  themselves 
on  my  attention.  But  there  are  also  other  matters,  especially  some  relating  to  the  centre 
of  percussion  that  are  in  it  merely  touched  upon  rather  than  dealt  with  thoroughly  ;  these 
came  to  me  later,  some  during  my  journey,  &  some  here  in  Tuscany,  where  the  business 
entrusted  to  me  has  kept  me  up  till  now.  These  matters  I  thought  should  be  sent  to  Your 
Reverence,  so  that,  if  perchance  they  should  reach  you  soon  enough,  they  might  be  added 
at  the  end  of  the  work  ;  for  they  deal  with  the  further  development  of  those  things  which 
I  have  expounded  therein,  &  open  the  road  to  more  sublime  &  useful  matters  for  inquiry. 

94.  First  of  all,  I  there  indeed  considered  the  directions  of  the  forces  in  the  same  The 
plane  as  that  in  which  the  masses  were  situated  ;  &,  therefore,  when  I  applied  the  theorems  *h£t 
to  the  centre  of  equilibrium  &  oscillation  even  for  several  masses,  I  restricted  the  Theory  tion'from*  the  case 
to  the  case  in  which  all  the  masses  were  lying  in  the  same  plane,  perpendicular  to  the  axis  ^  ^e^n^11  p^ne 
of  rotation.  Only  in  some  notes  did  I  mention  that  the  matter  could  be  developed  for  to  masses  lying 
masses  that  were  disposed  in  any  manner,  if  these  were  reduced  to  that  plane  by  perpendi-  any™here,  .  mer^ 
culars  to  the  plane.  But  I  gave  no  demonstration  of  this  application  by  means  of  such  a  work  itself,  is  here 
reduction  ;  &  I  asserted  that  the  consideration  of  a  system  of  four  masses  would  be  necessary  to  be  Proved- 
before  the  matter  could  be  dealt  with  thoroughly,  &  in  general. 

OS.  But  it  is  quite  easily  proved  that  such  a  reduction  can  be  correctly  made  ;    &  a  T.he     forces  .    f°r 

...  .   .  '  ......  r        ,.  .  ,     three  masses  m  the 

general  application,  without  any  special  fresh  theory  for  lour  masses,  can  take  place,  with  same  plane  as  that 
a  very  slight  extension  of  the  theory  for  three  masses.     Thus,  if  any  plane  is  taken  &  each  m .  whlch  they  he, 

•  11-  e       -L  •   i  •  j-      i        o      i  11   i  belnS      transferred 

force  is  resolved  into  two  forces,  of  which  one  is  perpendicular  &  the  other  parallel  to  the  to  another,  the 
plane  ;  then  the  sum  of  all  the  first  will  be  eliminated,  since  they  arise  from  mutual  forces  tnins  ^  done- 
that  are  equal  &  opposite  to  one  another  ;  for,  these  when  reduced  to  any  given  direction 
whatever  will  still  remain  equal  &  opposite  to  one  another,  &  their  sum  will  vanish  (£).  Also 
the  latter  will  be  compounded  in  exactly  the  same  manner  as  they  would  have  been  com- 
pounded, if  the  masses,  by  means  of  those  perpendicular  forces,  had  been  reduced  to  that 
plane,  &  were  really  in  it,  &  had  there  equal  forces  reduced  to  the  direction  of  that  plane  : 
the  equal  &  opposite  nature  of  these  forces  would  give  the  same  figure,  &  the  same  theorems 
as  were  proved  in  the  work  itself  for  forces  in  the  same  plane  as  that  in  which  the  masses 
were  lying.  Further,  this  way  of  looking  at  the  matter  will  extend  the  Theory  of  the 
centre  of  equilibrium  &  oscillation  to  all  cases,  in  which  any  system  is  supposed  to  be  con- 
nected with  a  single  point  on  the  axis  of  rotation,  as  when  a  sphere,  or  a  system  of  any 
number  of  masses  connected  together  oscillates  under  suspension  from  a  single  point. 

06.  Now  if  there  are  four  masses,  &  a  plane  is  taken  perpendicular  to  the  straight  If  there    are  fou,^ 

, .         7  .    .  ,    .  f    .        .  r  .  r  .  . .  °,        masses,  they  are  all 

line  joining  any  two  of  them,  &  the  same  resolution  is  made  as  in  the  preceding  paragraph  ;  to  be  reduced  to  a 
then,  the  matter  will  again  come  to  the  same  thing.     For,  those  two  masses,  being  thus  plane  perpendicular 

,  .....  .°    ,  .        .  11.  to  the  straight   line 

thrown  into  the  same  plane,  will  coalesce  into  a  single  mass ;    &  the  forces  belonging  to  joining      two     of 

them ;     hence    the 

transition    to    any 

(h)  This  is  what  I  said  in  the  letter.  To  it  may  be  added  the  point  that,  when  no  external  force  is  applied  acting  number  of  masses. 
in  one  direction  on  one  part,  y  the  opposite  direction  on  another  part,  of  the  system,  this  kind  of  force  must  also  he  zero 
for  each  of  the  points  of  the  system.  For,  a  change  of  mutual  position  is  prevented  by  the  mutual  connection  ;  y  at 
any  rate  this  would  be  induced,  if  in  any  of  the  parts  of  it  there  but  remained  a  force  that  was  not  checked  by  external 
forces.  Further,  when  dealing  with  the  centre  of  oscillation,  &  of  percussion,  W  with  equilibrium,  no  external  force  is 
supposed  to  act  in  the  direction  of  the  axis  of  rotation  or  conversion.  Hence,  in  these  cases,  for  which  the  theory  is 
here  extended,  it  is  sufficient  to  consider  these  other  forces,  which  act  in  the  direction  of  the  plane  perpendicular  to  the 
axis  i  £jf  this  is  done  in  what  follows. 

443 


444  PHILOSOPHIC  NATURALTS  THEORIA 

reliquas  binas  massas  pertinentes  habebunt  ad  se  invicem  eas  rationes,  quae  pro  systemate 
trium  massarum  deductae  sunt.  Hinc  ubi  systema  massarum  utcunque  dispersarum 
convert!  debet  circa  axem  aliquem,  sive  de  aequilibrii  centre  agatur,  sive  de  centra  oscilla- 
tionis^ sive  de  centre  percussionis,  licebit  considerare  massas  singulas  connexas  cum  binis 
punctis  utcunque  assumptis  in  axe,  &  cum  alio  puncto,  vel  massa  quavis  utcunque  assumpta, 
vel  concepta  intra  idem  systema,  &  habebitur  omnium  massarum  nexus  mutuus,  ac 
applicatio  ad  omnia  ejusmodi  centra  habebitur  eadem,  concipiendo  tantummodo  massas 
singulas  redactas  ad  planum  perpendiculare  per  rectas  ipsi  axi  parallelas. 

Applicatio     ad  97.  Sic  ex.  gr.  ubi  agitur  de  centre  oscillationis,  quae  pro  massis  existentibus  in  unico 

centn    oscillationis      t  j-      i      •       i  • 

generaiem     deter-  piano  perpendiculan  ad  axem  rotationis  proposui,  ac  demonstravi  respectu  puncti  suspen- 
minationem.  sionis,  centri  gravitatis,  traducentur  ad  massas  quascunque,  utcunque  dispersas  respectu 

axis,  &  respectu  rectae  parallelae  axi  ductae  per  centrum  gravitatis,  quam  rectam  Hugenius 
appellat  axem  gravitatis.  Nimirum  centrum  oscillationis  jacebit  in  recta  perpendiculari 
axi  rotationis  transeunte  per  centrum  gravitatis,  ac  ad  habendam  ejus  distantiam  ab  axe 
eodem,  si-[299]-ve  longitudinem  penduli  isochroni,  satis  erit  ducere  massas  singulas  in 
quadrata  suarum  distantiarum  perpendicularium  ab  eodem  axe,  &  productorum  summam 
dividere  per  factum  ex  summa  massarum,  &  distantia  perpendiculari  centri  gravitatis 
communis  ab  ipso  axe.  Rectangulum  autem  sub  binis  distantiis  centri  gravitatis  ab  axe 
conversionis,  &  a  centre  oscillationis  erit  aequale  summse  omnium  productorum,  quae 
habentur,  si  massae  singulae  ducantur  in  quadrata  suarum  distantiarum  perpendicularium 
ab  axe  gravitatis,  divisae  per  summam  massarum.  Si  enim  omnes  massae  reducantur  ad 
unicum  planum  perpendiculare  axi  conversionis,  abit  is  totus  axis  in  punctum  suspensionis, 
totus  axis  gravitatis  in  centrum  gravitatis,  &  singulas  distantiae  perpendiculares  ab  iis 
axibus  evadunt  distantiae  ab  iis  punctis  :  unde  patet  generaiem  Theoriam  reddi  omnem 
per  solam  applicationem  systematis  massarum  trium  rite  adhibitam. 


Ahud  utile    coroi-  ng.  Quod  ad  centrum  oscillationis  pertmet,  erui  potest  almd  Corollanum,  praeter  ilia, 

larmm  pertmens  ad  •  ,  .       r  •  j  •        c  •     »      • 

centrum   osciiia-  q1136  proposui,  quod  summo  saepe  usui  esse  potest  :  est  autem  ejusmodi.     01  pLurium  -partium 
tionis-  systematis  compositarum   ex   massis  quotcunque,   utcunque  disperses   inventa  fuerint  seorsim 

centra  gravitatis,  tff  centra  oscillationis  respondentia  data  puncto  suspensionis,  vel  dato  axi 
conversionis ;  inveniri  poterit  centrum  oscillationis  commune,  ducendo  singularum  partium 
massas  in  distantias  perpendiculares  sui  cujusque  centri  gravitatis  ab  axe  conversionis,  & 
centri  oscillationis  cujusvis  ab  eodem,  W  dividendo  productorum  summam  per  massam  totius 
systematis  ductam  in  distantiam  centri  gravitatis  communis  ab  eodem  axe.  Hoc  corollarium 
deducitur  ex  formula  generali  eruta  in  ipso  opere  num.  334  pro  centre  oscillationis,  quae 
respondet  figurae  63  exprimenti  unicam  massam  A  ex  pluribus  quotcunque,  quae  concipi 
possint  ubicunque  :  exprimit  autem  ibidem  P  punctum  suspensionis,  vel  axem  conversionis, 
G  centrum  gravitatis,  Q  centrum  oscillationis,  M  summam  massarum  A  +  B  +  C  &c, 

a    t          i            r>r\       A  X    AP2  +  B  X  BP2  +  &c. 
&  formula  est  PQ  = M  X  GP  ' 

Ejus  demonstrate.  99.  Nam    ex    ejusmodi    formula    est    M  X  GP  X  PQ  =  A  X  AP2  +  B  X  BP2     &c. 

Quare  si  singularum  partium  massae  M  ducantur  in  suas  binas  distantias  GP,  PQ  ;  habetur 
in  singulis  summa  omnium  A  X  AP2  +  B  X  BP2  &c.  Summa  autem  omnium  ejusmodi 
summarum  debet  esse  numerator  pro  formula  pertinente  ad  totum  systema,  cum  oporteat 
singulas  totius  systematis  massas  ducere  in  sua  cujusque  quadrata  distantiarum  ab  axe. 
Igitur  patet  numeratorem  ipsum  rite  haberi  per  summam  productorum  M  X  GP  X  PQ 
pertinentium  ad  singulas  systematis  partes,  uti  in  hoc  novo  Corollario  enunciatur. 

i 

Usus    pro     longi-  IOO.  Usus  hujus  Corollarii  facile  patebit.     Pendeat  ex.  gr.  globus  aliquis  suspensus 

com^osit^Uso-1  per  filum  quoddam.      Pro  globo  jam  constat  centrum  gravitatis  esse  in  ipso  centra  globi, 

chroni     facilius  &  constat  [soo]  itidem,  ac  e  superioribus  etiam  Theorematis  facile  deducitur,  centrum 

oscillationis  jacere  infra  centrum  globi,  per  f  tertiae  proportionalis  post  distantiam  puncti 

suspensionis  a  centra  globi,  &  radium  ;  pro  filo  autem  considerate  ut  recta  quadam  habetur 

centrum  gravitatis  in  medio  ipso  filo,  &  centrum  oscillationis,  suspensione  facta  per  fili 

extremum  est  in  fine  secundi  trientis  longitudinis  ejusdem  fili,  quod  itidem  ex  formula 


LETTER  TO   FR.   SCHERFFER  445 

the  other  two  masses  will  have  to  one  another  those  ratios  that  have  already  been  deter- 
mined for  a  system  of  three  masses.  Hence,  when  a  system  of  masses  arranged  in  any  manner 
must  rotate  about  some  axis,  whether  it  is  a  question  of  the  centre  of  equilibrium,  or  of  the 
centre  of  oscillation,  or  of  the  centre  of  percussion,  we  may  consider  each  of  the  masses 
as  being  connected  with  a  pair  of  points  chosen  anywhere  on  the  axis,  &  with  some  other 
point,  whether  this  is  some  mass  taken  in  any  manner  or  assumed  to  be  within  the  same 
system  ;  &  then,  there  will  be  a  mutual  connection  between  all  the  masses,  &  the  same 
application  can  be  made  to  all  such  centres,  by  merely  considering  that  each  of  the  masses 
is  reduced  to  a  perpendicular  plane  by  means  of  straight  lines  parallel  to  the  axis. 

97.  Thus,  for  example,  when  we  are  concerned  with  the  centre  of  oscillation,  the  Application  to  the 
results  which  I  enunciated  for  masses  existing  in  a  single  plane  perpendicular  to  the  axis  f^6™,1 

of  rotation,  and  proved,  with  respect  to  the  point  of  suspension  &  the  centre  of  gravity,  may  of  oscillation. 

be  applied  to  any  masses,  however  disposed  with  respect  to  the  axis,  &  with  respect  to  a 

straight  line  drawn  parallel  to  the  axis  through  the  centre  of  gravity  ;   this  straight  line  is 

called  the  axis  of  gravity  by  Huyghens.     That  is  to  say,  the  centre  of  oscillation  will  lie 

in  a  straight  line  perpendicular  to  the  axis  of  rotation  drawn  through  the  centre  of  gravity  ; 

&  to  obtain  the  distance  of  this  centre  of  oscillation  from  the  axis,  or  the  length  of  the  iso- 

chronous pendulum,  it  will  be  sufficient  to  multiply  each  of  the  masses  by  the  square  of 

its  distance  measured  perpendicular  to  the  same  axis,  &  to  divide  the  sum  of  the  products  by 

the  product  of  the  sum  of  the  masses  &  the  perpendicular  distance  of  the  common  centre 

of  gravity  from  the  axis.     Also  the  rectangle  contained  by  the  two  distances  of  the  centre 

of  gravity  from  the  axis  of  rotation  &  the  centre  of  oscillation  will  be  equal  to  the  sum 

of  all  the  products,  which  are  obtained  by  multiplying  each  of  the  masses  by  the  square 

of  its  perpendicular  distance  from  the  axis  of  gravity,  divided  by  the  sum  of  the  masses. 

For,  if  all  the  masses  are  reduced  to  a  single  plane  perpendicular  to  the  axis  of  rotation, 

the  whole  axis  merely  becomes  the  point  of  suspension,  the  whole  axis  of  gravity  becomes 

the  centre  of  gravity,  &  each  of  the  perpendicular  distances  from  these  axes  becomes  a 

distance  from  these  points.     Thus,  it  will  be  clear  that  the  whole  of  the  general  theory 

is  obtained  by  the  application  of  the  system  of  three  masses  alone,  if  this  is  correctly  done. 

98.  As  regards  the  centre  of  oscillation,  there  can  be  derived  another  corollary,  besides  Another     useful 
the  one  that  I  have  enunciated  ;  &  this  has  often  been  of  great  service  to  me  ;  it  is  as  fol-  1°^°"* 

lows.  //,  for  two  or  more  parts  of  a  system  composed  of  any  number  of  masses,  situated  in  any  oscillation. 
manner,  the  centres  of  gravity,  &  the  centres  of  oscillation  corresponding  to  a  given  point  of 
suspension,  or  a  given  axis  of  rotation,  have  been  separately  determined  ;  then,  the  common 
centre  of  oscillation  can  be  determined  by  multiplying  the  mass  of  each  of  the  parts  by  the  per- 
pendicular distance  of  its  centre  of  gravity  from  the  axis  of  rotation,  &  the  perpendicular  distance 
of  the  centre  of  oscillation  from  the  same  axis  ;  W  dividing  the  sum  of  these  products  by  the 
mass  of  the  whole  system,  y  the  distance  of  the  common  centre  of  gravity  from  the  same  axis. 
This  corollary  is  derived  from  the  general  formula  derived  in  the  work  itself,  Art.  334,  for 
the  centre  of  oscillation,  which  corresponds  to  Fig.  63,  showing  a  single  mass  A  out  of  any 
number  whatever  that  might  be  conceived  anywhere  ;  also  in  the  same  diagram,  the 
point  P  is  the  point  of  suspension,  or  the  axis  of  rotation,  G  the  centre  of  gravity,  Q  the 
centre  of  oscillation,  M  the  sum  of  the  masses  A  +  B  +  C,  &c,  and  the  formula  is 

po  _  A  x  AP*  -t-  B  x  BP*  +  &c. 
M  x  GP 

99.  Thus,  from  the  formula  given,  we  have  Demonstration    of 

M  x  GP  x  PQ  =  A  x  AP»  +  B  x  BP*  +  &c. 


Hence,  if  the  mass,  M,  of  each  of  the  parts  is  multiplied  each  by  its  own  two  distances  GP, 
PQ,  we  have  for  each  the  total  sum  A  X  AP2  +  B  X  BP2  -}-  &c.  But  the  sum  of  all  such 
sums  as  these  must  be  the  numerator  belonging  to  the  formula  for  the  whole  system,  since 
we  have  to  multiply  each  of  the  masses  of  the  whole  system  by  the  square  of  its  distance 
from  the  axis.  Therefore,  it  is  plain  that  the  numerator  can  be  correctly  taken  to  be  the 
sum  of  the  products  M  X  GP  X  PQ  belonging  to  the  several  parts  of  the  system,  as  we  have 
stated  in  this  new  corollary. 

100.  The  use  of  this  corollary  will  be  easily  seen.     For  example,  suppose  we  have  a  its  use  in  providing 

,     ,   ,  ,  .  /_  .  •:     .  .,   ,  r     '        rr  -  .         an  easy  determma- 

sphere  suspended  by  a  thin  rod.     ror  a  sphere,  it  is  well-known  that  the  centre  01  gravity  tion  of  the  length 
is  at  the  centre  of  the  sphere  ;  and  it  is  also  well-known,  &  indeed  it  can  be  easily  deduced  of     a     pendulum 
from  the  theorems  given  above,  that  the  centre  of  oscillation  lies  below  the  centre  of  the  given      composite 
sphere,  at  a  distance  from  it  equal  to  two-fifths  of  the  third  proportional  to  the  distance  of  pendulum. 
the  point  of  suspension  from  the  centre  &  the  radius.     For  the  rod,  considered  as  a  straight 
line,  the  centre  of  gravity  is  at  the  middle  point  of  the  rod  ;   &  the  centre  of  oscillation, 
when  the  suspension  is  made  from  one  end  of  the  rod,  is  two-thirds  of  the  length  of  the 
rod  from  that  end  ;  &  this  can  also  be  deduced  quite  easily  from  the  general  formula.     Hence 


446  PHILOSOPHIC  NATURALIS  THEORIA 

general!  facillime  deducitur.  Inde  centrum  oscillationis  commune  globi,  &  fili  nullo 
negotio  definietur  per  corollarium  superius. 

roenoo  I01'  ^  Longitude  fili  a,  massa  seu  pondus  b,  radius  globi  r,  massa  seu  pondus  p  :  erit 

pendentis  e  filo.  distantia  centri  gravitatis  fili  ab  axe  conversionis  erit  %a,  distantia  centri  oscillationis 
ejusdem  ftf.  Quare  productum  illud  pertinens  ad  filum  erit  J  a*b.  Pro  globo  erit 
distantia  centri  gravitatis  a  +  r,  quas  ponatur  =  m  ;  Distantia  centri  oscillationis  erit 

TT 

m  +  f  X  — .     Quare  productum  pertinens  ad  globum  erit  m?p  +  f  rrp.     Horum  summa 

est  mzp  +  |  rrp  +  J  d*b.  Porro  cum  centra  gravitatis  fili,  &  globi  jaceant  in  directum 
cum  puncto  suspensionis,  ad  habendam  distantiam  centri  gravitatis  communis  ductam 
in  summam  massarum  satis  erit  ducere  singularum  partium  massas  in  suorum  centrorum 
distantias,  ac  habebitur  mp  -f-  i  ab.  Quare  formula  pro  centro  oscillationis  utriusque 
simul,  erit 

m*p  -f  f  rrp  -f  %azb 
mp  +  \ab 

Non  Hcere  hie  con-  102.  Hie  autem  notandum  illud,  ad  centrum  oscillationis  commune  habendum  non 

uiaf6  uT^oDectf s"  licere  singularem  partium  massas  concipere,  ut  collectas  in  suis  singulas  aut  centris  oscilla- 

in  suis  centris  oscii-  tionis,  aut  centris  gravitatis.     In  primo  casu  numerator  colligeretur  ex  summa  omnium 

tati°nlS'aaut  ^ailis  Productorum,   quae   fierent   ducendo   singulas   massas    in    quadrata    distantiarum    centri 

inter'mediis    docu-  oscillationis  sui ;  in  secundo  in  quadrata  distantiarum  sui  centri  gravitatis.     In  illo  nimirum 

haberetur  plus  justo,  in  hoc  minus.     Sed  nee  possunt  concipi  ut  collects  in  aliquo  puncto 

intermedio,  cujus  distantia  sit  media  continue  proportionalis  inter  illas  distantias  ;    nam 

in  eo  casu  numerator  maneret  idem,  at  denominator  non  esset  idem,  qui  ut  idem  perseveraret, 

oporteret  concipere  massas  singulas  collectas  in  suis  centris  gravitatis,  non  ultra  ipsa.     Inde 

autem  patet,  non  semper  licere  concipere  massas  ingentes  in  suo  gravitatis  centro,  &  idcirco, 

ubi  in  Theoria  centri  oscillationis,  vel  percussionis  dico  massam  existentem  in  quodam 

puncto,  intelligi  debet,  ut  monui  in  ipso  opere,  tota  massa  ibi  compenetrata  vel  concipi 

massula  extensionis  infinitesimse  ut  massae  compenetratae  in  unico  suo  puncto  aequivaleat. 


Transitus  ad  cen-  [30  il  jo1?.  Quod   atthiet   ad    centrum  percussionis,  id  attigi  tantummodo   determinando 

trum  percussionis  :    "•  J       J         J  .  .  .        r.  p   ...  .  . 

ejusnotioneshaberi  punctum  systematis  massarum  jacentium  in  recta  quadam,  &  hbere  gyrantis,  cujus  puncti 

posse  piures.  impedito  motu  sistitur  motus  totius  systematis.      Porro  aeque  facile  determinatur  centrum 

.    percussionis  in  eo  sensu  acceptum  pro  quovis  systemate  massarum  utcunque  dispositarum, 

&  res  itidem  facile  perficitur,  si  aliae  diversae  etiam  centri  percussionis  ideas  adhibeantur. 

Rem  hie  paullo  diligentius  persequar. 


adhibita  in  'o^ere'-  I04"  InPrimis   ut   agamus    de   eadem   centri    percussionis   notione,    moveatur   libere 

centri      gravitatis  systema  quodcunque  ita  inter  se  connexum,  ut  ejus  partes  mutare  non  possint  distantias  a  se 

inamcrtu^beroVatUS  mvicem-     Centrum  gravitatis  totius  systematis  vel  quiescet,  vel  movebitur  uniformiter 

in  directum,  cum  per  theorema  inventum  a  Newtono,  &  a  me  demonstratum  in  ipso  Opere 

num.  250,  actiones  mutuae  non  turbent  statum  ipsius  :  systema  autem  totum  sibi  relictum 

vel  movebitur  motu  eodem  parallelo,  vel  convertetur  motu  aequali  circa    axem  datum 

transeuntem  per  ipsum  centrum  gravitatis,  &  vel  quiescentem  cum  ipso  centro,  vel  ejusdem 

uniformi  motu  parallelo  delatum  simul,  quod  itidem  demonstrari  potest  haud  difficulter. 

temlte™  traLiato  IO5'  Inde  autem  colligitur  illud,  in  motu  totius  systematis  composite  ex  motu  uniformi 

cam  nrtatione,  few  in  directum,  &  ex  rotatione  circular!  circa  axem  itidem  translatum  haberi  semper  rectam 

rectam     cum     eo  quandam  pertinentem  ad  systema,  nimirum  cum  eo  connexam,  pro  quovis  tempusculo 

bneemam  q  u  c"  vTs  suam,  qu82  illo  tempusculo  maneat  immota,  &  circa  quam,  ut  circa  quendam  axem  immotum 

tempusculo  suam  ;  COnvertatur  eo  tempusculo  totum  systema.     Concipiatur  enim  planum  quodvis  transiens 

poSit.a'  '  per  axem  rotationis  circularis,  &  in  eo  piano  sit  recta  quaevis  axi  parallela  ;   ea  convertetur 

circa  axem  velocitate  eo  majore,  quo  magis  ab  ipso  distat.      Erit  igitur  aliqua  distantia 

ejus  rectae  ejusmodi,  ut  velocitas  conversionis  aequetur  ibi  velocitati,  quam  habet  centrum 

gravitatis  cum  axe  translate  ;   &  in  altero  e  binis  appulsibus  ipsius  rectae  parallelae  gyrantis 


LETTER  TO   FR.   SCHERFFER 


447 


the  common  centre  of  oscillation  for  the  sphere  &  the  rod  together  can  with  little  difficulty 
be  determined  from  the  corollary  given  above. 

101.  Let  the  length  of  the  rod  be  a,  its  mass  or  weight  b,  the  radius  of  the  sphere  r, 
and  p  its  mass  or  weight.  The  distance  of  the  centre  of  gravity  of  the  rod  from  the  axis 
of  rotation  will  be  %a,  &  the  distance  ot  its  centre  of  oscillation  will  be  fa.  Hence,  the 
product  required  in  the  case  of  the  rod  is  ^a*b.  For  the  sphere,  the  distance  of  the  centre 
of  gravity  will  be  a  -f-  r  ;  call  this  m.  Then  the  distance  of  the  centre  of  oscillation  will  be 

r~ 
m  +  f  X  —     Hence,  the  product  for  the  sphere  will  be  m2p  -{-  %r*p.    The  sum  of  these  is 


Calculation  giving 
the  formula  for  a 
pendulum  formed 
of  a  sphere  hanging 
at  the  end  of  a  thin 
rod. 


m 


m*P  +  5 r*P  ~t~  s^b.     Further,  since  the  centres  of  gravity  of  the  rod  &  of  the  sphere  lie 
in  a  straight  line  through  the  point  of  suspension,  to  obtain  the  distance  of  the  common 
centre  of  gravity  multiplied  by  the  sum  of  the  masses,  it  is  enough  to  multiply  the  mass  of 
each  part  by  the  distance  of  its  own  centre  ;  in  this  way  we  obtain  mp  -f-  i#b.     Hence  the 
formula  for  the  centre  of  oscillation  for  both  together  will  be 

m*P  ~\~  7irZP  "4-  3-  &*b. 
mp  +  f  ab 

102.  Now,  here  we  have  to  observe  that,  in  order  to  find  the  common  centre  of  oscilla- 
tion, it  will  not  be  permissible  to  suppose  that  the  mass  of  each  part  is  condensed  at  either 
its  centre  of  oscillation  or  its  centre  of  gravity.     In  the  first  case,  the  numerator  would 
be  formed  of  the  sum  of  all  the  products,  obtained  by  multiplying  each  mass  by  the  square 
of  the  distance  of  its  centre  of  oscillation  ;    &  in  the  second  case,  by  multiplying  by  the 
square  of  the  distance  of  its  centre  of  gravity.     Thus,  in  the  former,  the  numerator  found 
would  be  greater  than  it  ought  to  be  ;    &  in  the  latter,  less.     Further,  the  masses  cannot 
be  considered  to  be  condensed  in  any  point  intermediate  to  these  centres,  such  that  its 
distance  is  some  term  of  a  continued  proportion  between  their  distances.     For,  in  that 
case,  the  numerator  would  remain  the  same  when  the  denominator  was  not  the  same  ;  for, 
in  order  that  the  latter  should  remain  the  same,  it  would  be  necessary  to  suppose  that  each 
mass  was  condensed  at  its  centre  of  gravity,  &  not  beyond  it.     From  this  it  is  also  evident 
that  it  is  not  always  permissible  to  suppose  that  huge  masses  can  be  at  their  centre  of  gravity  ; 
&,  on  this  account,  when  in  the  theory  of  the  centre  of  oscillation  or  percussion  I  say  that 
there  is  a  mass  at  a  certain  point,  it  must  be  understood,  as  I  mentioned  in  the  work  itself, 
that  the  whole  mass  is  compenetrated  at  the  point,  or  supposed  to  be  a  small  mass  of 
infinitesimal  extension,  so  as  to  be  equivalent  to  a  mass  compenetrated  at  a  single  point. 

103.  Now,  as  regards  the  centre  of  percussion,  I  merely  touched  upon  this  point,  when 
I  determined  its  position  for  the  case  of  a  system  of  masses  lying  in  a  straight  line  &  gyrating 
freely  ;  using  the  idea  that  the  point  was  such  that,  if  its  motion  was  prevented,  the  whole 
system  was  brought  to  rest.     Further,  the  centre  of  percussion  is  determined  with  equal 
facility,  when  considered  in  this  way,  for  any  system  of  masses  no  matter  how  they  are 
arranged.     The  matter  is  also  easily  accomplished,  even  if  diverse  other  ideas  of  the  centre 
of  percussion  are  adopted.     In  what  follows  here,  I  will  investigate  the  matter  a  little  more 
carefully. 

104.  First  of  all,  to  use  the  same  notion  of  the  centre  of  percussion  as  above,  let  the 
system  be  in  free  motion  of  any  sort  so  long  as  it  is  so  self-connected  that  its  parts  cannot 
change  their  distances  from  one  another.     Then,  the  centre  of  gravity  of  the  whole  system 
will  either  be  at  rest,  or  will  move  uniformly  in  a  straight  line  ;  for,  according  to  a  theorem, 
discovered  by  Newton,  and  demonstrated  by  myself  in  Art.  250  of  the  work,  the  mutual 
actions  will  not  disturb  the  state  of  the  centre  of  gravity.     Also  the  whole  system,  if  left 
to  itself,  will  either  move  with  the  same  parallel  motion,  or  will  rotate  with  uniform  motion 
about  a  given  axis  passing  through  the  centre  of  gravity  ;  this  axis  either  remains  at  rest 
along  with  the  centre  of  gravity,  or  moves  together  with  it  with  the  same  parallel  uniform 
motion,  as  also  can  be  proved  without  much  difficulty. 

105.  Also  from  this  it  can  be  deduced  that,  in  a  motion  of  the  whole  system,  com- 
pounded of  an  uniform  motion  in  a  straight  line  and  a  circular  motion  about  an  axis  that  is 
also  translated,  there  will  always  be  found  a  certain  straight  line  belonging  to  the  system,  that 
is  to  say,  connected  with  it,  corresponding  to  every  small  interval  of  time  ;  &  this  straight 
line  for  that  small  interval  of  time  remains  motionless,  and  about  it,  as  about  an  immovable 
axis,  the  whole  system  is  turned  in  that  short  interval  of  time.     For,  let  any  plane  be  taken 
passing  through  the  axis  of  circular  motion,  and  in  that  plane  take  any  straight  line  parallel 
to  the  axis  •  then  this  straight  line  will  be  turned  about  the  axis  with  a  velocity  that  is  greater 
in  proportion  as  its  distance  from  the  axis  is  increased.     There  will  therefore  be  some 
distance  for  such  a  straight  line,  such  that  in  that  position  the  velocity  of  turning  will  be 
equal  to  that  velocity  of  the  centre  of  gravity  &  the  axis  carried  along  with  it ;  &  in  one  or 
other  of  the  two  positions  of  the  parallel  straight  line,  gyrating  with  the  system,  when  it 


We  cannot  in  this 
consider  each  mass 
as  being  condensed 
at  either  its  centre 
of  oscillation  or  its 
centre  of  gravity, 
or  other  points 
inter  mediate;  a 
serviceable  warn- 
ing, to  be  taken 
from  the  example 
above. 


Passing  on  to  the 
centre  of  per- 
cussion ;  several 
different  ideas  of 
this  point  are 
possible. 


We  will  start  with 
the  same  idea  as 
that  used  in  the 
work  itself  ;  the 
state  of  the  centre 
of  gravity  is  con- 
served in  free 
motion. 


Hence  we  derive 
the  fact  that,  when 
a  system  is  trans- 
lated with  rotation, 
there  will  be,  corres- 
ponding to  any  short 
interval  of  time,  a 
certain  straight  line 
connected  with  the 
system,  which  is 
motionless ;  &  this 
straight  line  can 
easily  be  deter- 
mined. 


448  PHILOSOPHISE  NATURALIS  THEORIA 

cum  systemate  ad  planum  perpendiculare  ei  piano,  quod  axis  uniformiter  progrediens 
describit,  ejus  rectae  motus  circularis  net  contrarius  motui  axis  ipsius,  adeoque  motui,  quo 
ipsa  axem  comitatur,  cui  cum  ibi  &  aequalis  sit,  motu  altero  per  alterum  eliso,  ea  recta 
quiescet  illo  tempusculo,  &  systema  totum  motu  composite  gyrabit  circa  ipsam.  Nee 
erit  difficile  dato  motu  centri  gravitatis,  &  binarum  massarum  non  jacentium  in  eodem 
piano  transeunte  per  axem  rotationis,  invenire  positionem  axis,  &  hujus  rectae  immotae  pro 
quovis  dato  momento  temporis. 
Propositio  proble-  106.  Quaeratur  jam  in  eiusmodi  systemate  punctum  aliquod,  cuius  motus,  si  per  aliquam 

matis,  & praaparatio       •  •  j-  j    i  •       -i  •     •  •  r      •       -1         , 

ad  soiutior.em.  vim  externam  impediatur,  debeat  mutuis  actiombus  sisti  motus  totius  systematis,  quod 
punctum,  si  uspiam  fuerit,  dicatur  centrum  percussionis.  Concipiantur  autem  massas 
omnes  translatae  per  rectas  parallelas  rectae  [302]  illi  manenti  immotae  tempusculo,  quo 
motus  sistitur,  quam  rectam  hie  appellabimus  axem  rotationis,  in  planum  ipsi  perpendiculare 
transiens  per  centrum  gravitatis,  &  in  figura  64  exprimatur  id  planum  ipso  piano  schematis  : 
sit  autem  ibidem  P  centrum  rotationis,  per  quod  transeat  axis  ille,  sit  G  centrum  gravitatis, 
&  A  una  ex  massis.  Consideretur  quoddam  punctum  Q  assumptum  in  ipsa  recta  PG,  & 
aliud  extra  ipsam,  ac  singularum  massarum  motus  concipiatur  resolutus  in  duos,  alterum 
perpendicularem  rectae  PQ  agentem  directione  Aa,  alterum  ipsi  parallelum  agentem 
directione  PG,  ac  velocitas  absoluta  puncta  Q  dicatur  V. 


PA  v  V 

totfsnateoiuute°C&  I07>  Erit  pQ  •  PA  :  :  V  •     P        ,  quae  erit  velocitas  absoluta  massae  A.     Erit  autem 

relativaru 

vis  massae. 


relativarum    cujus-  p.  p 

o 


pa   .   .          x  y  _   x    y         s   erjt   velocjtas  secundum  directionem  Aa,  & 

QA  QA 

PA  Aa 

PA  .   Aa   :   :  ^-   X  V    .    =-^    X    V,    quae    erit    velocitas    secundum    directionem    PG. 


Nam  in  compositione,  &  resolutione  motuum,  si  rectae  perpendiculares  directionibus 
motus  compositi,  &  binorum  componentium  constituant  triangulum,  sunt  motus  ipsi,  ut 
latera  ejus  trianguli  ipsis  respondentia,  velocitas  autem  absoluta  est  perpendicularis  ad 
AP.  Inde  vero  bini  motus  secundum  eas  duas  directiones  erunt 


Evan  esc  entia  lo$f  jam  Vero  summa   f^~X  Ax  V   est    zero,    cum    ob    naturam    centri    gravitatis 

summae       determi-  J  PQ 

nans  problema.  y 

summa  omnium  Aa  X  A  sit  aequalis  zero,  &  =^r  sit  quantitas  data.  Quare  si  per  vim 
externam  applicatam  cuidam  puncto  Q,  &  mutuas  actiones  sistatur  summa  omnium  motuum 
=^-X  A  X  V,  sistetur  totus  systematis  motus,  reliqua  summa  elisa  per  solas  vires  mutuas, 

quarum  nimirum  summa  est  itidem  zero. 

inventio     summae  109.  Ut  habeatur  id  ipsum  punctum  Q,  concipiatur  quaevis  massa  A  connexa  cum  eo, 

nihUo.  a;quands  &  cum  puncto  P,  vel  cum  massis  ibidem  conceptis,  &  summa  omnium  motuum,  qui   ex 

nexu  derivantur  in  Q,  dum  extinguitur  is  motus  in  omnibus  A,  debet  elidi  per  vim  externam, 

summa  vero  omnium  provenientium  in  P,  ubi  nulla  vis  externa  agit,  debet  elidi  per  sese. 

Haec  igitur  posterior  summa  erit  investiganda,  &  ponenda  =  o. 

Calculus,  &  formula  [303]  no.  Porro  posito  radio  =  I,  est  ex  Theoremate  trium  massarum  ut  P  X  PQ  X  i 
derivata.  ad  A  X  AQ  X  sin  QAa,  sive  ut  P  X  PQ  ad  A  X  Qa,  ita   actio   in  A  perpendicularis   ad 

PQ  =  -^  X  V  ad  actionem  in  P  secundum  eandem  directionem,  quae  evadit — = ^- —  X  V: 

1\£  A      X     A  \2 

•  i     s~\          T*/~\        T»  «  •      •      T>         **  X  A  vj  X  A#  —  A  X  JL  &  TT       r* 

nimirum  ob  Qa  =  PQ  —  Pa,  erit  actio  in  P  =  -  P  v  PQ2 

V 

harum  summa  debat  aequari  zero  demptis  communibus  = pnv    sequabuntur     positiva 

-L       />,     Jt  \^. 


negativis,  nimirum  posita    pro  characteristica  summae,  habebitur/.AxPQxPtf  =/.AxP02, 

sive   PQ  =  -f'  A  ^g~">  ve^  °b  /-  A  X  Pa  —  M  X  PG,  posito  ut  prius  M    pro    summa 
j  •  AX  ±a 

massarum,  fiet  PQ  =  -^ ^ ,  qui  valor  datur  ob  datas  omnes  massas  A,  datas  omnes 

rectas  Pa,  datam  PG.     Q.E.F. 


LETTER  TO  FR.   SCHERFFER  449 

arrives  in  a  plane  perpendicular  to  the  plane  which  the  uniformly  progressing  axis  describes, 
the  circular  motion  of  the  straight  line  will  be  in  the  opposite  direction  to  that  of  the  axis 
itself,  and  thus  of  the  motion  with  which  it  accompanies  the  axis  ;  &  since  it  is  also  equal 
to  it  there,  the  one  motion  cancels  the  other,  &  the  straight  line  will  be  at  rest  for  the 
small  interval  of  time,  &  the  whole  system  will  gyrate  about  it  with  a  compound  motion. 
Nor  will  it  be  difficult,  given  the  motion  of  the  centre  of  gravity,  &  of  two  masses  not  lying 
in  the  same  plane  passing  through  the  axis  of  rotation,  to  find  the  position  of  this  axis  & 
that  of  the  motionless  straight  line  for  any  given  instant  of  time. 

106.  Now  let  it  be  required  to  find  in  such  a  system  a  point,  such  that,  if  its  motion  Enunciation  of   a 
is  prevented  by  some  external  force,  the  motion  of  the  whole  system  is  thereby  checked  by  problem  &  prepar- 
mutual  actions  ;  this  point,  if  there  is  one,  will  be  called  the  centre  of  percussion.     Suppose  solution*01     the 
all  the  masses  to  be  translated  along  straight  lines  parallel  to  the  straight  line  that  remains 

motionless  for  the  small  interval  of  time  in  which  the  motion  is  checked  ;  this  straight  line 
we  will  now  call  the  axis  of  rotation  ;  &  suppose  that  by  this  translation  they  are  all  brought 
into  a  plane  perpendicular  to  the  axis  of  rotation  &  passing  through  the  centre  of  gravity. 
In  Fig.  64,  let  this  plane  be  represented  by  the  plane  of  the  diagram  ;  &  there  also  let  P 
stand  for  the  centre  of  rotation  through  which  the  axis  passes  ;  let  G  be  the  centre  of  gra- 
vity, &  A  one  of  the  masses.  Consider  any  point  Q,  taken  in  the  straight  line  PG,  &  another 
point  that  is  not  on  this  line  ;  &  let  the  motion  of  each  mass  be  resolved  into  two,  of  which 
one  is  perpendicular  to  the  straight  line  PG  &  acts  in  the  direction  Aa,  &  the  other  is 
parallel  to  it  &  acts  in  the  direction  PG  ;  let  the  absolute  velocity  of  the  point  Q  be 
called  V. 

107.  If  v  is  the  absolute  velocity  of  the  mass  A,  we  have  PQ  :  PA  =  V  :  v  ;  therefore  Determination    of 
v  =  V  X  PA/PQ.      Similarly,  since  we  have  PA  :  Pa  =  V  X  PA/PQ  :    V  X  Pa/PQ  ;  teiocity.t  l^Vhe 
therefore  V  X  Pa/PQ  will  be    the  velocity  in  the  direction  Aa.     Also,  since    we    have  relative'  velocities, 
PA  :  A*=V  X  PA/PQ  :  V  X  Aa/PQ  •    hence,  V  X  Aa/PQ  will  be  the  velocity  in  the  of  any  mass" 
direction  PG.     For,  in  composition  and  resolution  of  motion,  if  straight  lines  perpendicular 

to  the  directions  of  the  resultant  motion  &  its  two  components  form  a  triangle,  then  the 
motions  are  proportional  to  the  corresponding  sides  of  the  triangle  ;  &  the  absolute  velocity 
is  perpendicular  to  AP.  Hence,  the  two  motions  in  these  two  directions  will  be 

equal  to  *±    X  A  X   V,    and          X  A  X  V. 


108.  Now,  the  sum  of  all  such  as  =^-  X  A  X  V  is  equal  to  zero,  since,  on  account  Evanescence  of  this 

JrQ  sum    which    deter- 

,    .  .    ,  ...  ..  mines  the  problem. 

ot  the  nature  of  the  centre  of  gravity,  the  sum  of  all  such  as  Aa  X  A  is  equal  to  zero,  and 
V/PQ  is  a  given  quantity.     Hence,  if  by  means  of  an  external  torce  applied  at  any  point  Q, 

&  the  mutual  actions,  the  sum  of  all  the  motions         X  A  X  V    is    checked,    then    the 


whole  motion  of  the  system  is  checked  also  ;    for  the  remaining  sum  is  cancelled  by  the 
mutual  forces  only,  of  which  indeed  the  sum  is  also  zero. 

109.  In  order  to  find  the  point  Q,  take  any  mass  A  connected  with  it  &  the  point  P,  The  determination 
or  with  masses  supposed  to  be  situated  at  these  points  ;   then  the  sum  of  all  the  motions,  u^be  eq^atld  '  to 
which  are  derived  from  the  connection  for  Q,  when  this  motion  is  destroyed  for  every  A,  zero- 
must  be  cancelled  by  the  external  force  ;    but  the  sum  of  all  these  that  arise  for  P,  upon 
which  no  external  force  acts,  must  cancel  one  another.     Hence  it  is  the  latter  sum  that  will 
have  to  be  investigated  &  put  equal  to  zero. 

no.  Now,  if  the  radius  is  made  the  unit,  then,  from  the  theorem  for  three  masses,  The    calculation, 
we  have  the  ratio  of  P  X  PQ  X  I  to  A  X  AQ  X  sinQAa,  or  P  X  PQ  to  A  X  Qa,  equal  " 

to  the  ratio  of  the  action  at  A  perpendicular  to  PQ  (which  is  equal  to  ^  X  V)  to  the  action 


at  P  in  the  same  direction;    &  therefore  the  latter  is  equal  to       ^      pr»     x  V.  that 

X     /\    X  \£ 

is  to  say,   since   Qa  =  PQ  —  Pa,   the  action  at  P  =    Ax  PQ  X  Pf  —  A  X  Pa*     x  v. 

i   X  A  \2 

Since  the  sum  of  all  of  these  has  to  be  equated  to  zero,  on  cancelling  the  common  factor 
V/(P  X  PQ2),  the  positives  will  be  equal  to  the  negatives  ;  hence,  using  the  symbol  / 
as  the  characteristic  of  a  sum,  we  have  /.  A  X  PQ  X  Pa  =  /.  A  X  Pa*  ;  that  is, 
PQ  =/.  A  XPa*/(f.A  X  Pa).  Now,  if  as  before  we  put  M  for  the  sum  of  all  the  masses, 
then/.  A  X  Pa=  M  X  PG,  &  we  have  PQ  =/.  A  X  Pa*/(M  X  PG).  This  value  can  be 
determined  ;  for  all  the  masses  like  A  are  given,  also  all  the  straight  lines  such  as  Pa 
are  given,  &  PG  is  given.  Q.E.F. 

GG 


45°  PHILOSOPHIC  NATURALIS  THEORIA 

Thf°rmuia  erutum  ni-  Corollarium  I.     Quoniam  aP  aequatur  distantiae  perpendicular!  A  a  piano  trans- 

eunte  per  P  perpendicular!  ad  rectam  PG,  habebitur  hujusmodi  Theorema.  Distantia 
centri  percussionis  ab  axe  rotationis  in  recta  ipsi  axi  perpendiculari  transeunte  •per  centrum 
gravitatis  habebitur,  ducendo  singulas  massas  in  quadrata  suarum  distantiarum  perpendicularium 
a  -piano  perpendiculari  eidem  rectce  transeunte  per  axem  ipsum  rotationis,  ac  dividendo  summam 
omnium  ejusmodi  productorum  per  factum  ex  summa  massarum  in  distantiam  perpendicularem 
centri  gravitatis  communis  ab  eodem  plano.ty 

Deductio    casus,  [304]  112.  Corollarium  II.     Si  massae  jaceant  in  eodem  unico  piano  quovis  transeunte 
quo  jaceant  omnes  per  axem  ;  A,  &  a  congruunt,  adeoque  distantiae  Pa  sunt  ipsae  distantise  ab  axe.     Quamobrem 

massae     in   eodem    f     ,  ',  ,   °,  .'  .  r  .  .  ,  , 

piano.  in  hoc  casu  formula  haec  inventa  pro  centro  percussionis  congruit  prorsus  cum  iormula 

inventa  pro  centro  oscillationis,  &  ea  duo  centra  sunt  idem  punctum,  si  axis  rotationis  sit 
idem,  adeoque  in  eo  casu  transferenda  sunt  ad  centrum  percussionis,  qucecunque  pro  centro 
oscillationis  sunt  demonstrata. 

Si   qua  massa  sit  j™.  Corollarium  III.     Si  aliqua  massa  iaceat  extra  eiusmodi  planum  pertinens  ad  aliam 

extra   :     discnmen  ?  •      -i  •    T>  •  T>  A        j  •      •      j  •  *    L  •*  Z. 

centri  oscillationis,  quampiam  ;    erit  ibi  ra  minor,  quam  r  A,  adeoque  centrum  percussionis  aistabit  minus  ab 

a    centro    percus-  axe  rotationis,  quam  distet  centrum  oscillationis. 

Slonis-  r  AvPrf2 

Formula  deducts  114.  Corollarium  IV.     In  formula  generali  PG  =  V-        *::     habetur   Pa2  =  PG2  + 

pro    pluribus    aliis  M  X  GP 

theorematis.  Gat  _  2pg  x  Q^      pQrro  y_  ^  x  aPQ  X  Ga  evanescit    ob    evanescentem  /.  A  X  Ga,  & 


. 

deducuntur  sequentia  Theoremata  affinia  similibus  pertinentibus  ad  centrum  oscillationis 
deductis  in  ipso  opere. 

Theorema  de  posi-  115.  Si  impressio  ad  sistendum  motum  fiat  in  recta  perpendiculari  axi  rotationis  transeunte 

tatis*  centri  gravi"  per  centrum  gravitatis,  centrum  gravitatis  jacet  inter  centrum  percussionis,  W  axem  rotationis. 

Nam  PQ  evasit  major  quam  PG. 

Theorema      de  n  6.  Productum  sub  binis  distantiis  illius  ab  his  est  constant,  ubi  axis  rotationis  sit  in 

an^  TOoductoantI~  fodem  piano  quovis  transeunte  per  centrum  gravitatis  cum  eadem  directione  in  quacunque  distantia 

,     ~,~       f.AxGa*      .     ^n  ^,  -or*       J.Ax*Ga* 
ab  ipso  centro  gravitatis.     Nam  ob  GQ  =  *-*  -  pp-  erit  GQ  X  PG  = 

deductum"1   inde  HJ.  In  eo  casu  punctum  axis  pertinens  ad  id  planum,  &  centrum  percussionis  recipro- 

cantur  ;    cum  nimirum  productum  sub  binis  eorum  distantiis  a  constants  centro  gravitatis  sit 
constans. 

Axe  rotationis  ab-  1  1  8.  Abeunte  axe  rotat  ionis  in  infinitum,  ubi  nimirum  totum  systema  movetur  tantummodo 

eunte  in  infinitum.  mgfu  paraiiei0  centrum  percussionis  abit  in  centrum  gravitatis.     Nam  altera  e  binis  distantiis 

centrum  p  crcussio~  /  •       •    *•    *  •%    *  i  T>  •  *  j  * 

nis  abire  in  cent-  excrescente  in  infinitum,  debet  altera  evanescere.     rorro  is  casus  accidit  semper  etiam, 
rum  gravitatis.         ^  omnes  massae  abeunt  in  unum  punctum,  quod  erit  turn  ipsum  gravitatis  centrum  to- 
[305]-tius  systematis,  &  progredietur  sine  rotatione  ante  percussionem. 

si  axis   rotationis  nq    Abeunte  axe  rotationis  in  centrum  gravitatis,  nimirum  quiescente  ipso  gravitatis 

transeat     per   cen-  ........  •«  •  ?•  • 

trum    gravitatis,  centra,  centrum  percussionis  abit  in  infinitum,  nee  ulla  percussione  appiicata  unico  puncto  motus 
motum  sisti     non  s-st^  potestt     Nam  e  COntrario  altera  distantia  evanescente,  altera  abit  in  infinitum. 

osse  •  * 


posse 


Centri  percussionis  I2O    Corollarium  V.     Centrum  percussionis  debet  jacere  in  recta  perpendiculari  ad  axem 

positio  notabihs.  t.       .,...'  ••>        r^-, 

rotationis  transeunte  per  centrum  gravitatis.  Id  evmcitur  per  quartum  e  supenonbus  ineo- 
rematis.  Solutio  problematis  adhibita  exhibet  solam  distantiam  centri  percussionis  ab  axe 
illo  rotationis.  Nam  demonstratio  manet  eadem,  ad  quodcunque  planum  perpendiculare 

(i)  Facile  deducitur  ex  hoc  primo  corollario,  ad  habendum  centrum  percussionis  massarum  utcunque  dispersarum  satis 
esse  singulas  massas  reducere  ad  rectam  transeuntem  per  centrum  gravitatis,  W  perpendicularem  axi  rotationis  per  rectas 
ipsi  axi  perpendiculares,  W  invenire  massarum  ita  reductarum  centrum  oscillationis,  habito  puncto  rotationis  pro  puncto 
suspensionis  ;  id  enim  erit  ipsum  centrum  percussionis  qutesitum.  Nam  distantite  ab  ipso  piano  perpendiculari  illi  recite, 
quarum  distantiarum  fit  mentio  in  hoc  corollario,  manent  eeelem  in  ejusmodi  translatione  massarum,  W  evadunt  distantia 
a  puncto  suspensionis.  Theorema  autem  post  substitutionem  distantiarum  a  puncto  suspensionis  pro  Us  ipsis  distantiis  ab 
illo  piano  exhibet  ipsam  formulam  distantia:  centri  oscillationis  a  puncto  suspensionis,  qua  habetur  num.  334.  Hinc  autem 
consequitur  generalis  reciprocatio  puncti  rotationis,  W  centri  percussionis,  ac  alia  plura  in  sequentibus  deducta  multo 
immediatius  deducuntur  e  proprietatibus  centri  oscillationis  jam  demonstratis. 


LETTER  TO  FR.   SCHERFFER  451 

111.  Collar  ary  I.     Since  aP  is  equal  to  the  perpendicular  distance  of  A  from  a  plane  A  Theorem  derived 
passing  through  P  perpendicular  to  the  straight  line  PG,  we  have  the  following  theorem.  from  this  formula. 
The  distance  of  the  centre  of  percussion  from  the  axis  of  rotation  in  a  straight  line  perpendicular 

to  it  passing  through  the  centre  of  gravity,  will  be  obtained  by  multiplying  each  mass  by  the 
square  of  its  perpendicular  distance  from  a  plane  passing  through  the  axis  of  rotation,  y  perpen- 
dicular to  the  straight  line  ;  W  then  dividing  the  sum  of  all  such  products  by  the  product  of 
the  sum  of  all  the  masses  multiplied  by  the  perpendicular  distance  of  the  common  centre  of  gravity 
from  the  same  plane.  (') 

112.  Corollary  II.     If  the  masses  lie  in  any  the  same  single  plane  passing  through  the  Dedwstfcmol   «w 

e    °  a 


axis,  A  &  a  coincide,  &  therefore  the  distances  fa  become  the  distances  of  the  masses  from 

the  axis.     Hence,  in  this  case,  the  formula  here  found  for  the  centre  of  percussion  agrees  the  same  plane 

in  every  way  with  the  formula  found  for  the  centre  of  oscillation  ;    thus  the  two  centres 

are  the  same  point,  if  the  axis  of  rotation  is  the  same.     Hence,  in  this  case,  everything  that 

has  been  proved  for  the  centre  of  oscillation,  holds  good  for  the  centre  of  percussion. 

113.  Corollary  III.     If  any  mass  lies  outside  the  plane  belonging  to  any  other,  then  if  any  mass  does 
Pa  will  be  less  than  PA  ;  hence,  the  centre  of  percussion  will  be  at  a  less  distance  from  the  axis  **  in 


is    a 


of  rotation  than  the  centre  of  oscillation.  distinction  between 

f  A  V  P/72  tne  centre  of  oscilla- 

114.  Corollary  IF.     In  the  general  formula  PQ  =   J',        "  we   have    P^  =  PG2  tion  &  the  centre  of 

M  X  GP  percussion. 


+  Ga*  —  2  PQ  X  Ga.    Also,  the  sum/.A  XzPQxGa  vanishes,  since/.A  X  Ga  vanishes  5 

&/.A  X  PG2/(M  X  PG)=  PG.     Hence  we  have  theorems. 


PO      pr  i  /•  A  x  Ga2    .  rf)     /.A  x  Ga* 
-  M  x  PG  '  *  MxPG- 

From  this  can  be  deduced  the  following  theorems  like  to  similar  theorems  pertaining  to  the 
centre  of  oscillation  deduced  in  the  work  itself. 

115.  If  the  impressed  force  applied  for  the  purpose  of  checking  motion  is  in  a  straight  line  Theorem 

perpendicular  to  the  axis  of  rotation  &  passing  through  the  centre  of  gravity,  the  centre  of  gravity  J£f  th®e 

will  lie  between  the  centre  of  percussion  W  the  axis  of  rotation.     For  PQ  is  greater  than  PG.  gravity. 

1  1  6.   The  product  of  the  two  distances  of  the  former  from  the  two  latter  is  constant,  when  Theorem 

the  axis  of  rotation  is  in  any  the  same  plane  passing  through  the  centre  of  gravity,  the  direction  0"8  ^ 

of  measurement  being  the   same  for   any   distance  from   the   centre   of  gravity.     For,  since  tances. 


CQ  -  -'  <h«efore  GQ  x  PG  - 


117.  In   that  case,  the  point  on   the  axis  corresponding  to  the  plane   y  the  centre  of  Corollary    derived 

percussion  will  be  interchangeable  ;  for,  the  product  of  their  two  distances  from  a  constant  centre  from  this' 
of  gravity  is  constant. 

1  1  8.  //  the  axis  of  rotation  goes  off  to  infinity,  that  is  to  say,  when  the  whole  system  is  trans-  if    the    axis     of 

lated  with  simply  a  parallel  motion,  the  centre  of  percussion  will  become  coincident  with  the  centre  l^i°y,  tTe^centre 

of  gravity.     For,  if  one  of  the  two  distances  increases  indefinitely,  the  other  must  become  of  percussion  win 

evanescent.     Also,  this  will  always  happen,  when  all  the  masses  coincide  at  a  single  point  ;  wlthTh 

this  point  will  then  be  the  centre  of  gravity  of  the  whole  system,  &  it  will  be  moving  without  gravity. 
rotation  before  percussion. 

119.  //  the  axis  of  rotation  passes  through  the  centre  of  gravity,  the  centre  of  percussion  "^^ 

passes  of  to  infinity,  W"  the  motion  cannot  be  checked  by  any  blow  applied  at  a  single  point,  through  the  cente 

For,  on  the  contrary,  when  the  finite  distance  vanishes,  the  other  distance  must  become  of  ,  .  s^'ty-     *e 

.    ,.    .  J  motion    cannot  be 

innnite.  checked. 

120.  Corollary  F.     The  centre  of  percussion  must  lie  in  the  straight  line  perpendicular  A   noteworthy 
to  the  axis  of  rotation  &  passing  through  the  centre  of  gravity.     This  is  proved  by  the  fourth  centre^of  °percuse 
of  the  theorems  given  above.     The  method  of  solution  of  the  problem  that  was  employed  sion. 

shows  the  unique  distance  of  the  centre  of  percussion  from  the  axis  of  rotation.     For,  the 
demonstration  remains  the  same,  no  matter  to  what  plane  perpendicular  to  the  axis  all  the 

(i)  /*  is  easily  deduced  from  this  first  corollary  that,  in  order  to  obtain  the  centre  of  percussion  of  any  masses 
however  arranged,  it  is  sufficient  to  reduce  each  of  the  masses  to  a  straight  line  •passing  through  the  centre  of  gravity 
y  perpendicular  to  the  axis  of  rotation,  by  means  of  straight  lines  perpendicular  to  the  axis  ;  &  then  to  find  the  centre 
of  oscillation  of  the  masses  thus  reduced,  the  point  of  rotation  being  taken  as  the  point  of  suspension.  This  will  be  the 
centre  of  percussion  required.  For,  the  distances  from  the  plane  perpendicular  to  the  straight  line,  such  as  are  men- 
tioned in  this  corollary,  remain  the  same  in  this  kind  of  translation  of  the  masses  y  become  the  distances  from  the 
point  of  suspension.  Moreover,  the  theorem,  after  the  substitution  of  the  distances  from  the  point  of  suspension  for  the 
distances  from  the  plane,  gives  the  same  formula  for  the  distance  of  the  centre  of  oscillation  from  the  point  of  suspension, 
which  was  obtained  in  Art.  334.  From  it  also  there  follows  the  general  reciprocity  of  the  point  of  rotation  y  the  centre 
of  percussion  ;  y  many  other  things  deduced  in  what  follows  can  be  more  easily  derived  from  the  properties  of  the  centre 
of  oscillation  already  proved. 


45  1 


PHILOSOPHIC  NATURALIS  THEORIA 


axi  reducantur  per  rectas  ipsi  axi  parallelas  &  massae  omnes,  &  ipsum  centrum  gravitatis 
commune,  adeoque  inde  non  haberetur  unicum  centrum  percussionis,  sed  series  eorum 
continua  parallela  axi  ipsi,  quae  abeunte  axe  rotationis  ejus  directionis  in  infinitum,  nimirum 
cessante  conversione  respectu  ejus  directionis,  transit  per  centrum  gravitatis  juxta  id 
Theorema.  Porro  si  concipiatur  planum  quodvis  perpendiculare  axi  rotationis,  omnes 
massae  respectu  rectarum  perpendicularium  axi  priori  in  eo  jacentium  rotationem  nullam 
habent,  cum  distantiam  ab  eo  piano  non  mutent,  sed  ferantur  secundum  ejus  directionem, 
adeoque  respectu  omnium  directionum  priori  axi  perpendicularium  jacentium  in  eo  piano 
res  eodem  modo  se  habet,  ac  si  axis  rotationis  cujusdam  ipsas  respicientis  in  infinitum  distet 
ab  earum  singulis,  &  proinde  respectu  ipsarum  debet  centrum  percussionis  abire  ad  distan- 
tiam, in  qua  est  centrum  gravitatis,  nimiium  jacere  in  eo  planorum  parallelorum  omnes 
ejusmodi  directiones  continentium,  quod  transit  per  ipsum  centrum  gravitatis  :  adeoque 
ad  sistendum  penitus  omnem  motum,  &  ne  pars  altera  procurrat  ultra  alteram,  &  earn 
vincat,  debet  centrum  percussionis  jacere  in  piano  perpendiculari  ad  axem  transeunte  per 
centrum  gravitatis,  &  debent  in  solutione  problematis  omnes  massae  reduci  ad  id  ipsum 
planum,  ut  praestitimus,  non  ad  aliud  quodpiam  ipsi  parallelum  :  ac  eo  pacto  habebitur 
aequilibrium  massarum,  hinc  &  inde  positarum,  quarum  ductarum  in  suas  distantias  ab 
eodem  piano  summae  hinc,  &  inde  acceptae  aequabuntur  inter  se.  Porro  eo  piano  ad  solu- 
tionem  adhibito,  patet  ex  ipsa  solutione,  centrum  percussionis  jacere  in  recta  perpendiculari 
axi  ducta  per  centrum  gravitatis  :  jacet  enim  in  recta,  quae  a  centro  gravitatis  ducitur  ad 
illud  punctum  in  quo  axis  id  planum  secat,  quae  recta  ipsi  axi  perpendicularis  toti  illi  piano 
perpendicularis  esse  debet. 


impactus   m  cen-  I2i.  Corollarium  VI.      Impactus  in  centro  percussionis  in  corpus  externa  vi  eius  motum 

trum      percussionis       .  .,  .  .     .        '.  .  .      .  '.....' 

qui  sit.  sistens  est  idem,  qui  esset,  si  singulce  masses  incurrerent  in  ipsum  cum  suis  velocitatibus  respecti- 

[$o6\-vis  redactis  ad  directionem  perpendicularem  piano  transeunti  per  axem  rotationis,  £ff 
centrum  gravitatis,  sive  si  massarum  summa  in  ipsum  incurreret  directione,  &  velocitate  motus, 
qua  fertur  centrum  gravitatis. 

Demonstratio    pri-  122.  Patet  primum,  quia  debet  in  Q  haberi  vis  contraria    directioni    illius    motus 

mae  partis.  perpendicularis  piano   transeunti  per  axem,  &  PG,  par  extinguendis    omnibus  omnium 

massarum  velocitatibus  ad  earn  directionem  redactis,  quae  vis  itidem  requireretur,  si  omnes 

massse  eo  immediate  devenirent  cum  ejusmodi  velocitatibus. 

p 

patet  secundum  ex  eo,  quod  velocitas  ilia  pro  massa  A  sit  pf)  x    V,  adeoque 


Demonstratio 


motus 


A  x 


X  V,  quorum  motuum  summa  est 


M  x  PG 


X  V. 


PG 

Est  autem  .g-^-J  x  V, 


n  —          ,  -        n 

velocitas  puncti  G,  quod  punctum  movetur  solo  motu  perpendiculari  ad  PG,  adeoqiie  si 
massa  totalis  M  incurrat  in  Q  cum  directione,  &  celeritate,  qua  fertur  centrum  gravitatis  G, 
faciet  impressionem  eandem. 


impressio  ubi  fieri  124.  Corollarium  VII.  Potest  motus  sisti  impressione  facta  etiam  extra  rectam  PG, 

trum  percussionis  seu  extra  planum  transiens  per  axem  rotationis,  &  centrum  gravitatis,  nimirum  si  impressio 
cum  eodem  effectu.  flat  in  quodvis  punctum  rectce  eidem  piano  perpendicularis,  &  transeuntis  per  Q,  directione 

rectee  ipsius.     Nam  per  nexum  inter  id  punctum,  &  Q  statim  impressio  per   earn  rectam 

transfertur  ab  eo  puncto  ad  ipsum  Q. 


Motus     communi- 
qU1 


quiescenti. 


125.  Corollarium  VIII.  Contra  vero  si  imprimatur  dato  cuidam  puncto  systematis 
sys1temati  quiescentis  vis  qucedam  matrix  ;  invenietur  facile  motus  inde  communicandus  ipsi  systemati. 
Nam  ejusmodi  motus  erit  is,  qui  contrario  aequali  impactu  sisteretur.  Determinatio  autem 
regressu  facto  per  ipsam  problematis  solutionem  erit  hujusmodi.  Centrum  gravitatis 
commune  movebitur  directione,  qua  egit  vis,  &  velocitate,  quam  ea  potest  imprimere  massae 
totius  systematis,  quae  ad  earn,  quam  potest  imprimere  massae  cuivis,  est  ut  haec  posterior 
massa  ad  illam  priorem,  &  si  vis  ipsa  applicata  fuerit  ad  centrum  gravitatis,  vel  immediate, 
vel  per  rectam  tendentem  ad  ipsum  ;  systema  sine  ulla  rotatione  movebitur  eadem  veloci- 
tate :  sin  autem  applicetur  ad  aliud  punctum  quodvis  directione  non  tendente  ad  ipsum 
centrum  gravitatis,  praeterea  habebitur  conversio,  cujus  axis,  &  celeritas  sic  invenietur. 
Per  centrum  gravitatis  G  agatur  planum  perpendiculare  rectae,  secundum  quam  fit  impactus, 
&  notetur  punctum  Q,  in  quo  eidem  piano  occurrit  eadem  recta.  Per  ipsum  punctum  G 
ducatur  in  eo  piano  recta  perpendicularis  ad  QG,  quae  erit  axis  quaesitus.  Per  punctum  Q 
concipiatur  alterum  planum  perpendiculare  rectae  GQ,  ca-[307]-piantur  omnes  distantiae 
perpendiculares  omnium  massarum  A  ab  ejusmodi  piano,  aequales  nimirum  suis  aQ  : 


LETTER  TO  FR.   SCHERFFER  453 

masses  &  their  common  centre  of  gravity  are  reduced  by  straight  lines  parallel  to  the  axis. 
Thus,  from  it,  we  should  not  obtain  a  single  centre  of  percussion,  but  a  continuous  series 
of  them  parallel  to  the  axis  ;  &  this,  when  the  axis  of  rotation  goes  off  to  infinity  for  this 
direction,  that  is,  when  turning  ceases  for  this  direction,  will  pass  through  the  centre  of 
gravity,  according  to  the  theorem.  Further,  if  any  plane  perpendicular  to  the  axis  of 
rotation  is  taken,  all  the  masses  have  no  rotation  with  regard  to  straight  lines  perpendicular 
to  the  former  axis  which  lie  in  the  plane  ;  for  they  will  not  change  their  distances  from 
that  plane,  but  are  carried  in  its  direction.  Hence,  with  regard  to  all  directions  perpendi- 
cular to  the  former  axis  which  lie  in  that  plane,  the  matter  comes  out  in  the  same  way  ;  &, 
if  the  axis  of  rotation  for  any  one  of  the  former  is  infinitely  distant  from  each  of  the  latter, 
and  therefore  with  respect  to  the  former,  the  centre  of  percussion  has  to  pass  to  that  distance 
at  which  is  the  centre  of  gravity,  that  is  to  say,  has  to  lie  in  that  one  of  the  parallel  planes  con- 
taining all  such  directions,  which  passes  through  the  centre  of  gravity.  Thus,  to  stop  all 
motion  entirely,  &  to  prevent  one  part  outrunning  another  part  &  overcoming  it,  the  centre 
of  percussion  must  lie  in  a  plane  perpendicular  to  the  axis  &  passing  through  the  centre  of 
gravity  ;  &  in  the  solution  of  the  problem,  all  the  masses  are  bound  to  be  reduced  to  that 
plane,  as  we  have  shown,  &  not  to  any  other  that  is  parallel  to  it.  In  this  way,  we  shall 
obtain  equilibrium  of  the  masses,  situated  on  either  side  of  it  ;  &  the  sums  of  these  multiplied 
by  their  distances  from  this  plane,  taken  together  on  one  side  &  on  the  other,  will  be  equal 
to  one  another.  Moreover,  if  this  plane  is  used  for  the  solution,  it  is  clear  from  the  solution 
itself,  that  the  centre  of  percussion  lies  in  a  straight  line  perpendicular  to  the  axis,  drawn 
through  the  centre  of  gravity.  For,  it  will  lie  in  the  straight  line  that  is  drawn  from  the 
centre  of  gravity  to  that  point  in  which  the  axis  cuts  the  plane,  &  this  straight  line  must 
be  perpendicular  to  the  axis,  since  the  axis  is  perpendicular  to  the  whole  of  the  plane. 

121.  Corollary  VI.      The  impact  at  the  centre  of  percussion  on  a  body  by  an  external  The  nature  of  the 
force,  which  checks  its  motion,  is  the  same  as  we  should  have,  if  each  mass  were  to  collide  with 

it  with  its  velocity  resolved  in  the  direction  perpendicular  to  the  plane  passing  through  the  axis 
of  rotation  &  the  centre  of  gravity  ;  or  if  the  sum  of  the  masses  collided  with  it  with  the  direction 
6f  velocity  of  motion,  with  which  the  centre  of  gravity  is  moving. 

122.  The  first  part  is  evident,  because  there  must  be  at  Q  a  force  opposite  in  direction  Proof  of  the  first 
to  the  motion  perpendicular  to  the  plane  passing  through  the  axis  &  PG,  capable  of  destroying  part' 

all  the  velocities  of  all  the  masses  resolved  in  that  direction  ;  &  this  force  would  also  be 
required,  if  all  the  masses  collided  with  it  directly  with  such  velocities. 

123.  The  second  part  is  evident  from  the  fact  that  the  velocity  for  the  mass  A  is  Proof  of  the  second 

part. 

X   V  ;    &   thus,  the    motion  is  —  pf\-~-  X    V  ;     &    the   sum   of   these   motions    is 


1VT  v  Pf*  Pf 

-  X  V.     But  pp-  X  V  is  the  velocity  of  the  point  G,  &  the  sole  motion  of  this 


point  is  perpendicular  to  PG  ;  &  thus,  if  the  total  mass  M  collided  with  Q  with  the 
direction  &  speed  with  which  the  centre  of  gravity  G  moves,  it  would  produce  the  same 
effect. 

1  24.  Corollary  VII.     The  motion  may  be   checked  even  by   a  blow  applied  without  the  when  the  blow  can 
straight  line  PG,  or  without  the  plane  passing  through  the  axis  of  rotation  W  the  centre  of  gravity  ;  {j^Sntre  £f  yperd 
that  is,  if  it  is  applied  at  any  point  of  a  straight  line  perpendicular  to  the  same  plane,  &  passing  cussion    with  the 
through  Q,  in  the  direction  of  this  straight  line.     For,  through  the  connection  between  that  same  effect' 
point  &  Q,  the  blow  is  immediately  transferred  along  the  straight  line  from  the  point  to  Q 
itself. 

125.  Corollary  VIII.  On  the  other  hand,  if  any  motive  force  is  impressed  upon  any  Motion  communi- 
given  point  of  a  system  at  rest,  it  is  easy  to  find  the  motion  thereby  communicated  to  the  system.  System  at  rest.  ° 
For  such  motion  will  be  that  which  would  be  checked  by  an  equal  &  opposite  blow.  The 
determination  of  the  motion,  made  by  retracing  our  steps  through  the  solution  of  that 
problem,  would  proceed  as  follows.  The  common  centre  of  gravity  will  be  moved  in  the 
direction  in  which  the  force  acts,  &  with  a  velocity  which  it  can  give  to  the  mass  of  the  whole 
system  ;  this  velocity  is  to  that  which  it  could  give  to  any  mass  as  the  latter  mass  is  to  the 
former.  If  the  force  were  applied  at  the  centre  of  gravity,  either  directly,  or  along  a  straight 
line  tending  to  it,  then  the  system,  without  any  rotation,  would  move  with  the  same  velocity. 
But  if  it  were  applied  at  any  other  point  in  a  direction  not  tending  towards  the  centre  of 
gravity,  we  should  have  in  addition  a  rotation,  of  which  the  axis  &  the  velocity  will  be  found 
thus.  Let  a  plane  be  drawn  through  the  centre  of  gravity  G  perpendicular  to  the  straight 
line  along  which  the  blow  is  impressed,  &  let  the  point  in  which  the  straight  line  meets  this 
plane  be  denoted  as  the  point  Q.  Through  G  draw  in  this  plane  a  straight  line  perpendicular 
to  QG  ;  this  will  be  the  axis  required.  Draw  another  plane  through  the  point  Q,  per- 
pendicular to  the  straight  line  QG  ;  take  all  the  perpendicular  distances  of  all  the  masses  A 


454  PHILOSOPHIC  NATURALIS  THEORIA 

singularum  quadrata  ducantur  in  suas  massas,  &  factorum  summa  dividatur  per  summam 
massarum,  turn  in  recta  GQ  producta  capiantur  GP  aequalis  ;  ei  quoto  diviso  per  ipsam 
QG,  &  celeritas  puncti  P  revolventis  circa  axem  inventum  in  circulo,  cujus  radius  GP,  erit 
aequalis  celeritati  inventae  centri  gravitatis,  directio  autem  motus  contraria  eidem.  Unde 
habetur  directio,  &  celeritas  motus  punctorum  reliquorum  systematis. 

Demonstrate.  126.  Patet  constructio  ex  eo,  quod  ita  motu  composito  movebitur  systema  circa  axem 

immotum  transeuntem  per  P,  qui  motus  regressu  facto  a  constructione  tradita  ad  inventi- 
onem  praemissam  centri  percussionis  sisteretur  impressione  contraria,  &  aequali  impressioni 
datae. 

Aditus  ad  perquisi-  127-  Scholium.     Hoc    postremo    corollario     definitur     motus     vi    externa    impressus 

tiones      uiteriores  systemati  quiescenti.     Quod  si  jam  systema  habuerit  aliquem  motum    progressivum,  & 

motu      impresso     <        ,  •>..•',.  ,,^-  .  .  . 

systemati  moto.  circularem,  novus  motus  externa  vi  inductus  juxta  corollanum  ipsum  componendus  erit 
cum  priore,  quod,  quo  pacto  fieri  debeat,  hie  non  inquiram,  ubi  centrum  percussionis 
persequor  tantummodo.  Ea  perquisitio  ex  iisdem  principiis  perfici  potest,  &  ejus  ope 
patet,  aperiri  aditum  ad  inquirendas  etiam  mutationes,  quae  ab  inaequali  actione  Solis, 
&  Lunae  in  partes  supra  globi  formam  extantes  inducuntur  in  diurnum  motum,  adeoque 
ad  definiendam  ex  genuinis  principiis  prascessionem  aequinoctiorum,  &  nutationem  axis  : 
sed  ea  investigatio  peculiarem  tractionem  requirit. 

Transitus  ad  aliam  I28-  Interea   gradum  hie    faciam   ad   aliam  notionem   quandam   centri  percussionis, 

notionem  ejus  cen-  nihilo  minus,  imo  etiam  magis  aptam  ipsi  nomini.     Ad  earn  perquisitionem  sic  progrediar. 

tri. 

Probiema  conti-  129.  Problema.     Si  systema  datum  gyrans  data  velocitate  circa  axem  datum  externa  vi 

nens  hanc  ideam.      immotum  incurrat  in  dato  suo  -puncto  in  massam  datam,  delatam  velocitate  data  in  directions 

motus  •puncti  ejusdem,  quam  massam  debeat  abripere  secum  ;   queeritur  velocitas,  quam  ei  masses 

im-primet,  &  ipsum  systema  retinebit  post  im-pactum. 

Soiutio  :    formula  130  Concipiatur  totum  systema  projectum  in  planum  perpendiculare  axi  rotationis 

continents  motum  transiens  per  centrum  gravitatis  G,  in  quo  piano  punctum  conversionis  sit  P,  massa  autem 

massse     in      quam    .  —t—  .      **.**?,  ,?.r  •"••  IT  i  •  n 

incidit,    &     suum  in  recta  PG  in  Q.     Velocitas  puncti  cujusvis  systematis,  quod  distet  ab  axe  per  intervallum 
reiiquum.  _  I?  ante  incursum  sit  =  a,  velocitas  ab  eodem  amissa  sit  =  x,  adeoque  velocitas  post 

impactum  =  a  —  x,  velocitas  autem  massae  Q  ante  impactum  sit  =  PQ  X  b.  Erit  ut  I 
ad  AP,  ita  x  ad  velocitatem  amissam  a  massa  A,  quae  erit  AP  X  x.  Erit  autem  ut  I  ad 
a  —  x  ita  PQ  ad  velocitatem  residua  m  in  puncto  systematis  Q,  quas  net  PQ  X  (a  —  #), 
&  ea  erit  itidem  velocitas  massse  Q  post  [308]  impactum,  adeoque  massa  Q  acquiret  veloci- 
tatem PQ  X  (a  —  b  —  x),  sive  posito  a  —  b=c,  habebitur  PQ  X  (c  —  x).  Porro  ex 
mutuo  nexu  massae  A  cum  P,  &  Q  erit  Q  X  PQ  ad  A  X  AP,  ut  effectus  ad  velocitatem 

A  X  AP2 

pertinens  in  A  =  AP  X  x  ad  effectum  in  Q  =  j~—  \  np   X  x.          Summa  horum  effec- 

(j.  X  vj.r 

tuum  provenientium  e  massis  omnibus  erit  aequalis  velocitati  acquisitae  in  Q.      Nimirum 

X     c,    * 


x  =  _  Q  X  °-F*  -  x  c  .    Dato  autem  x  datur  a  —  x,  &  is  valor  ductus  in  distantiam 

/.  A  x  AP2  +  Q  x  QP2 
puncti  cujusvis  systematis,  vel  etiam  massae  Q,  exhibebit  velocitatem  quaesitam.     Q.E.F. 


Casus  particulares,  1 31.  Scholium.     Formula  habet  locum  etiam  pro  casu,  quo  massa  Q  quiescat,  vel  quo 

ad  quos  appiicari  feratur  contra  motum  systematis,  dummodo  in  primo  casu  fiat  b  =  o,  &  c  =  a,  ac  in  secundo 

valor  b  mutetur  in  negativum,  adeoque  sit  c  =  a  +  b.     Posset  etiam  facile  appiicari  ad 

casum,  quo  in  conflictu  ageret  elasticas  perfecta  vel  imperfecta.      Determinatio  tradita 

exhiberet  partem  effectus  in  collisione  facti  tempore  amissas  figurae,  ex  quo  effectus  debitus 


LETTER  TO  FR.   SCHERFFER  455 

from  this  plane,  each  equal  to  the  corresponding  aQ  ;  multiply  the  square  of  each  of  these 
by  the  corresponding  mass,  &  divide  the  sum  of  all  the  products  by  the  sum  of  the  masses. 
Then  in  the  straight  line  QG  produced  take  GP  equal  to  this  quotient  divided  by  QG. 
The  velocity  of  the  point  P  rotating  in  a  circle  about  the  axis  which  has  been  found,  of 
which  the  radius  is  GP,  will  be  equal  to  the  velocity  of  the  centre  of  gravity  which  has  also 
been  found,  but  the  direction  of  the  motion  will  be  in  the  opposite  direction.  From  this, 
we  have  the  direction  &  the  velocity  for  all  the  other  points  of  the  system. 

126.  The  correctness  of  the  construction  is  evident  from  the  fact  that  in  this  way  Demonstration. 
the  system  will  move  with  a  compound  motion  in  a  circle  about  a  motionless  axis  passing 

through  P  ;  &  this  motion,  by  retracing  our  steps  from  the  construction  for  finding  the 
centre  of  percussion,  already  given,  would  be  checked  by  a  blow  equal  &  opposite  to  the 
given  blow. 

127.  Scholium.      In  the  last  corollary  the  motion  impressed  by  an  external  force  on  The   way  is  open 
a  system  at  rest  is  determined.     But  if  now  the  system  should  have  some  motion,  progressive  gationf  w^e^mo- 
&  circular,  the  new  motion  induced  by  the  external  force  in  accordance  with  the  corollary  tion  is  impressed  on 
will  have  to  be  compounded  with  what  it  already  has.     I  do  not  inquire  here,  how  this  will  a  movms  system. 
happen,  for  here  I  am  only  concerned  with  the  centre  of  percussion.     The  investigation  can 

be  carried  out  by  means  of  the  very  same  principles  ;  &  by  the  help  of  this  investigation, 
it  is  clear  that  the  door  would  be  opened  also  for  the  investigation  of  the  variations  which  are 
induced  in  the  daily  motion  by  the  unequal  actions  of  the  Sun,  &  of  the  Moon,  on  parts  of 
the  Earth  that  jut  out  beyond  the  figure  of  the  sphere  ;  &  thus  for  determining  from 
real  principles  the  precession  of  the  equinoxes  &  the  nutation  of  the  axis.  But  this  investiga- 
tion requires  a  special  treatise. 

128.  Meanwhile,  I  will  now  go  on  to  another  idea  of  the  centre  of  percussion,  which  Passing  on  to  an- 
is  no  less,  nay  it  is  even  more,  fit  to  have  that  name  given  to  it.     To  this   investigation  oth"  ldea  of  thls 

'   .     '  ,       ....       .  centre. 

I  proceed  in  the  following  manner. 

129.  Problem.     //  a  given  system,  gyrating  with  given  velocity  about  a  given  axis,  not  Problem   embody- 
acted  upon  by  an  external  force,  collides  at  a  given  point  of  itself  with  a  given  mass,  which 

is  moving  with  a  given  velocity  in  the  direction  of  the  motion  of  this  point,  the  mass  being  of 
necessity  borne  along  with  the  system  ;  it  is  required  to  find  the  velocity  impressed  on  the  mass, 
y  retained  by  the  system  after  impact. 

130.  Suppose  that  the  whole  system  is  projected  on  a  plane  perpendicular  to  the  axis  Solution  ;  formulae 
of  rotation  passing  through  the  centre  of  gravity  G  ;   in  this  plane  let  the  point  of  rotation  containing      the 

,       .°      ,  .    .      ..  °   —.  .^         XT  f          i      •  r  •  r     i        motion  of  the  mass 

be  P,  &  let  the  mass  be  in  the  straight  line  PG  at  Q.     Let  the  velocity  of  any  point  of  the  with  which  it  col- 
system,  whose  distance  from  the  axis  is  unity,  before  the  impact  be  a,  &  let  the  velocity  Udes-    a  n  d    *  n  e 

i         i       •     i  t      •         f         •  MI  u  AI       i        i          i      •         r    i       motion   left  in    it- 

lost  by  it  be  x  ;  &  thus,  the  velocity  alter  impact  will  be  a—  x.  Also  let  the  velocity  of  the  seif. 
mass  at  Q  before  impact  be  PQ  X  b.  Then,  as  I  is  to  AP  so  is  x  to  the  velocity  lost  by  the 
mass  at  A,  which  will  therefore  be  AP  X  x.  Also,  as  I  is  to  a  —  x  so  is  PQ  to  the  velocity 
that  remains  in  the  point  Q  of  the  system  ;  &  therefore  this  is  PQ  X  (a  —  x)  ;  this  will 
also  be  the  velocity  of  the  mass  Q  after  impact.  Hence,  the  mass  Q  will  acquire  a  velocity 
PQ  X  (a  —  b  —  x}  ;  or,  if  we  put  a  —  b  =  c  ,  it  will  be  PQ  X  (c  —  x).  Further,  from 
the  mutual  connection  between  the  mass  A  &  P  &  Q,  we  shall  have  the  ratio  of  Q  X  PQ  to 
A  X  AP  equal  to  that  of  the  effect  pertaining  to  the  velocity  at  A,  which  is  equal  to 

A  x  AP* 

AP  X  x,  to  the  effect  at  Q,  which  is  therefore  equal  to  _  X  x.     The    sum   of 

Q  X  Qir 

these  effects,  arising  from  all  the  masses,  will  be  equal  to  the  velocity  acquired  at  Q.  That 
is  to  say,  we  have 


Q  x  QPZ 
1  *  =  /.Ax  AP'  +  QxQP2 

But,  if  we  are  given  x,  we  are  also  given  a  —  x  ;  and  this  value,  multiplied  by  the  distance 
of  any  point  of  the  system,  or  also  that  of  the  mass  Q,  will  give  the  velocity  required.     Q.E.F. 

131.  Scholium.     The  formula  holds  good  even  when  the  mass  Q  is  at  rest,  or  when  Particular  cases  to 
it  moves  in  the  opposite  direction  to  the  system  ;  so  long  as,  in  the  first  case,  b  is  made  ^hlj£d  li    can  be 
equal  to  zero,  or  c  =  a  ;   &  in  the  second  case,  the  value  of  b  is  changed  from  positive  to 
negative,  so  that  c  =  a  -j-  b.     It  might  also  easily  be  applied  to  the  case  in  which  elasticity, 
either  perfect  or  imperfect,  would  take  a  part  in  the  collision.     The  determination  given 
would  represent  that  part  of  the  effect  of  the  collision  which  was  produced  during  the 
interval  of  time  corresponding  to  loss  of  shape  ;  &  from  this  the  proper  effect  for  the  whole 


456  PHILOSOPHIC  NATURALIS  THEORIA 

tempori  totus  collisionis  usque  ad  finem  recuperatae  figurae  colligitur  facile,  duplicando 
priorem,  vel  augendo  in  ratione  data  uti  fit  in  colKsionibus. 

ulterior  132.  Itidem  locum  habet  pro  casu,  quo  massa  nova  non  jaceat  in  Q  in  recta  PG,  sed 

in  quovis  alio  puncto  plani  perpendicularis  axi  transeuntis  per  G,  ex  quo  si  intelligatur 
perpendiculum  in  PG  ei  occurrens  in  Q  ;  idem  prorsus  erit  impactus  ibi,  qui  esset  in  Q, 
translata  actione  per  illam  systematis  rectam.  Qui  imo  si  Q  non  jaceat  in  eo  piano  perpen- 
diculari  ad  axem,  quod  transit  per  centrum  gravitatis,  sed  ubivis  extra,  res  eodem  redit, 
dummodo  per  id  punctum  concipiatur  planum  perpendiculare  axi  illi  immoto  per  vim 
externam  ad  quod  planum  reducatur  centrum  gravitatis,  &  qusevis  massa  A ;  vel  si  ipsa 
massa  Q  cum  reliquis  reducatur  ad  quodvis  aliud  planum  perpendiculare  axi.  Omnia 
eodem  recidunt  ob  id  ipsum,  quod  axis  externa  vi  immotus  sit.  Sed  jam  ex  generali 
solutione  problematis  deducimus  plura  Corollaria. 


Reiatio    ad     cen-  133.  Corollarium  I.     Si  distantia  centri  oscillationis  totius  systematis  ab  axe  P  dicatur 

trum  oscillationis.    R}  distantia  centri  gravitatis  G,  massa  tota  M,  habebitur 

_  _  Qx  PQ2  v       .  ro,  f_MxGxR  , 

~  MxGxR  +  QxPQ2       Ct      l3  9J  x~     Q  xPQ* 

f  A  x  AP2 
Patet    ex   eo,    quod    ex  natura    centri    oscillationis    habetur    R  =  -A^  -  -=-  ,  adeoque 

M  x  G 
/.  A  x  AP'  =  M  x  G  x  R. 


Expressio     veioci-  ,,.    Corollarium  II.     Velocitas  acquisita  a  massa  Q  erit  J^  *  G  *  R  *  PQ-    X  c. 

tatis  m  massa  sim-  M  X  G  X  R+  Q  X  PQ2 

phcior  ope  dims.  Q         pQl 

Est  enim  ea  velocitas  PQ  X  (e  -  x),  sive  PQ  (c  --  M  x  G  ^  R  +  Q  x  PQ«  X  f)»  1uod 
reductum  ad  eundem  denominatorem  elisis  terminis  contrariis  eo  redit. 


UbicoiHgendumes-  j-jr    Corollarium   III.     Si   manente   velocitate   circular!   systematis   tota   ejus    massa 

set  totum  systema  .    ,J  ....  .  .  i       .       „  ...      .  J   . 

ad  eandem  veioci-  concipiatur  collecta  m  unico  puncto  jacente  inter  centra  gravitatis,  &  oscillationis,  cujus 
tatem     impri-  distantia  a  puncto  conversionis  sit  media   geometrice   proportionalis   inter   distantias   reli- 

mendam  massae.  ,  .  ,          ,.  ,      .  , 

quorum  punctorum,  vel  m  eadem  distantia  ex  parte  opposita  ;  velocitas  eadem  impnmeretur 
novae  massae  in  quovis  puncto  sitae.  Tune  enim  abiret  in  illud  punctum  utrumque  centrum, 
&  valor  G  X  R  esset  idem,  ac  prius,  nimirum  aequalis  quadrate  ejus  distantiae  ab  axe,  quod 
quadratum  est  positivum  etiam,  si  distantia  accepta  ex  parte  opposita  fiat  negativa. 

in  quot,  &  quibus  136.  CorollariumlV.     Si  capiatur  hinc,  vel  inde  in  PG  segmentum,  quod  ad  distantiam 

massa^eandem  ^x  ejus  puncti  ab  axe  sit  in  subduplicata  ratione  massae  totius  systematis  ad  massam  Q  ;  ipsa 
impactu  veiotita-  massa  Q  in  -quatuor  distantiis  ab  axe,  binis  hinc,  &  binis  inde,  quarum  binarum  producta 
'  aequentur  singula  quadrato  ejus  segmenti,  acquiret  velocitatem  in  omnibus  eandem 
magnitudine,  licet  in  binis  directionis  contrariae,  &  ea  net  maxima,  ubi  ipsa  massa  sit  in 
fine  ejus  segmenti  ex  parte  axis  ultralibet.  Erit  enim  velocitas  acquisita  directe  ut 

'•  vel  dividendo  per  const""em  ~^  x  '-  * 


M 
endo    illud    segmentum  =  ±  T,    cujus    quadratum   T*   debet   esse  =  ^  X  G  X  R,    erit 

PQ  ^a 

directe  ut         _I_PO«'  adeoque  reciproce  ut  ^^  +  PQ.     Is    autem    [310]    valor    manet 

idem,  si  pro  PQ  ponantur  bini  valores,  quorum  productum  aequatur  T*,  migrante  tantum- 
modo  altera  binomii  parte  in  alteram      Si  enim  alter  valor  sit  m,  erit  alter  —  ;  &  posito 


m 


W 

illo  pro  PQ  :    habetur  —  +  m,  posito  hoc  habetur  -=—  +  —  ,  sive  m  +  —  .      Sed  cum 

m  1  m  m 

T2  ... 

eae  distantiae  abeunt  ad  partes  oppositas,  fiunt  —  w,  &  —  ,  migrante  in  negativum  etiam 

tn 


LETTER  TO  FR.   SCHERFFER  457 

time  of  collision,  up  to  the  end  of  recovery  of  shape  could  be  easily  derived,  by  doubling 
in  the  first  case,  &  by  increasing  in  a  given  ratio  in  the  second  case  ;  just  as  was  done  when 
we  considered  collisions. 

132.  The  formula  also  holds  good  for  the  case  in  which  the  new  mass  does  not  lie  at  S^  jde|*tei 
the  point  Q  in  the  straight  line  PG,  but  at  some  other  point  of  a  plane  perpendicular  to 

the  axis  &  passing  through  G  ;  if  from  this  point  a  perpendicular  is  supposed  to  be  drawn 
to  PG,  meeting  it  in  Q,  then  the  effect  will  be  exactly  the  same  as  if  the  impact  had  been 
at  Q,  the  action  being  transferred  by  this  straight  line  of  the  system.  Indeed,  if  Q  does  not 
lie  in  the  plane  perpendicular  to  the  axis,  which  passes  through  the  centre  of  gravity,  but 
somewhere  without  it,  it  all  comes  to  the  same  thing,  so  long  as  through  that  point  a  plane 
is  supposed  to  be  drawn  perpendicular  to  the  axis  that  is  unmoved  by  the  external  force, 
and  the  centre  of  gravity  is  reduced  to  this  plane,  together  with  any  mass  A  ;  or  if  the 
mass  Q,  together  with  the  rest,  is  reduced  to  any  plane  perpendicular  to  the  axis.  It  all 
comes  to  the  same  thing,  on  account  of  the  fact  that  there  is  an  axis  that  is  unmoved  by  the 
external  force.  But  now  we  will  deduce  several  corollaries  from  the  general  solution  of 
the  problem. 

133.  Corollary  I.     If  the  distance  of  the  centre  of  oscillation  of  the  whole  system  from  Relation    to    the 
the  axis  P  is  denoted  by  R,  the  distance  of  the  centre  of  gravity  by  G,  &  the  total  mass  by  "" 

M,  then  we  have  „  =  M  x     ****Q  x  Qp.  X  ,  i    &  |=  +  «•     It   is 


evident  from  the  fact  that,  from  the  nature  of  the  centre  of  oscillation,  we  have 


R  =    •  .  &  thus 

1VL  /\    \J 

1  34.  Corollary  II.   The  velocity  acquired  by  the  mass  Q  will  be  A  simpler  expres- 

sion for  the  velo- 

MXGXRXPQ..  city    in   the    mass 

MXGXR  +  QXPQ«X  by  its  help. 

for,  this  is  the  velocity  PQ  (c  -  x),  or  PQ  (c  -  ^--^^W^-  p.Qi-  x  c)  ; 

and  this,  when  reduced  to  the  same  denominator,  comes  to  that  which  was  given,  after 
cancelling  terms  of  opposite  sign. 

135.  Corollary  III.     If,  while  the  circular  velocity  remained  unaltered,  the  whole  mass  The  point  in  which 
of  the  system  is  supposed  to  be  collected  at  a  single  point  lying  between  the  centres  of  ^ouid^veTo^ 
gravity  &  oscillation,  the  distance  of  which  from  the  point  of  rotation  is  a  geometrical  mean  collected    in  order 
between  the  distances  of  the  other  points,  or  at  the  same  distance  on  the  other  side  of  the  ^meim^^St    ^n 
point  of  rotation  ;    then,  the  same  velocity  would  be  impressed  on  the  new  mass  situated  the  mass. 

at  any  point.  For,  in  that  case,  each  centre  would  coincide  with  that  point,  &  the  value 
of  G  X  R  would  be  the  same  as  before,  namely,  equal  to  the  square  of  its  distance  from 
the  axis  ;  &  this  square  is  positive,  even  if  the  distance,  when  taken  on  the  other  side  of  the 
point  of  rotation,  is  negative. 

136.  If,  on  one  side  or  the  other,  in  PG  a  segment  is  taken,  which  is  to  the  distance  Th.e     number    of 

t  .1    '        •        r  ,  ...  i    i       v  '        r    i          11  r    i  i       points,     and    their 

ot  the  point  from  the  axis  in  the  subduphcate  ratio  of  the  whole  mass  of  the  system  to  the  distances  from  the 

mass  Q  ;  then,  the  mass  Q,  if  placed  at  one  of  four  distances  from  the  axis,  two  on  one  side  axis-  for  wh^h  the 

&  two  on  the  other,  so  that  the  products  for  each  pair  should  be  equal  to  the  square  of  the  quire     the  same 

segment,  would  at  each  distance  acquire  a  velocity  of  the  same  magnitude  although  in  yelocities  from  the 

.         ,  .          .  ,  .  A  i          i  •          i      •  iii  T  impact  ;   where  the 

opposite  directions  tor  the  two  pairs.     Also  this  velocity  would  be  greatest,  when  the  mass  velocity  would   be 
was  placed  at  the  end  of  the  segment  on  either  side  of  the-  axis.     For,  the  velocity  acquired  greatest. 

varies    directly    as  M  x  Q  x  R    .    Q  x        a     x   c  '>     dividing    this    by    the    constant 


-  X  c  ,  and  denoting  the  segment  by  ±  T,  of  which  the  square,  T1,  must  be 

equal  to  ^-  X  G  X  R,  the  velocity  will  vary  directly  as         _\_T>(\*>  ^  therefore,  inversely 

T* 

as     7=c  +  PQ-     Now,  this  value  remains  the  same,  if  for  PQ  we  substitute  either  of  the 


pair  of  values  whose  product  is  T*,  the  first  part  of  the  binomial  expression  merely 
interchanging  with  the  second.  For,  if  either  value  is  denoted  by  m,  the  other  will  be 
T2/«  ;  &,  if  the  former  is  substituted  for  PQ,  we  get  T*/m  +  m  ;  or,  if  the  latter,  we 
have  T*wi/T*  -f-  T*/m,  i.e.,  m  +  T*/m.  But,  when  these  distances  are  taken  on  the 
opposite  side,  they  become  —  m  &  —  T*/m,  &  the  value  also  of  the  formula  becomes  nega- 
tive ;  this  shows  that  the  direction  of  the  motion  is  opposite  to  what  it  was  before  ;  in 


458  PHILOSOPHIC  NATURALIS  THEORIA 

valore  formulae,  quod  ostendit  directionem  motus  contrariam  priori,  systemate  nimirum 
hinc,  &  inde  ab  axe  in  partibus  oppositis  habente  directiones  motuum  oppositas. 

rpa 

Demonstratio    de-  137.  Quoniam  autem  assumpto  quovis  valore  finite  pro  PQ,  formula  ~>c+PQ  est 

terminationis  max- 


finita,  &  evadit  infinita  facto  PQ  tarn  infinite,  quam  =  o  ;  patet  in  hisce  postremis  duobus 
casibus  velocitatem  e  contrario  evanescere,  in  reliquis  esse  finitam,  adeoque  alicubi  debere 
esse  maximam.  Non  potest  autem  esse  maxima,  nisi  ubi  ad  eandem  magnitudinem  redit, 
quod  accidit  in  transitu  PQ  per  utrumvis  valorem  ±  T,  circa  quern  hinc  &  inde  valores 
aequales  sunt.  Ibi  igitur  id  habetur  maximum. 

^g    Scholium  2.     Libuit  sine  calculo  differential!  invenire  illud  maximum,  quod  ope 

difierentiaiem.         C3L\C}3k  ipsius  admodum  facile  definitur.     Ponantur  T  =  *,  &  PQ  =  z.     Fiet  formula  -+z, 

z 

&  differentiando  --  -  +  dz  —  o,    sive  —  **  +  z*  —  o,   vel  z*  =  /*,    &  z  =  ±  t  ,    sive 
zz 

PQ  =  ±  T,  ut  in  corollario  4  inventum  est. 
DUX   aliae    accep-  j,g    Ljcebit  autem  iam  ex  postremis  duobus  corollariis  deducere  alias  duas  notiones 

tiones    centn    per  .•>*..  J  .  .,  „  .  „     . 

cussionis,    &  ejus  centn  percussioms,  cum  suis  eorundem  determinatiombus.     rotest  pnmo  appellan  centrum 
determinatio    ex  percussionis  illud  punctum,  in  quo  tota  systematis  massa  collecta  eandem  velocitatem 

superionbus.  f  r.,          .     •  *.  '      ,  ,  1 

impnmeret  massae  eidem  incurrendo  m  earn  eodem  suo  puncto  cum  eadem  velocitate,  quae 
videtur  omnium  aptissima  centri  percussionis  notio.  Centrum  percussionis  in  ea  acceptione 
determinatur  admodum  eleganter  ope  corollarii  3  :  jacet  nimirum  inter  centrum  gravitatis, 
&  centrum  oscillationis  ita,  ut  ejus  distantia  ab  axe  rotationis  sit  media  geometrice  pro- 
portionalis  inter  illorum  distantias,  vel  ubivis  in  recta  axi  parallela  ducta  per  punctum  ita 
inventum.  Potest  secundo  appellari  centrum  percussionis  illud  punctum,  per  quod  si  fiat 
percussio,  imprimitur  velocitas  omnium  maxima  massae,  in  [311]  quam  incurritur.  In 
hac  acceptione  centrum  percussionis  itidem  eleganter  determinatur  per  corollarium 
quartum,  mutando  earn  distantiam  in  ratione  subduplicata  massae,  in  quam  incurritur, 
ad  massam  totius  systematis. 

watum  lta&0npro  I4°-  ^n  ^oc  secundo  sensu  acceptum,  &  investigatum  esse    centrum  percussionis  a 

particular!      casu  summo   Geometra    Celeberrimo   Pisano   Professore  Perrellio,   nuper  mihi  significavit  Vir 

determmatum.         itidem  Doctissimus,  &  geometra  insignis  Eques  Mozzius,  qui  &  suam    mihi   ejus    centri 

determinationem  exhibuit  pro  casu  systematis  continentis    unicam  massam  in  rectilinea 

virga  inflexili. 

?u  t  ernedftermi  if  I4I>  Libuit   rem  longe  alia   methodo  hie  erutam  generaliter,  &  cum    superioribus 

atum  ad  foecundi-  omnibus  conspirantem,  ac  ex  iis  sponte  propemodum  profluentem  proponere,  ut  innotescat 
ta*tejn  A  Theoriae  mira     sane    foecunditas     Theorematis     simplicissimi    pertinentis     ad     rationem     virium 

ostendendam.  .  r     „     .    .     .r.  ... 

compositarum  in  systemate  massarum  tnum.     bed  de  his  omnibus  jam  satis. 
DABAM  FLORENTIAE,  17  Junii,  1758. 


FINIS. 


LETTER  TO  FR.   SCHERFFER 


459 


other  words,  the  system  has  opposite  directions  for  motions  of  opposite  parts  on  either  side 
of  the  axis. 

137.  Now,  since,  for  any  assumed  finite  value  of  PQ,  the  formula  T2/PQ  -f  PQ  is 
finite,  &  comes  out  infinite  both  when  PQ  is  made  infinite  &  when  it  is  made  zero,  it  is 
clear  that  the  velocity,  which  varies  inversely  as  the  formula,  must  vanish  in  these  two 
extreme  cases,  &  be  finite  in  all  other  cases ;  hence,  at  some  time  there  must  be  a  maxi- 
mum.   But  it  cannot  be  a  maximum,  except  when  the  two  parts  of  the  formula  become 
equal ;    &  this  happens  as  PQ  passes  through  either  of  the  values  ±  T,  about  which,  on 
either  side,  the  values  are  equal.     Hence  there  is  a  maximum  there. 

138.  Scholium  ^.     I  have  preferred  to  find  this  maximum   without  the  help  of  the 
differential  calculus  ;  but  with  the  help  of  the  calculus,  it  can  be  determined  very  easily. 
Put  T  =  t,  &  PQ  =  z  ;    then  the  formula  becomes  t*/z  +  z.      Differentiating,  we  have 

-  t  dz/z*  +  dz  =  o,  or      -  t*  +  z2  =  o,  or    z2  =  t* ;    &  z  =  ±  /,  or   PQ  =  ±  T,  as 
was  found  in  corollary  IV. 

139.  We  may  now,  from  the  last  two  corollaries,  deduce  two  other  ideas  of  the  centre 
of  percussion,  together  with  the  determination  of  each.      In  the  first  place,  we  may  call 
the  centre  of  percussion  that  point  which  is  such  that  if  the  whole  mass  of  the  system  were 
collected  therein,  it  would  impress  the  same  velocity  on  the  same  mass  by  colliding  with  it 
with  this  same  point  of  itself  with  the  same  velocity ;  &  it  seems  that  this  is  the  most  apt 
idea  of  all  for  the  centre  of  percussion.     The  centre  of  percussion,  in  this  acceptation,  is 
determined  in  a  very  elegant  manner  by  the  aid  of  corollary  III.     Thus,  it  will  lie  between 
the  centre  of  gravity  &  the  centre  of  oscillation,  in  such  a  manner  that  its  distance  from 
the  axis  of  rotation  is  a  geometrical  mean  between  those  two  distances,  or  anywhere  in  a 
straight  line  parallel  to  the  axis  drawn  through  the  point  thus  found.     Again,  the  name 
centre  of  percussion  may  be  given  to  that  point  which  is  such  that,  if  the  blow  is  delivered 
through  it,  it  will  give  to  the  mass  on  which  it  falls  the  greatest  possible  velocity.     In  this 
acceptation,  the  centre  of  percussion  is  also  elegantly  determined  by  the  fourth  corollary, 
by  changing  the  distance  in  the  subduplicate  ratio  of  the  mass  struck  to  the  whole  mass  of 
the  system. 

140.  That  learned  man  &  fine  geometer,  Signer  Mozzi,  has  but  lately  acquainted 
me  with  the  fact  that  the  centre  of  percussion  was  taken,  in  this  second  sense,  &  investigated 
by  that  excellent  geometer,  the  well-known  Professor  at  Pisa,  Perrelli ;  &  Mozzi  also  showed 
me  his  own  determination  for  the  case  of  a  system  consisting  of  a  single  mass  in  the  form 
of  a  rectilinear  inflexible  rod. 

141.  I  have  preferred  to  set  forth  the  matter  here  derived  in  general  in  a  far  different 
manner,  agreeing  as  it  does  with  all  that  has  gone  before,  &  arising  from  it  almost  automa- 
tically, so  as  to   make  known  the  truly  wonderful  fertility  of  that  very  simple  theorem 
dealing  with  the  ratio  of  the  composite  forces  in  a  system  of  three  masses.      But  now  I 
have  said  enough  about  all  these  things. 

FLORENCE, 

17 'th  June,  1758. 


Demonstration  that 
the  maximum  is 
correctly  given. 


Determination  of 
the  maximum  by 
means  of  the  dif- 
ferential calculus. 


Two  other  accep- 
tations of  the  term 
centre  of  percus- 
sion ;  and  its 
determination  by 
means  of  what  has 
been  given  above. 


By  whom  so  con- 
sidered, and  deter- 
mined in  a  parti- 
cular case. 


Here  a  more  gen- 
eral determination 
from  other  prin- 
ciples has  been 
given,  in  order  to 
show  the  fertility 
of  the  theory. 


THE   END. 


INDEX 

PARS  I 

Pag.  Num. 

Introductio        ..............         i  i 

Expositio  Theorise     .............         4  7 

Occasio  inveniendae,  &  ordo,  ac  analytica  deductio  invents  Theoriae     .          .          .          .8  16 

Lex  continuitatis  quid  sit           .          .          .          .          .          .          .          .          .          .          .13  32 

Ejus  probatio  ab  inductione ;    vis  inductionis       ........       16  39 

Ejusdem  probatio  metaphysica    ...........       22  48 

Ejus  applicatio  ad  excludendum  immediatum  contactum      ......       28  63 

Deductio  legis  virium,  &  determinatio  curvae  earn  exprimentis      .....       33  73 

Primorum  elementorum  materiae  indivisibilitas,  &  inextensio          .          .          .          .          -37  81 

Eorundem  homogeneitas    .          .          .          .          .          .          .          .          .          .          .          -41  91 

Objectiones  contra  vires  in  genere,  &  contra  hanc  virium  legem            .          .          .          -45  100 

Objectiones  contra  hanc  constitutionem  primorum  elementorum  materiae      .         .         -59  I3l 

PARS   II 

Applicatio  Theories  ad.  Mechanicam 

Argumentum  hujus  partis           .         .         .         .         .         .         .         .         .         .         -77  166 

Consideratio  curvae  virium          .         .         .         .         .         .         .         .         .         .         .       77  167 

De  arcubus       ..............       77  168 

De  areis  ...............       79  172 

De  appulsibus  ad  axem,  &  recessibus  in  innnitum,  ubi  de  limitibus  virium   .          .          .          .82  179 

De  combinationibus  punctorum,  &  primo  quidem  de  systemate  punctorum  duorum       .       86  189 

De  systemate  punctorum  trium           .          .          .          .          .          .          .          .          .                 92  204 

De  systemate  punctorum  quatuor       .         .         .         .         .         .         .         .         .         .no  238 

De  massis,  &  primo  quidem  de  centro  gravitatis,  ubi  etiam  de  viribus  quotcunque  generaliter 

componendis       .         .         .         .         .         .         .         .         .'••'.         .         .         .111  240 

De  aequalitate  actionis,  ac  reactionis  .          .          .          .          .          .          .          .          .          .124  265 

De  collisionibus  corporum,  &  incursu  in  planum  immobile  .          .          .          .          .          .125  266 

Exclusio  verae  virium  resolutionis        ......                                           132  279 

De  compositione,  &  imaginaria  resolutione  virium,  ubi  aliquid  etiam  de  Viribus  vivis  .  136  289 
De  continuitate  servata  in  variis  motibus,  ubi  quaedam  de  collisionibus,  de  reflexionibus,  & 

refractionibus  motuum         ....                                                                  •     J39  297 

De  systemate  trium  massarum    .....                                                           .     143  3°7 

Theoremata  pertinentia  ad  directiones  virium  compositarum  in  singulis          .                    .     143  308 

Theoremata  pertinentia  ad  ipsarum  virium  magnitudines      .         .                                      .     145  313 

Centrum  asquilibrii,  &  vis  in  fulcrum  inde.          .....                              .     148  321 

Momenta  pro  machinis,  &  omnia  vectium  genera  inde  itidem      .         .                            .150  325 

Centrum  itidem  oscillationis         .....                                                        •     J52  32^ 

Centrum  etiam  percussionis        .....                                                        •     JS7  344 

Multa  huic  Theorize  communia  cum  aliis  hie  tantummodo  indicata      .                            .158  347 

De  fluidorum  pressione      ......                                                        •     '59  34^ 

De  velocitate  fluidi  erumpentis  .....  •  162  354 

PARS   III 

Applicatio  Theories  ad  Physicam 

Argumentum  hujus  partis           ....                                                                  •     164  358 

Impenetrabilitas         ....                                                                                         .     164  360 

Extensio  cujusmodi  sit  in  hac  Theoria,  ubi  de  Geometria    .                                                 .     169  371 

Figurabilitas,  ubi  de  mole,  massa,  densitate                                                                               •     I72  375 

Mobilitas,  &  continuitas  motuum        .         .                                                                           •     J75  3^3 

jEqualitas  actionis,  &  reactionis           .         .                                                                           •     17%  3^8 

Divisibilitas  quae  sit :  componibilitas  aequivalens  divisibilitati  in  innnitum                              .     179  391 

460 


INDEX 

PART  I 

Page  Art. 

Introduction      ..............  35  I 

Statement  of  the  theory 37  7 

What  led  to  its  discovery,  the  order,  &  the  analytical  deduction  of  the  theory  found     .  45  16 

What  the  Law  of  Continuity  is 51  32 

Proof  of  this  law  by  induction  ;    the  power  of  induction     ......  55  39 

Metaphysical  proof  of  the  same  thing          .........  63  48 

Its  application  for  the  purpose  of  eliminating  the  idea  of  immediate  contact         .          .  71  63 

Deduction  of  the  law  of  forces  &  the  determination  of  the  curve  representing  this  law     .  77  73 

Indivisibility  &  non -extension  of  the  primary  elements  of  matter          .          .          .  83  81 

Their  homogeneity    .          .          .          .          .          .          .          .          .          .          .          .  89  91 

Objections  against  forces  in  general,  &  against  this  particular  law  of  forces        ...  95  100 

Objections  against  this  particular  constitution  of  the  primary  elements  of  matter           .  in  131 

PART  II 

Application  of  the  Theory  to  Mechanics 

The  theme  of  the  second  part  .         .         .         . 135  166 

Consideration  of  the  curve  of  forces 135  167 

The  arcs 135  168 

The  areas .  139  172 

Approach  towards  &  recession  to  an  infinite  distance  from  the  axis  ;  the  limits  of  the  forces  143  179 

Combinations  of  points  ;    firstly,  a  system  of  two  points 149  189 

System  of  three  points       ............  155  204 

System  of  four  points         ............  187  238 

Masses  ;  firstly,  the  centre  of  gravity  ;&  the  general  composition  of  any  number  of  forces     .  189  240 

Equality  of  action  &  reaction 203  265 

Collision  of  solid  bodies,  &  impact  on  a  fixed  plane 205  266 

Exclusion  of  the  idea  of  real  resolution  of  forces .213  279 

Composition,  &  hypothetical  resolution,  of  forces  ;    remarks  on  "  living  forces."      .          .  223  289 
Continuity  observed  in  various  motions ;    also  some  remarks  upon  collisions,  reflections  & 

refractions  of  motions 227  297 

System  of  three  masses 237  307 

Theorems  relating  to  the  directions  of  the  resultant  forces  on  each  of  the  systems         .  237  308 

Theorems  relating  to  the  magnitude  of  these  forces    .......  241  313 

The  centre  of  equilibrium,  &  the  force  on  the  fulcrum  derived  from  it        ...  247  321 

Theory  of  moments  for  machines,  &  hence  all  kinds  of  levers  also         .                   .          .  249  325 

The  centre  of  oscillation  .         .         .         .         .         .         .         .         .         .         .         .251  328 

The  centre  of  percussion  ............  257  344 

Many  points  of  this  Theory  common  to  others,  merely  mentioned       ....  259  347 

Pressure  of  fluids       ..........          ...  259  348 

Velocity  of  a  fluid  issuing  from  a  vessel     ......          ...  263  354 

PART  III 

Application  of  the  Theory  to  Physics 

The  theme  of  the  third  part 267  358 

Impenetrability          .............  267  360 

What  kind  of  extension  is  admitted  in  this  Theory ;    geometry    .....  273  371 

Figurability  ;    also  volume,  mass  &  density  .........  275  375 

Mobility,  &  continuity  of  motions      .          .                   .          .          .          .          .          .          .  281  383 

Equality  of   action  &  reaction 283  388 

What  divisibility  is  ;    componibility  equivalent  to  infinite  divisibility     ....  285  391 

461 


462  INDEX 

Pag.  Num. 

Immutabihtas  pnmorum  matenae  elementorum   .         .         .         .         .         .         .         .181  398 

Gravitas                      182  399 

Cohssio  .                            185  406 

Discrimen  inter  particulas.         ...........     191  419 

Soliditas,  &  fluiditas            ............     194  426 

Virgx  rigidae,  flexiles,  elasticae,  fragiles         .........     199  436 

Viscositas _ 200  438 

Certas  quorundam  corporum  figurae    ..........     200  439 

De  fluidorum  resistentia    ............     203  442 

De  elasticis,  &  mollibus     . 204  446 

Ductilitas,  &  Malleabilitas 205  448 

Densitas  indifferens  ad  omnes  proprietates            ........     206  449 

Vulgaria  4  elementa  quid  sint    ...........     206  450 

De  operationibus  chemicis  singillatim           .........     207  451 

De  natura  ignis         .............     215  467 

De  lumine,  ubi  de  omnibus  ejus  proprietatibus,  ac  de  Phosphoris         ....     217  472 

De  sapore,  &  odore  .............     234  503 

De  sono            . 235  504 

De  tactu,  ubi  de  frigore,  &  calore     ..........     237  507 

De  electricitate,  ubi  de  analogia,  &  differentia  materiae  electricae,  &  ignese    .          .          .     239  511 

De  Magnetismo         .............     242  514 

Quid  sit  materia,  forma,  corruptio,  alteratio 243  516 

APPENDIX 

Ad  Metapbysicam  pertinens 

De  Anima         ..............     248  526 

De  DEO ..     254  539 

SUPPLEMENTA 

§  I    De  Spatio,  &  Tempore              ..........     264  i 

§  II     De  Spatio,  ac  Tempore,  ut  a  nobis  cognoscuntur 273  18 

§  III     Solutio  analytica  Problematis  determinantis  naturam  legis  virium         .          .          .     277  25 

§  IV     Contra  vires  in  minimis  distantiis  attractivas,  &  excrescentes  in  infinitum     .          .     289  77 

§    V     De  Jiquilibrio  binarum  massarum  connexarum  invicem  per  bina  alia  puncta        .     293  86 

§  VI     Epistola  ad  P.  Scherffer 297  93 

NOI    RIFORMATORI 

Dello  Studio  di  Padova. 

AVENDO  veduto  per  la  Fede  di  Revisione,  ed  Approvazione  del  P.  F.  Gio.  Paolo  Zapparella, 
Inquisitor  Generale  del  Santo  Officio  di  Fenizia,  del  Libro  intitolato  Philosophic  Naturalis  Theoria 

redacta  ad  unicam  legem  virium  in  natura  existentium,  Auctore  P.  Rogerio  Josepho  Boscovicb  &c.  non  v'esser 

cosa  alcuna  contro  la  Santa  Fede  Cattolica,  e  parimente  per  attestato  del  Segretario  Nostro,  niente  contro 
Principi,  e  buoni  costumi  concediamo  licenza  a  Giambattista  Remondini  Stampator  di  Venezia,  che  possa 
essere  stampato,  osservando  gli  ordini  in  materia  di  stampe,  e  presentando  le  solite  Copie  alk  Publiche 
Librerie  di  Venizia,  e  di  Padova. 

Dat.  li  7.  Settembre  1758. 
(Gio.  Emo,  Procurator,  Rif. 
(Z.  Alvise  Mocenigo,  Rif. 

( 
Registrato  in  Libro  a  carte  47.  al  num.  383. 

Gio.  Girolamo  Zuccato,  Segretario. 
Adi  18  Settembre  1758. 

Registrato  nel  Magistr.  Eccellentiss.  degli  Esec.  contro  la  Bestemmia. 

Gio.  Pietro  Dolfin,  Segretario. 


INDEX  463 

Page  Art. 

Immutability  of  the  primary  elements  of  matter 287  398 

Gravity 287  399 

Cohesion ...............  291  406 

Distinction  between  particles 299  419 

Solidity  &  fluidity 303  426 

Rigid,  flexible,  elastic  &  fragile  rods 309  436 

Viscosity 311  438 

Definite  shapes  of  certain  bodies         .         .         .         .         .         .         .         .         .         .311  439 

Resistance  of  fluids 315  442 

Elastic,  &  soft,  bodies .          .          .317  446 

Ductility  &  malleability 317  448 

Density  unrelated  to  all  other  properties    .          .          .          .          .          .          .          .          .317  449 

The  so-called  "four  elements"           .         .         . -3*9  45° 

Chemical  operations,  each  in  turn      .         .         .         .         .         .         .         .         .         .  319  451 

The  nature  of  fire 329  467 

Light,  its  properties,  &  light -giving  bodies •  331  472 

Taste  and  smell 353  503 

Sound 355  504 

Touch ;    cold  &  heat 357  507 

Electricity;    the  resemblances  &  differences  between  electric  matter  &  fire    .         .         .  361  511 

Magnetism        ..............  363  514 

Matter,  form,  corruption,  alteration  .          ........          .  365  516 

APPENDIX 

Relating  to  Metaphysics 

The  Soul           ..............  373  526 

GOD                                                                                                                              .  379  539 

SUPPLEMENTS 

I  On  Space  &  Time  ............  393  i 

II  On  Space  &  Time,  as  we  know  them 405  18 

III  Analytical  solution  of  the  problem  to  determine  the  nature  of  the  law  of  forces  .  411  25 

IV  Arguments  against  forces  that  are  attracti  ve  at  very  small  distances,  &  increase  indefinitely  427  77 

V  Equilibrium  of  two  masses  connected  together  by  two  other  points  .  .  .  437  86 

VI  A  letter  from  the  Author  to  Father  Scherffer 443  93 


WE,  as  Censors  of  the  College  of  Padua,  having  seen,  through  trust  in  the  revision  &  approval  of 
Father  F.  Gio.  Paolo  Zapparella,  Inquisitor  General  of  the  Holy  Office  in  Venice,  that  there 
is  nothing  in  the  book,  entitled  Philosophic  Naturalis  Iheoria  redacta  ad  unicam  legem  virium  in  natura 
existentium,  by  P.  Rogerius  Josephus  Boscovich,  that  is  contrary  to  the  Holy  Catholic  Faith ;    &  also, 
on  the  testimony  of  our  Secretary,  that  there  is  nothing  contrary  to  our  Rules,  according  to  good  usance, 
give  leave  to  Giambattista  Remondinus,  printer  in  Venice,  to  print  the  book ;   provided  that  he  observe 
the  regulations  governing  the  press,  &  present  the  usual  copies  to  the  Public  Libraries  of  Venice  &  Padua. 
Given  this  7th  of  September,  1758. 

GIG.  EMO,  Procurator,  Censor. 
Z.  ALVISE  MOCENIGO,  Censor. 

??* 
Registered  in  Book,  p.  47,  no.  383. 

September  iSth,  1758.  Gio.  Girolamo  Zuccato,  Secretary. 

Registered  in  the  High  Court  for  the  Prevention  of  Blasphemy. 

Gio.  Pietro  Dolfin,  Secretary. 
*  There  is  here  a  space  for  another  name  that  was  not  filled  in. 


CATALOGUS     OPERUM 
P.    ROGERII   JOSEPHI    BOSCOV1CH,    S.J. 

imfressorum   usque  ad  initium  anni   1763. 

Annus  prima 
Opera,  fcf  opuscula  justa  molis.  edition, 

Sopra  il  Turbine,  che  la  notte  tra  gli  1 1,  e  12  Giugno  del  1749  danneggio  una  gran  parte  di  Roma.    Dis-       1749 
sertazione  del  P.  Ruggiero  Giuseppe  Boscovich  della  Comp.  di  Gesii.     In  Roma   appresso  Nicolo, 
e  Marco  Pagliarini,  in  8. 

Elementorum  Matheseos  tomi  tres,  in  4.  Prodierunt  anno  1752  sub  titulo,  Elementorum  Matheseos  ad  1752 
usum  studiosae  juventutis,  tomi  primi  pars  prima  complectens  Geometriam  planam,  Arithmeticam 
vulgarem,  Geometriam  Solidorum,  &  Trigonometriam  cum  planam,  turn  sphaericam.  Pars  altera, 
in  qua  Algebrae  finite  elementa  traduntur.  Romae  :  excudebat  Generosus  Salomoni.  Us  binis  tomis 
sine  nova  eorum  impressionf  mutatus  est  titulus  anno  1754  in  hunc,  Elementorum  Universe  Matheseos 
Auctore  P.  Rogerio  Josepho  Boscovich  Soc.  Jesu  Publico  Matheseos  Professore  Tomus  I  continens 
&c.  Tomus  II  continens  &c,  y  adjectus  est  sequens. 

Tomus  III  continens  Sectionum  Conicarum  Elementa  nova  quadam  methodo  concinnata,  &  Disserta-       1754 
tionem  de  Transformatione  locorum  Geometricorum,  ubi  de  Continuitatis  lege,  ac  de  quibusdam 
Infiniti  mysteriis  :   Typis  iisdem  ejusdem  Generosi  Salomoni  omnes  in  8.     Extat   eorundem  impressio 
Veneta  anni  1758,  sed  typorum  mendis  deformatissima. 

De  Litteraria  Expeditione  per  Pontificiam  ditionem  ad  dimetiendos  duos  Meridian!  gradus,  &  corrigendam  1755 
mappam  geographicam,  jussu,  &  auspiciis  Benedicti  XIV.  P.M.  suscepta  Patribus  Soc.  Jesu  Chris- 
tophoro  Maire,  &  Rogerio  Josepho  Boscovich,  Romae  1755.  In  Typographic  Palladis :  excudebant 
Nicolaus,  &  Marcus  Palearini,  in  4.  Quidquid  eo  volumine  continetur,  est  Patris  Boscovich  prater  bina 
brevia  opuscula  Patris  Maire,  qua  ipse  P.  Boscovich  inseruit.  Prostat  etiam  Mappa  Geographica 
ditionis  Pontificia  delineata  P.  Maire  ex  observationibus  utrique  communibus. 

De  Inasqualitatibus,  quas  Saturnus,  &  Jupiter  sibi  mutuo  videntur  inducere,  praesertim  circa  tempus       1756 
conjunctionis.    Opusculum   ad  Parisiensem  Academiam   transmissum,  &  nunc  primum   editum. 
Auctore  P.  Rogerio  Josepho  Boscovich  Soc.  Jesu  ;  Romae  ;  ex  Typographia  Generosi  Salomoni,  in  8. 

Philosophic  Naturalis  Theoria  redacta  ad  unicam  legem  virium  in  Natura  existentium  Auctore  P.  Rogerio       I7*fi 
Jos.  Boscovich  S.J.  publico  Matheseos  Professore  in  Collegio  Romano.    Prostat  Vienna  Austria  in 
Officina  libraria  Kalivvodiana  :   in  4.     In  fine  accedit  Epistola  ad  P.  Carolum    Scherffer  Soc.  Jesu. 
Habetur  secunda  editio  Viennensis  paullo  posterior :    tertia  hie  exhibetur :    Epistola  babetur  in  ejus 
Supplements. 

Adnotationes  in  aliorum  Opera. 

Caroli  Noceti  e  Societate  Jesu  de  Iride,  &  Aurora  Boreali  Carmina  .  .  .  cum  notis  Josephi  Rogerii  Bosco-       X747 
vich  ex  eadem  Societate.    Romae :    excudebant  Nicolaus,  &  Marcus  Palearini,  in  4.     Perperam 
nomen  Josephi  antepositum  est  ibi  nomini  Rogerii. 

Philosophise  Recentioris  a  Benedicto  Stay  in  Romano  Archigymnasio  Publico  Eloquentiae  Professore  .  .  .       1755 
cum  adnotationibus,  &  Supplementis  P.  Rogerii  Joseph!  Boscovich  S.J.  in  Collegio  Rom.  Publici 
Matheseos  Professoris.    Tomus  I.  Romae  :    Typis,  &  sumptibus  Nicolai,  &  Marci  Palearini,  in  8. 
Duce  ejus  editiones  prodierunt  simul. 

Tomus  II  Romae  :   Typis,  &  sumptibus  Nicolai,  &  Marci  Palearini,  in  8.    In  singulis  hisce  voluminibus       J7°° 
ea,  qua  ad  P.  Boscovich  pertinent,  efficerent  per  se  ipsa  justum  volumen.     In  solis  primi  Stayani  tomi 
supplements  occurrunt  39  ipsius  Dissertationes  de  variis  argumentis  pertinentibus  potissimum  ad  Meta- 
physicam  {3  Mechanicam. 

Dissertationes  impresses  pro  exercitationibus  annuis,  y  publics  propugnattz :    omnes 

in  4. 

De  Maculis  Solaribus,  Exercitatio  Astronomica  habita  in  Collegio  Romano  Soc.  Jesu.     Romae  :   ex  Typo-       J73^ 

graphia  Komarek. 
De  Mercurii  novissimo  infra  Solem  transitu.    Dissertatio  habita  in  Seminario  Romano.    Romse,  Typis       J737 

Antonii  de  Rubeis. 
Constructio  Geometrica  Trigonometric  sphaericae.     Romae,  ex  Typographia  Komarek.    Hujus  titulus  vel 

est  hie  ipse,  vel  parum  ab  hoc  differt. 
De  Aurora  Boreali  Dissertatio  habita  in  Seminario  Romano.    Romas  :  Typis  Antonii  de  Rubeis.    Eadem       *738 

eodem  anno  edita  fuit  etiam  typis  Komarek. 
De  Novo  Telescopii  usu  ad  objecta  caelestia  determinanda.    Dissertatio  habenda  a  PP.  Soc.  Jesu  in  Collegio       J739 

Romano.     Romae,  ex  Typographia  Komarek.    Extat  recusa  sine  ulla  mutatione  in  Actis  Lipsiensibus 

ad  annum  1740. 

HH 


Annus  primee 

edition.        £)e  Veterum  argumentis  pro  Telluris  sphaericitate.     Dissertatio  habita  in  Seminario  Romano  Soc.  Jesu. 

Romae  :   Typis  Antonii  de  Rubeis. 

Dissertatio  de  Telluris  Figura  habita  in  Seminario  Romano  Soc.  Jesu.  Romae  :  Typis  Antonii  de  Rubeis. 
Eadem  prodiit  in  8,  anno  1744  in  opere,  cui  titulus  Memorie  &c.  In  Lucca  per  li  Salani,  e  Giuntini, 
y  in  titulo  additur  :  nunc  primum  aucta,  &  illustrata  ab  ipsomet  Auctore  ; "sed  ea  editio  scatet  typorum 
erroribus,  ut  y  reliqua  inferius  nominanda  in  eadem  collectione  inserta. 

1740  De  Circulis  Osculatoribus.     Dissertatio   habenda  a  PP.  Societatis  Jesu  in  Collegio  Romano.     Romae : 

ex  Typographia  Komarek. 

De  Motu  corporum  projectorum  in  spatio  non  resistente.  Dissertatio  habita  in  Seminario  Romano  Soc. 
Jesu.  Romae  :  Typis  Antonii  de  Rubeis. 

1741  De  Natura,  &  usu  infinitorum,  &  infinite  parvorum.    Dissertatio  habita  in  Collegio  Romano  Soc.  Jesu. 

Romae  :    ex  Typographia  Komarek. 

De  Inaequalitate  gravitatis  in  diversis  Terrae  locis.  Dissertatio  habita  in  Seminario  Romano  Soc.  Jesu. 
Romae :  Typis  Antonii  de  Rubeis. 

1742  De  Annuis  Fixarum  aberrationibus.     Dissertatio  habita  in  Collegio  Romano  Societatis  Jesu.     Romae  : 

ex  Typographia  Komarek. 
De  Observationibus  Astronomicis,  &  quo  pertingat  earundem  certitudo.     Dissertatio  habita  in  Seminario 

Romano  Soc.  Jesu.  Romae  :    Typis  Antonii  de  Rubeis. 
Disquisitio  in  Universam  Astronomiam  publicae  Disputation!  proposita  in  Collegio  Romano  Soc.  Jesu. 

Romae  :    ex  Typographia  Komarek. 

1743  De  Motu  Corporis  attracti  in  centrum  immobile  viribus  decrescentibus  in  ratione  distantiarum  reciproca 

duplicata  in  spatiis  non  resistentibus.  Dissertatio  habita  in  Collegio  Romano.  Romae  :  Typis 
Komarek.  Eadem  prodiit  anno  1747  sine  ulla  mutatione  in  Commentariis  A  cad.  Bononiensis  Tom.  II. 
•par.  III. 

1744  Nova  methodus  adhibendi  phasium  observationes  in  Eclipsibus  Lunaribus  ad  exercendam  Geometriam, 

&  promovendam  Astronomiam.  Dissertatio  habita  in  Collegio  Romano.  Romae  :  ex  Typographia 
Komarek.  Eadem  prodiit  in  8,  anno  1747  cum  exigua  mutatione,  vel  additamento  in  Opere  superius 
memorato,  cui  titulus  Memorie  &c.  In  Lucca  per  li  Salani,  e  Giuntini. 

X745  De  Viribus  Vivis.  Dissertatio  habita  in  Collegio  Romano  Soc.  Jesu.  Romae  :  Typis  Komarek.  Eadem 
prodiit  anno  1747  sine  ulla  mutatione  in  Commentariis  A  cad.  Bonon.  To.  II.  par.  Ill,  y  in  Germania 
pluribus  vicibus  est  recusa. 

174"       De  Cometis.     Dissertatio  habita  a  PP.  Soc.  Jesu  in  Collegio  Rom.    Romae  :   ex  Typographia  Komarek. 

J747  De  .<Estu  Maris.  Dissertatio  habita  a  PP.  Soc.  Jesu  in  Collegio  Romano.  Romas :  ex  Typographia 
Komarek.  Ea  est  Dissertationis  pars  I  ;  secunda  pars  nunquam  prodiit.  Quee  pro  ilia  fuerant  des- 
tinata,  habentur  in  Opere  De  Expeditione  Litteraria,  y  in  supplements  Philosophies  Stayanes  tomo  II. 

1748  Dissertationis  de  Lumine  pars  prima  publice  propugnata  in  Seminario  Romano  Soc.  Jesu.     Romae  :  Typis 

Antonii  de  Rubeis. 

Dissertationis  de  Lumine  pars  secunda  publice  propugnata  a  PP.  Soc.  Jesu  in  Collegio  Romano.  Romae  : 
ex  Typographia  Komarek. 

1749  De  Determinanda  Orbita  Planetae  ope  Catoptricae,  ex  datis  vi,  celeritate,  &  directione  motus  in  dato 

puncto.     Exercitatio  habita  a  PP.  Soc.  Jesu  in  Collegio  Romano.     Romae  :  ex  Typographia  Komarek. 

'7S1  De  Centre  Gravitatis.  Dissertatio  habita  in  Collegio  Romano  Soc.  Jesu.  Romae:  ex  Typographia 
Komarek.  Eadem  paullo  post  prodiit  iterum  cum  sequenti  titulo,  y  additamento.  De  Centre  Gravitatis. 
Dissertatio  publice  propugnata  in  Collegio  Romano  Soc.  Jesu  Auctore  P.  Rogerio  Josepho  Boscovich 
Societatis  ejusdem.  Editio  altera.  Accedit  Disquisitio  in  centrum  Magnitudinis,  qua  quasdam  in 
ea  Dissertatione  proposita,  atque  alia  iis  affinia  demonstrantur.  Romae,  Typis,  &  sumptibus  Nicolai, 
&  Marci  Palearini. 

I7S3  De  Lunae  Atmosphaera.  Dissertatio  habita  a  PP.  Soc.  Jesu  in  Collegio  Romano.  Romae:  ex  Typographia 
Generosi  Salomoni.  Multa  eorundem  typorum  exemplaria  prodierunt  paullo  post  cum  nomine  Auctoris 
in  ipso  titulo,  y  cum  exigua  unius  loci  mutatione. 

X754  De  Continuitatis  Lege,  &  Consectariis  pertinentibus  ad  prima  materiae  elementa,  eorumque  vires.  Disser- 
tatio habita  a  PP.  Societatis  Jesu  in  Collegio  Romano.  Romae  :  ex  Typographia  Generosi  Salomoni. 

J7SS       De  Lege  virium  in  Natura  existentium.     Dissertatio  habita  a  PP.  Soc.  Jesu  in  Collegio  Romano.     Romae  : 

Typis  Generosi  Salomoni. 

De  Lentibus,  &  Telescopiis  dioptricis.  Dissertatio  habita  in  Seminario  Romano.  Romae  :  ex  Typographia 
Antonii  de  Rubeis. 

Plures  ex  hisce  Dissertationibus  prodierunt  etiam  iisdem  typis,  sed  cum  alia  titulo,  habente  non  locum,  ubi 
sunt  habitee,  vel  propugnatee,  sed  tantummodo  nomen  Auctoris.  In  hac  postrema  mutates  sunt  bines  pagince, 
posteaquam  plurima  exemplaria  fuerant  distracta.  In  prioribus  tribus  sunt  pauca  qucedam  mutata,  vel 
addita  a  P.  Horatio  Burgundio  adhuc  Professore  Matheseos  in  Collegio  Romano,  qui  fuerat  ejus  Prce- 
ceptor  ;  sed  eo  jam  ad  Dissertationes  ejusmodi  conscribendas  utebatur. 

Eee  omnes,  quce  -pertinent  ad.  Seminarium  Romanum,  habent  in  ipso  titulo  adscripta  nomina  Nobilium  Con- 
victorum,  qui  illas  propugnarunt,  y  sub  eorum  nomine  referuntur  plures  ex  iis  in  Actis  Lipsiensibus. 

Multa  pertinentia  ad  ipsum  P.  Boscovich  babentur  in  binis  Dissertationibus,  quarum  tituli,  Synopsis  Physicae 
Generalis,  &  De  Lumine,  quarum  utraque  est  edita  Romae  anno  1754,  Typis  Antonii  de  Rubeis,  in  4. 
Id  ibidem  testatur  earundem  Auctor  (is  est  P.  Carolus  Benvenutus  Soc.  ejusdem)  affirmans,  ea  sibi  ab 
eodem  P.  Boscovich  fuisse  communicata. 

Habetur  etiam  ampliatio  solutionis  cujusdam  problematis  pertinentis  ad  Auroram  Borealem,  soluti  in  adnota- 
tionibus  ad  Carmen  P.  Noceti,  inserta  in  quadam  Dissertatione  impressa  Romcs  circa  annum  1756,  y 


Annus  prinue 

publice  propugnata,  cujus  Auctor  est  P.  Lunardi  Soc .  Jesu,  qui  affirmat  ibidem,  se  ea.nd.em  acceptam  ab        edition, 
ipso  P.  Boscovich  proponere  ejusdem  verbis. 

Subjiciemus  jam  bina  opuscula  Italica,  quee  communi  nomine  PP.»m  Le  Seur,  Jacquier,  ac  suo  conscripsit 
ipse  P.  Boscovich.  Utrumque  est  sine  loco  impressionis,  &  nomine  Typography ;  impresserunt  autem 
Palearini  Fratres  Romce  jussu  Prasulis,  qui  turn  curabat  Fabricam  5.  Petri,  a  quo  &  publice  distributa 
sunt  per  Urbem. 

Parere  di  tre  Matematici,  sopra  i  danni,  che  si  sono  trovati  nella  Cupola  di  S.  Pietro  sul  fine  del  1742,  dato       1742 
per  ordine  di  Nostro  Signore  Benedetto  XIV,  in  4.     In  fine  opusculi  habentur  subscripta  omnium  trio, 
nomina. 

Riflessioni  de'  PP.  Tomaso  Le  Seur,  Francesco  Jacquier  dell'  Ordine  de'  Minimi,  e  Ruggerio  Giuseppe       1743 
Boscovich  della  Comp.  di  Gesu  sopra  alcune  difficolta  spettanti  i  danni,  e  rifarcimenti  della  Cupola 
di  S.  Pietro  proposte  nella  Congregazione  tenutasi  nel  Quirinale  a'  20  Gennaro  1743,  e  sopra  alcune 
nuove  Ispezioni  fatte  dopo  la  medesima  Congregazione. 

Habentur  itidem  Italico  sermone  bina  ex  Us,  quas  Itali  vacant  Scritture,  -pro  quadam  lite  Ecclesia  S.  Agnetis       1757 
Romance,  pertinentes  ad  aquarum  cursum  Romce  editce  anno  1757. 

Inserta, 

Nunc  faciemus  gradum  ad  inserta  in  Publicis  Academiarum  monumentis,  in  diariis,  in  collectionibus,  y  in 
privatorum  Auctorum  Operibus. 

In  Monumentis  Acad.  Bononiensis. 

Prater  reimpressionem  binarum  Dissertationum  in  To.  II,  de  quibus  supra,  habetur  in  To.  IV  De  Litteraria       1 757 
Expeditione  per  Pontificiam  ditionem.    Est  Synopsis  amplioris  Operis,  ac  habentur  plura  ejus  exemplaria 
etiam  seorsum  impressa. 

In  Romano  Litteratorum  diario  vulgo  Giornale  de'  Letterati 
appresso  i  Fratelli  Pagliarini. 

D'Un'  antica  villa  scoperta  sul  dosso  del  Tuscolo  :   d'un  antico  Orologio  a  Sole,  e  di  alcune  altre  rarita,       1746 
che  si  sono  tra  le  rovine  della  medesima  ritrovate.    Luogo  di  Vitruvio  illustrate.    Ibi  ejus  schedias- 
matis  Auctor  profert,  uti  ipse  profitetur,  quee  singillatim  audierat  ab  ipso  P.  Boscovich. 

Dimostrazione  facile  di  una  principale  proprieta  delle  Sezioni  Coniche,  la  quale  non  dipende  da  altri 
Teoremi  conici,  e  disegno  di  un  nuovo  metodo  di  trattare  questa  dottrina. 

Dissertazione  della  Tenuita  della  Luca  Solare,    Del  P.  RuggieroGi us.  Boscovich Matematico  del  Collegio       1747 
Romano. 

Dimostrazione  di  un  passo  spettante  all'  angolo  massimo,  e  minimo  dell'  Iride,  cavato  dalla  prop,  ix 
par.  2  del  libro  i  dell'  Ottica  del  Newton  con  altre  riflessioni  su  quel  capitolo.  Del  P.  Ruggiero 
Gius.  Boscovich  dell  Comp.  di  Gesu. 

Metodo  di  alzare  un  Infinitinomio  a  qualunque  potenza.    Del  P.  Ruggiero  Gius.  Boscovich. 

Parte  prima  delle  Riflessioni  sul  metodo  di  alzare  un  Infinitinomio  a  qualunque  potenza.     Del  P.  Ruggiero       1748 
Gius.  Boscovich  della  Comp.  di  Gesu. 

Parte  seconda  &c. 

Soluzione  Geometrica  di  un  Problema  spettante  1'ora  delle  alte,  e  basse  maree,  e  suo  confronto  con  una 
soluzione  algebraica  del  medesimo  data  dal  Sig.  Daniele  Bernoulli.  Del  P.  Ruggiero  Giuseppe 
Boscovich  della  Compagnia  di  Gesu. 

Dialogi  Pastorali  V  sull'  Aurora  Boreale  del  P.  Ruggiero  Gius.  Boscovich  della  Comp.  di  Gesu. 

Dimostrazione  di  un  metodo  dato  dall'  Eulero  per  dividere  una  frazione  razionale  in  piu  frazioni  piu  semplici       X749 
con  delle  altre  riflessioni  sulla  stessa  materia. 

Lettera  del  P.  Ruggiero  Gius.  Boscovich  della  Comp.  di  Gesu  al  Sig.  Ab.  Angelo  Bandini  in  risposta       '75° 
alia  lettera  del  Sig.  Ernesto  Freeman  sopra  L'Obelisco  d'Augusto.     Nomen  Freeman  est  fictitium, 
Auctorem  denotans  Neapoli  latentem,  y  aliis  Operibus  satis  notum.     Extat  eadem  etiam  in  folio. 

Altera  de  eodem  Obelisco  ad.mod.um  prolixa  Epistola,  Italice,  W  Latine  scripta  ad  eundem  Bandinium  suo 
nomine  ab  ipso  P.  Boscovich  habetur  in  ejusdem  Bandinii  Opere,  cui  titulus,  De  Obelisco  Caesaris  Augusti 
e  Campi  Martii  ruderibus  nuper  eruto.  Commentarius  Auctore  Angelo  Maria  Bandinio.  Romas 
apud  Fratres  Palearinos,  in  folio.  Ibidem  in  fine  habetur  alia  epistola  itidem  admodum  prolixa  de  eodem 
argumento  nomine  Stuarti,  e  cujus  schedis  relictis  apud  Cardinalem  Falentium  in  ejus  discessu  ab  Urbe 
earn  Epistolam  conscripsit,  ac  ejus  comperta  illustravit,  ac  auxit  ipse  P.  Boscovich. 

Osservazioni  dell'  ultimo  passaggio  di  Mercurio  sotto  il  Sole  seguito  a'  6.  di  Maggio  1753,  fatte  in  Roma,       1753 
e  raccolte  dal  P.  Ruggiero  Gius.  Boscovich  della  Comp.  di  Gesu  con  alcune  reflessioni  sulle  medesime. 

In  aliis  monumentis. 

In  Collectione  Opusculorum  Lucensi  cui  titulus :  Memorie  sopra  la  Fisica,  e  Istoria  naturale  di  diversi  Valen- 
tuomini.  In  Lucca  per  li  Salani,  e  Giuntini,  in  8,  Prater  binas  dissertationes,  de  quibus  supra,  habetur. 

Problema  Mechanicum  de  solido  maxima  attractions  solutum  a  P.  Rogerio  Josepho  Boscovich  Soc.  Jesu 
Publico  Professore  Matheseos  in  Collegio  Romano  :  Tomo  I. 

De  Materias  divisibilitate,  &  Principiis  corporum.     Dissertatio  conscripta  jam  ab  anno  1748,  &  nunc       J757 
primum  edita.     Auctore  P.  Rogerio  Jos.  Boscovich  Soc.  Jesu,  To.  IV. 

Omnium  horum  quatuor  Opusculorum  habentur  etiam  exemplaria  seorsum  impressa. 


Annus  primes 

edition.         jn  eJitione  Elementorum  Geometries  Patris  Tacqueti  facta  Roma  sumptibus  Venantii  Monaldini,  Typis 
1745  Hieronymi  Mainardi,  in  8.  habetur  Trigonometria  sphaerica  P.  Rogerii  Josephi  Boscovich,  qua:  deinde 

adhuc  magis  expolita  prodiit  Tomo  I.  ejus  Elementorum  Matheseos.     Habetur  praterea  ibidem  Trac- 
tatus  De  Cycloide,  &  Logistica,  qui  etiam  seorsum  impressus  est  iisdem  typis. 

J7S2       In  Opere  Comitis  Zoannis  Baptists  Soardi,  cui  titulus  Nuovi  instrument!  &c.  in  Brescia  dalle  stampe  di 
Gio.  Battista  Rizzardi,  in  4.,  habentur  bina  epistola  Italica  ipsius  P.  Boscovich  de  Curvis  quibusdam, 
cum  figuris,  &  demonstrationibus. 
!758       In  Optica  Abbatis  De  la  Caille  latine  reddita  a  P.  Carolo  Scherffer  Soc.  Jesu,  y  impressa  Vienna  in  Austria 

habetur  schediasma  Patris  Boscovich  de  Micrometre  objective. 

In  postremo  tomo  Commentar.  Academiae  Parisiensis  in  Historia,  &  in  uno  e  tomis  Correspondent tium 
ejusdem  Academics,  creditur  esse  breve  aliquid  pertinens  ad  ipsum  P.  Boscovich.  Est  aliquid  etiam  in 
diario  Gallico  Journal  des  javans,  y  fortasse  in  Anglicanis  Transactionibus,  atque  alibi  insertum  hisce 
itinerum  annis. 

Poetica. 

J7S3  P.  Rogerii  Josephi  Boscovich  Soc.  Jesu  inter  Arcades  Numenii  Anigrei  Ecloga  recitata  in  publico  Arcadum 
consessu  primo  Ludorum  Olympicorum  die,  quo  die  Michael  Joseph  Morejus  Generalis  Arcadias 
Gustos  illustrium  Poetarum  Arcadum  effigies  formandas  jaculorum  ludi  substitnerat.  Romae  in  8. 
Extat  eadem  iisdem  Typis  etiam  in  Collectione  turn  impressa  omnium,  qua  ea  occasione  sunt  recitata. 
Stanislai  Poloniae  Regis,  Lotharingiae,  ac  Barri  Ducis,  &  inter  Arcades  Euthimii  Aliphiraei,  dum  ejus  effigies 
in  publico  Arcadum  Ccetu  erigeretur,  Apotheosis.  Auctore  P.  Rogerio  Josepho  Boscovich  Soc. 
Jesu  inter  Arcades  Numenio  Anigreo.  Romas  ex  Typographia  Generosi  Salomon!,  in  8.  Est 
poema  versu  heroico.  Idem  autem  recusum  fuit  Nancei  cum  versione  Gallic  a  Domini  Cogolin, 

1757  Pro  Benedicto  XIV.  P.M.  Soteria.     Est  itidem  poema  Heroicum  ejusdem  P.  Boscovich  pertinens  vel  ad  hunc, 

vel  ad  superiorem  annum  :   est  autem  impressum  Romae  in  4,  apud  Fratres  Palearinos,  occasione  periculi 
mortis  imminentis,  evitati  a  Pontif.ce  convalescente. 

1758  In  Nuptiis  Joannis  Corrarii,  &  Andrianae  Pisauriae  e  nobilissimis  Venetae  Reip.  Senatoriis  familiis.     Carmen 

P.  -Rogerii  Jos.  Boscovich  S.J.  Public!   in  Romano   Collegio    Matheseos   Professoris.    Romae :    ex 

Typographic  Palladis :   excudebant  Nicolaus,  &  Marcus  Palearini,  in  4. 
1760       De  Solis,  ac  Lunse  defectibus  libri  V  P.  Rogerii  Josephi  Boscovich  Societatis  Jesu  ad  Regiam  Societatem 

Londinensem,  Londini  1760.  in  4.     Nan  habetur  nomen  Typographi,  qui  impressit,  sed  Bibliopolarum 

quorum  sumptibus  est  impressum  :  deest  hie  ejus  editionis  exemplar,  ex  quo  ea  nomina  corrects  describantur. 

Idem  recusum  fuit  anno  1761  Venetiis  apud  Zattam  in  8°.  cum  exiguo  additamento  in  fine,  y  cum  hoc 

catalogo,  quern  inde  hue  derivavimus.     Habentur  in  adnotationibus  bina  Epigrammata  cum  versionibus 

Italicis,  sive  Sonetti. 
Est  &  aliud  ejus  poema  Heroicum  anno  1756  impressum  Vienna  in  Austria   in   collectione  carminum  facta 

occasione  inaugurationis  novarum  Academic  Viennensis  cedium. 
Sunt  y  epigrammata  nonnulla  in  Collec tionibus  Arcadum,  inter  qua  unum  pro  recuperata  valetudine  Johannis 

V  Lusitania  Regis,  tf  unum  pro  Rege  turn  utriusque  Sicilia,  y  nunc  Hispania,  ac  pro  Regina  ejus 

conjuge. 
Extant  etiam  pauca  admodum  exemplaria  unius  ex  illis,  quas  in  Italia  appellamus  Cantatine,  impressa  Viterbii 

anno  1750  pro  Visitatione  B.  Maria  Virginis,  in  qua  sex,  quas  dicimus  Ariette,  prof  ana;   ad  sacrum 

argumentum  transferenda  erant,  manente  Musica,  y  inter  se  connectenda. 


ERRATA 


p.      2, 1.  II,  for  ac  omnem  read  ad  omnem 

p.      3,  1.    5,  for  has  been  read  should  be 

p.      4,  1.  18,  for  Venetisis  read  Venetiis 

p.      6, 1.    9  from  bottom,  for  exceres  read  cxerces 

1.    4    from  bottom,  for  eocatum  read  evocatum 
p.      7, 1.  18  from  bottom,  after  despatched  add  to  the 
Court  of  Spain 

1.  13  from  bottom,  for  befits  read  befit 
p.      8, 1.    i,  for  publico  read  publice 

1.  13,  for  utique  read  ubique 

1.  28,  for  infiliciter  read  infeliciter 
p.    10, 1.    8,  for  opportunam  read  opportunum 

1.    9,  for  mediocrum  read  mediocrium 
p.    12,  1.  13,  for  aliquando  read  aliquanto 

1.  10  from  bottom,  for  repulsivis  read  repulsivas 
p.    14,1.  13,  for  adhibitis  read  adhibitas 

1.  24,  for  postremo  read  postrema 
p.    1 8, 1.    2,  for  alter  read  altera 
p.    22, 1.  15,  after  vero  etiam  insert  leges 
p.    28,  1.  17,  for  acquiretur  read  acquireretur 

1.  28,  for  -menae  read  -mena 
p.    40, 1.  22,  for  Naturam  read  Natura 

1.  23,  for  quandem  read  quandam 

1.  29,  for  recidit  read  recedit 

1.  32,  for  postquam  read  post  quam 
p.    47, 1.  34,  for  many  lead  most 
p.    48,  1.  1 8,  for  lins  read  lineae 

1.  29,  for  genere  read  generis 
p.    50, 1.  26,  for  deferendam  read  deserendam 

1.  31,  for  viderimus  read  videremus 

1.  46,  for  nominandi  read  nominando 
p.    52, 11.  5,  6  of  marginal  note  to  §  7,  for  nihilmu  read 

nihilum. 
p.    54, 1.    i,  for  exhibit  read  exhibet 

1.    3,  for  opposite  read  opposita 

1.  12,  for  sit  read  fit 
p.    55, 1.    4,  after  &  add  then 
p.    56,  1.    3,  for  servat  read  servant 
p.    58, 1.  32,  for  crederit  read  crederet 
p.    60, 1.    3,  marg.  note,  Art.  46,  for  sit  read  fit 
p.    64, 1.    2,  for  terio  read  tertio 
p.    65, 1.  57,  for  fact  read  by  the  fact 
p.    66, 1.    9,  for  concipiantur  read  concipiatur 

1.  16,  for  ordinate  read  ordinatae 
p.    67, 1.  48,  for  before  &  read  previously 
p.    68,  1.  II,  for  in  GM'  read  in  GM 
p.    71, 1.  46,  for  and  this  read  and  that  this  is  found 

nowhere 
p.    72, 1.    i,  for  ejusmodi  read  hujusmodi 

1.    4   from     bottom,     for     potissimuim     read 

potissimum 

p.    74, 1.  38,  for  illo  read  ilia 

p.    76,  1.    3  from  bottom,  for  devenirent  read  devenerint 
p.    81,  1.  42,  for  is  read  ought  to  be 
p.    82,  1.    5,  marg.  note,  Art.  82,  for  se  read  sed 
p.    86,  Art.  89,  in  marg.  note,  for  densitatis  read  densitas 
p.    88, 1.  II,  for  adi  read  ad 

1.  1 6,  for  reliquent  read  relinquent 
p.    90, 1.  30,  for  diversimodo  read  diversimode 

1.  34,  for  distantia  read  distantiae 

Art.  95,  marg.  note,  for  de-  read  dif- 
p.    92,  1.  33,  for  apparent  read  apparerent 
p.    94,  1.  22,  for  incurrant  read  incurrunt 
p.    95,  Art.  103,  1.  I,  for  are  read  is ;    and  in  marg.  note 

insert  in  between  and  and  what 
p.    96,  1.    8,  for  potissimo  read  potissimum 

1.  16,  for  praecedentum  read  prascedentem 

marg.  note,  Art.  105,  for  transire  read  transiri 
p.    97,  1.    9  from  bottom,  for  quite  enough  read  better 


p.    99,  1.  40,  insert  a  comma  after  locus 

p.  loo,  marg.  note,  Art.  112,  for  recte  read  rectae 

1.  32,  for  ellipis  read  ellipsis 
p.  106,  marg.    note,   Art.    125,   for   perfectionum    read 

perfectiorum 
p.  107, 1.  23,  for  off  they  are  read  away  they  go 

1.  13  from  bottom,  for  have  read  has 
p.  109,'  1.  27,  after  tantummodo  add  admitto 
p.  no,  1.  17,  for  expandantur  read  expendantur 

1.  24,  for  a  read  & 

bottom  line,  for  distinctis  read  distinctas 
p.  112,  1.  27,  for  veteram  read  veterem 
p.  113, 1.    5  from  bottom,  for  because  read  that 

1.   4  from  bottom,  after  change  add  is  excluded  by 
p.  115, 1.  33,  for  and  read  et 

marg.  note,  Art.  139,  add  at  end  impugned 
p.  1 1 8, 1.    7  from  bottom  for  ali  read  alia 
p.  122, 1.  26,  for  justmodi  read  ejusmodi 
p.  125,  1.  29,  for  ignored  read  urged  in  reply 
p.  128, 1.  31,  for  ea  read  eas 
p.  129, 1.  16,  for  Principii  read  Principiis 
p.  139, 1.    8,  for  arm  E  read  arm  ED 

footnote,  1.  5,  for  DP  read  OP 
p.  140, 1.  34,  insert  cum  before  directione 

1.    4,  footnote,  for  ut  in  read  ut  n 
p.  148, 1.  10,  for  Expositas  read  Expositis,  for  curva  read 

curvam 
p.  156,  1.    I,  for  a  que  read  atque 

1.    7,  for  caculo  read  calculo 

1.  39,  for  Tam  read  Turn 

p.  158,  Art.  209,  marg.  note,  add  at  endLegum  multitudo 
&  varietas 

footnote,  1.  1 1  from  bottom  of  page,  for  obvenerit 

read  obveniret 
p.  160,  footnote,  1.  I,  for  sit  read  fit 

1.  12,  for  ed  read  sed 

p.  161,  footnote,  1.  20,  after  segment  add  DR 
p.  162, 1.    7   from  bottom,  for  reflexionis  read   reflex- 

iones 

p.  167, 1.  40,  for  ae  read  da 

p.  168, 1.    8,  from  bottom  for  zjC—'AC  read  27    C'AC 
p.  171, 1.  4  from  bottom  for  GL,  or  LI  read  GI,  or  IL 
p.  172, 1.  34,  for  compositas  read  compositis 
p.  175, 1.  13,  for  30  read  27 

1.  39,  insert  a  comma  after  approximately 
p.  176, 1.    7,  for  delatam  read  delatum 
p.  178,  marg.  note,  Art.  230,  1.  5,  for  foco  read  focos 
p.  188, 1.  31,  for  summa  read  summas 
p.  195,  1.  19,  insert  a  comma  after  point  P 
p.  197, 1.  35,  for  sum  of  the  (at  end  of  line)  read  sums  of  the 
p.  198,  Fig.  40,  insert  F  where  AE  cuts  CD 
p.  199,  1.  35,  for  ceases  read  cease 

1.  37,  after  all,  &  insert  I  assume 
p.  202, 1.    6,  for  summa  read  summam 

Art.  264, 1.  3,  for  quacunque  read  quancunque  and 
in  marg.  note  for  corallarium  read  corol- 
larium 

p.  205, 1.  21,  for  -recessions  read  recession 
p.  206, 1.    3,  for  globis  read  globus 

1.    2  from  bottom,  insert  motu  before  quodam 


p.  208,  last  line,  for 


m  -)-  n 


read 


m-(-n 


in  each  case 


n  m 

p.  209, 1.  11,  for  (2CQ  —  2Cq)  read  (2CQ  —  2cQ) 
p.  210.  1.    8,  for  quiescat  read  quiescit 
p.  211, 1.  25,  the  denominator  (Q  +  q)  should  be  (Q  +  q)2 
p.  215, 1.    5,  for  BP  read  BO 
p.  223, 1.  26,  for  50,61,62  read  50,51,52 
p.  227, 1.  21  from  bottom,  for  to  read  of 
p.  228, 1.    5,  from  bottom,  for  Angulum  read  Angulus 


ERRATA 


p.  233, 1.    8,  for  in  volute  read  evolute 

bottom  line,  after  vary  insert  inversely 

p.  241,  marg.  note,  Art.  313,  add  at  end  This  is  very  soon 
proved 

p.  242,  last  line,  for  denominator  AD  read  BD 

p.  247, 1.  15  from  bottom,  for  A  &  B  read  B  &  C 

p.  248,  1.  24,  for  conversione  read  conversionem 

p.  250, 1.  39,  for  justa  read  jurta 

p.  252, 1.    5  from  bottom,  for  gravitas  read  gravitatis 

p.  256,  1.  lo,  for  quassitum  read  quaesitam 

p.  270,  marg.  note,  Art.  366,  1.  5,  for  magnas  read  magna 

p.  278, 1.    7,  for  varior  read  rarior 

p.  280, 1.  12  from  bottom,  for  tranctanda  read  tractanda 

p.  284,  1.  15,  for  sit  read  fit 

p.  286, 1.  20,  for  multuplicetur  read  multiplicetur 

p.  288, 1.  21,  for  Solum  read  Solem 

p.  292, 1.  i,  for  quietam  read  quietem 
1.  3,  for  sit,  read  fit 

1.  25,  for  ullo,  read  ulla  deleting  the  comma 
1.  26,  for  ilia  read  ille 

p.  304,  1.  12  from  bottom,  for  propre  read  prope 

p.  310, 1.  15  from  bottom,  for  hsebebunt  read  hzrebunt 

p.  314,  1.    2,  insert  &  before  mutandam 

p.  319, 1.  1 8  from  bottom,  for  some  repel  read  and  repel 

p.  324, 1.  22,  for  pertinet  read  pertinent 

p.  329, 1.  25,  for  ethers  read  others 

p.  332, 1.  16,  for  aequabilis  read  aequalibus 

1.  21,  for  Benvenuti  read  Benvenutus 
1.    3  from  bottom,  for  qui  a  read  quia 

P-  336, 1.  35)  insert  utcunque  afar  circunquaque 

p.  346, 1.  19,  for  sit  read  fit 


p.  348,  1.  28,  for  irregularitur  read  irregulariter 
p.  350,  1.  13  from  bottom,  for  flexo  read  flexu 
P-  355)  !•  H>  /<""  with  read  to 
p.  356,  1.    9,  after  porro  aliud  insert  post  aliud 

1.  19,  after  accidit  insert  idem  accidit 
p.  366,  1.  32,  for  ordores  read  odores 
p.  394,  1.  31,  for  imaginarae  read  imaginariae 
p.  396,  1.  19  from  bottom,  after  solum  add  etiam 

1.  14  from  bottom,  for  sunt  read  sint 
p.  398,  1.  20,  after  est  insert  tota 

1.  35,  for  esses  read  esse 
p.  400,  1.  33,  after  omnino  insert  saltern 
p.  406,  1.    6,  for  congruant  read  congruunt 
p.  410,  1.    8,  for  bzma  read  bz™'3 

Art  27,  marg.  note,  for  quaesitum  read  quaesitam 
p.  412,  1.  II,  insert  positives  before  assumantur 
p.  422,  1.  22,  for  ab  read  ad 
p.  434,  Art  86,  marg.  note,  for  massa  read  massas 

1.    5  from  bottom,  for  contrarium  read  contrariam 
p.  444,  1.  n,  insert  &  before  centri 
p.  448,  1.  17,  for  PQ  read  PG 

Art.  107,  marg.  note,  for  absoluutae  read  absolutas 

1.    4  from  bottom,  after  posita  insert  J 
p.  454,  1.    2  from  bottom,  for  elasticas  read  elasticitas 

>J.J  rp2 

p.  456,  1.    5  from  bottom  for  —  read— 


p.  458,1.  n,  for  ^  read 

zz  zz 

p.  459,  1.  13,  for  —  tdz/.z2  read  —  t2dz/z2 
p.  464,  1.    4,  for  Discrimen  read  Discrimina 

1.  10  from  bottom,  for  Venizia  read  Venezia 


Printed  in  Great  Britain  by  Butler  &  Tanner,  Frame  and  London 


THIS   BOOK  IS  DUE  ON   THE   LAST  DATE 
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WAR  811975 
MAR  3  i  RECU" 


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RERLE: 


JUN  o  'i  1989 


DEC  1  7 


6  i 


KB  8 


IIBRARY,  UNIVERSITY  OF  CALIFORNIA,  DAVIS 

Book  Slip-35m-7,'62(D296s4)458 


d  QC 19 

£52, 

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